Volatility Analysis and the Hilbert-Huang Transform

University of WollongongResearch OnlineUniversity of Wollongong Thesis Collection1954-2016 University of Wollongong Thesis Collections

2016

Reading the Waves: Volatility Analysis and theHilbert-Huang TransformCarson DrummondUniversity of Wollongong

Unless otherwise indicated, the views expressed in this thesis are those of the author and donot necessarily represent the views of the University of Wollongong.

Research Online is the open access institutional repository for the University of Wollongong. For further information contact the UOW Library:[email protected]

Recommended CitationDrummond, Carson, Reading the Waves: Volatility Analysis and the Hilbert-Huang Transform, Doctor of Philosophy thesis, School ofMathematics and Applied Statistics, University of Wollongong, 2016. https://ro.uow.edu.au/theses/4831

https://ro.uow.edu.au

https://ro.uow.edu.au/theses

https://ro.uow.edu.au/theses

https://ro.uow.edu.au/thesesuow

University of Wollongong

Doctoral Thesis

Reading the Waves: Volatility Analysisand the Hilbert-Huang Transform

Author:

Carson Drummond

Supervisor:

Pam Davy

Co-Supervisor:

Chandra Gulati

A thesis submitted in fulfilment of the requirements

for the degree of Doctor of Philosophy

in the

National Institute for Applied Statistics Research Australia

School of Mathematics and Applied Statistics

August 2016

http://www.uow.edu.au

http://www.johnsmith.com

Research Group Web Site URL Here (include http://)

Department or School Web Site URL Here (include http://)

Declaration of Authorship

I, Carson Drummond, declare that this thesis titled, ’Reading the Waves: Volatility

Analysis and the Hilbert-Huang Transform’ and the work presented in it are my own. I

confirm that:

This work was done wholly or mainly while in candidature for a research degree

at this University.

Where any part of this thesis has previously been submitted for a degree or any

other qualification at this University or any other institution, this has been clearly

stated.

Where I have consulted the published work of others, this is always clearly at-

tributed.

Where I have quoted from the work of others, the source is always given. With

the exception of such quotations, this thesis is entirely my own work.

I have acknowledged all main sources of help.

Where the thesis is based on work done by myself jointly with others, I have made

clear exactly what was done by others and what I have contributed myself.

Signed:

Date:

i

“I don’t pretend we have all the answers. But the questions are certainly worth

thinking about.”

Arthur C. Clarke

Abstract

Reading the Waves: Volatility Analysis and the Hilbert-Huang Transform

by Carson Drummond

Countless processes, both natural and man made are driven by seemingly random pro-

cesses. Measuring the randomness which dominates everything from the movements of

stock market prices to the recordings on a seismograph is the first step towards un-

derstanding it. This thesis focused on measuring the volatility of financial time series,

however the potential applications of the techniques developed herein are not limited to

the financial realm.

The Hilbert-Huang Transform (HHT) is a powerful new tool which is well suited for

the analysis of non-stationary and nonlinear time series. The literature exploring its po-

tential applications to the analysis of financial data is surprisingly sparse considering its

apparent suitability. This thesis developed and tested several techniques for estimating

time series volatility, all of which use the HHT to break down financial data into simple

wave like structures which facilitate analysis.

The estimation techniques developed were tested on low frequency data, namely ten

years worth daily data for the NASDAQ and All Ords Indices. A simulated study to

emulate high frequency data was also carried out to study volatility estimation in the

presence of microstructure noise. Finally, several estimation techniques were used to give

one step ahead predictions on the AUD/USD, GBP/USD and EUR/USD exchanges, a

simulated options market was then set up in which differing predictions compete against

one another.

The HHT based estimators were found to be competitive with the alternatives tested

in the low frequency realm, with the added advantage that the technique is intuitive

and can handle unevenly spaced data with ease. The HHT procedure was used as a low

pass filter in order to sift off market microstructure and effectively measure volatility

when the true price was obfuscated by market frictions. Finally, the construction of a

simulated options market operating on real high frequency foreign exchange data showed

that this filtering approach was also effective when real data was used.

. . .

Acknowledgements

First and foremost I would like to thank my primary supervisor, Pam Davy. Her enthu-

siasm for research into the unknown is contagious and her contributions to this thesis

are many. I would also like to thank my co-supervisor, Chandra Gulati. Chandra’s great

attention to detail and broad knowledge of statistical literature has been invaluable. The

guidance, encouragement and many words of wisdom from both of my supervisors has

kept me going and this thesis is a result of their labors as much as my own.

My thanks also go to Song-Ping Zhu for encouraging me to pursue a PhD and giving

me the freedom to go where my research interests took me.

Lastly, my sincerest thanks go to my parents, Carol and Gary Drummond, for their

unwavering love, understanding and support.

iv

Contents

Declaration of Authorship i

Abstract iii

Acknowledgements iv

Contents v

List of Figures vii

List of Tables ix

Abbreviations x

1 Introduction 1

1.1 Defining and Measuring Volatility . . . . . . . . . . . . . . . . . . . . . . . 3

1.2 Low and High Frequency Volatility Dynamics and Measurement . . . . . 5

1.2.1 Low Frequency Volatility Dynamics . . . . . . . . . . . . . . . . . 5

1.2.2 High Frequency Volatility Dynamics . . . . . . . . . . . . . . . . . 8

1.3 A New Tool in the Toolbox . . . . . . . . . . . . . . . . . . . . . . . . . . 11

1.4 Thesis Outline . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14

2 Background 16

2.1 Volatility Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 16

2.1.1 The Moving Average Volatility Model . . . . . . . . . . . . . . . . 16

2.1.2 The Exponentially Weighted Moving Average Model . . . . . . . . 17

2.1.3 The GARCH(1,1) Model . . . . . . . . . . . . . . . . . . . . . . . . 20

2.1.4 Basic Realized Volatility . . . . . . . . . . . . . . . . . . . . . . . . 21

2.1.5 Overnight Returns . . . . . . . . . . . . . . . . . . . . . . . . . . . 22

2.1.6 Sparse Realized Volatility and the Averaged or Subsampled Real-ized Volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 24

2.1.7 Two Sample Realized Volatility . . . . . . . . . . . . . . . . . . . . 26

2.2 The Hilbert-Huang Transform . . . . . . . . . . . . . . . . . . . . . . . . . 26

2.2.1 Empirical Mode Decomposition . . . . . . . . . . . . . . . . . . . 27

2.2.2 The HHT as a Variability Measure . . . . . . . . . . . . . . . . . . 30

v

Contents vi

2.2.3 Monte Carlo Analysis of the EMD Procedure . . . . . . . . . . . . 30

2.2.4 The Hilbert Transform . . . . . . . . . . . . . . . . . . . . . . . . . 39

2.3 Comparison and Evaluation Methods . . . . . . . . . . . . . . . . . . . . . 41

3 Low Frequency Volatility Estimation & the HHT 44

3.1 Volatility Estimation using the HHT . . . . . . . . . . . . . . . . . . . . . 44

3.1.1 Generating a Proxy of the Returns . . . . . . . . . . . . . . . . . . 44

3.1.2 Extracting Volatility from the Returns Proxy . . . . . . . . . . . . 48

3.2 Results and Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 50

3.2.1 Data and Testing Methodology . . . . . . . . . . . . . . . . . . . . 50

3.2.2 Graphical Volatility Analysis . . . . . . . . . . . . . . . . . . . . . 52

3.2.3 Further Analysis of the HHT Procedure . . . . . . . . . . . . . . . 54

3.2.4 Numerical Results . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

3.3 Chapter Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 62

4 High Frequency Simulated Data and Microstructure Noise 66

4.1 Improved HHT Volatility Estimator . . . . . . . . . . . . . . . . . . . . . 66

4.2 Volatility Comparison Test . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.1 Test Design . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 69

4.2.2 Test Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 70

4.3 The Filtered-HHT Volatility Estimator for Microstructure Noise . . . . . 72

4.3.1 Adapting the HHT Estimator to the High Frequency Realm . . . . 73

4.4 Monte Carlo Analysis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 76

4.4.1 Spectral Analysis and Hurst Parameter Estimation . . . . . . . . . 78

4.4.2 Results and Comparisons . . . . . . . . . . . . . . . . . . . . . . . 79


5 Simulated Options Market & Real FX Data 90

5.1 The Simulated Options Market and Clarification of Methods . . . . . . . 91

5.1.1 Simulated Option Market Construction . . . . . . . . . . . . . . . 91

5.1.2 Data Specification and Clarification of Methods . . . . . . . . . . . 93

5.1.3 Modified HHT Estimator With Market Microstructure . . . . . . . 95

5.1.4 Results . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99


6 Discussion & Conclusion 112

6.1 Discussion . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 112

6.2 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 116

6.3 Concluding Remarks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 118

Bibliography 119

List of Figures

2.1 All Ords Index prices from 1st Jan 2000 to 31st Dec 2004 (bold) and asimple MA volatility estimate with a window length of 50 days (dashed). 18

2.2 EWMA volatility estimates with decay rates λ = 0.94 & λ = 0.7 for theAll Ords data set covering 1st Jan 2000 - 31st Dec 2004. . . . . . . . . . . 19

2.3 Super imposed sinusoids (solid blue), the local maxima and connectingspline (red circle and dot-dash line respectively) and the local minimaand connecting spline (green circle and dashed line respectively). . . . . . 29

2.4 The first two IMFs of S(t). . . . . . . . . . . . . . . . . . . . . . . . . . . 30

2.5 IMF number vs log2 IMF period for GWN. . . . . . . . . . . . . . . . . . 31

2.6 Log mean period vs log2 variance for GWN (top) & IMF number vs log2

IMF variance for GWN. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33

2.7 Power Spectra of 10 IMFs on GWN. . . . . . . . . . . . . . . . . . . . . . 34

2.8 Rescaled Power Spectra of 10 IMFs on GWN. . . . . . . . . . . . . . . . 35

2.9 Power Spectra of 10 IMFs on fGn for three values of H. . . . . . . . . . . 36

2.10 Log period vs log variance for 3 values of H (top). IMF number vs logvariance for 3 values of H (bottom). . . . . . . . . . . . . . . . . . . . . . 38

2.11 Instantaneous amplitude (left) and frequency (right) given by a Hilberttransform of the IMFs displayed in Figure 2.4. . . . . . . . . . . . . . . . 41

3.1 Frequency vs K statistic of a K-S test during a typical search for a returnproxy. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 48

3.2 Actual returns (top left). Proxy to the log returns P (t) (top right). QQplot of the the proxy P (t) against the actual returns (bottom left). CDFplot of the proxy P (t) against the actual returns (bottom right). . . . . . 49

3.3 HHT based and realized volatility estimates for the NASDAQ data set,1st Jan 2000 - 31st Dec 2004 (top). Corresponding GARCH(1,1) andrealized volatility estimates (middle). Corresponding EWMA and realizedvolatility estimates (bottom). . . . . . . . . . . . . . . . . . . . . . . . . 53

3.4 NASDAQ Index price for 1st Jan 2000 - 31st Dec 2001 and a measure ofthe local average eM(t). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55

3.5 The high frequency components which form the proxy (top). The remain-ing components which are a form of local mean (bottom). . . . . . . . . 56

3.6 The ‘variability’ measure proposed by Huang applied to the NASDAQdata set. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 56

3.7 Amplitudes of the first 5 IMFs scaled by the data for 10 years of theNASDAQ Index data set. . . . . . . . . . . . . . . . . . . . . . . . . . . . 57

3.8 Frequency information given by the Hilbert transform of the first 6 IMFsfor two years of the NASDAQ Index data set. . . . . . . . . . . . . . . . . 58

vii

List of Figures viii

3.9 First six IMFs for the log of the NASDAQ Index from 1st Jan 2000 - 31st

Dec 2001. The average period of each IMF is given in calendar days asopposed to trading days. . . . . . . . . . . . . . . . . . . . . . . . . . . . 59

3.10 QQ plots of the volatility distribution for the HHT method (top), EWMA(middle) and GARCH (bottom) for the NASDAQ Index covering theperiod 1st Jan 2000 - 31st Dec 2004. . . . . . . . . . . . . . . . . . . . . . 60

3.11 CDF plots of volatility distribution for the HHT method (top), EWMA(middle) and GARCH (bottom) for the NASDAQ Index covering 1st Jan2000 - 31st Dec 2004. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61

4.1 EWMA Response profiles for three values of λ. . . . . . . . . . . . . . . . 71

4.2 Response profiles for HHT estimator & EWMA with λ = 0.67 . . . . . . . 72

4.3 SNR vs response time for EWMA with 0.1 ≤ λ ≤ 0.98 (curve) and theHHT volatility estimate (circle). . . . . . . . . . . . . . . . . . . . . . . . 73

4.4 Volatility estimates for various sample rates across different levels of fil-tration. The ‘o’ symbol denotes the estimate provided by Equation (4.13).. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 77

4.5 Spectral analysis of IMFs from GARCH, Affine and Log volatility models.Top row: no microstructure noise. Middle row: 0.1% noise. Bottom row:0.5% noise. Note that the vertical axis is in terms of the Power DensitySpectra (PDS) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80

4.6 Volatility signature plot at 0.1% noise. M1 M2 M3 refers to the threevolatility models, GARCH, Two-Factor Affine and Log-Normal diffusionrespectively. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 81

4.7 Volatility vs Time plots for the GARCH (top), Two-Factor Affine (middle)and Log-Normal Diffusion (bottom) models at 0.1% noise. Each plotshows model volatility, the Adjusted-TSRV estimate and the Filtered-HHT volatility estimate. . . . . . . . . . . . . . . . . . . . . . . . . . . . 86

4.8 Volatility vs Time plots for the GARCH (top), Two-Factor Affine (middle)and Log-Normal Diffusion (bottom) models at 0.5% noise. Each plotshows model volatility, the Adjusted-TSRV estimate and the Filtered-HHT volatility estimate. . . . . . . . . . . . . . . . . . . . . . . . . . . . . 87

5.1 Volatility estimates with sample intervals of ∆t(1, . . . , 8) = [0.2, 0.25, 0.5, 1, 1.5, 2, 3, 4],calculated across different levels of filtration using FX data. The ‘o’ sym-bol denotes the estimate provided by the procedure outlined in this chap-ter. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 98

5.2 Instantaneous or spot volatility estimates for 24 hours of high frequencyAUD/USD data. . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 99

5.3 The ‘frictionless’ price (red), as extracted from 24 hours of high frequencyAUD/USD data (blue). . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100

List of Tables

3.1 Errors for various volatility estimators for the All Ords Index coveringthe period 1st Jan 2000 - 31st Dec 2004 . . . . . . . . . . . . . . . . . . . 63

3.2 Errors for various volatility estimators for the NASDAQ Index coveringthe 1st Jan 2000 - 31st Dec 2004 . . . . . . . . . . . . . . . . . . . . . . . 63

3.3 Errors for various volatility estimators for the All Ords Index coveringthe 1st Jan 2005 - 31st Dec 2009 . . . . . . . . . . . . . . . . . . . . . . . 63

3.4 Errors for various volatility estimators for the NASDAQ Index coveringthe 1st Jan 2005 - 31st Dec 2009 . . . . . . . . . . . . . . . . . . . . . . . 63

4.1 Hurst exponent estimates from the EMD procedure and their associatedstandard deviations for the three models at three noise levels. . . . . . . . 79

4.2 Sampling rate and equivalent sampling intervals in seconds. . . . . . . . . 81

4.3 Mean error analysis at γ = 0.1% noise. . . . . . . . . . . . . . . . . . . . . 83

4.4 Mean square error analysis at γ = 0.1% noise. . . . . . . . . . . . . . . . . 83

4.5 Mean absolute error analysis at γ = 0.1% noise. . . . . . . . . . . . . . . . 83

4.6 Mean error analysis at γ = 0.5% noise. . . . . . . . . . . . . . . . . . . . . 84

4.7 Mean square error analysis at γ = 0.5% noise. . . . . . . . . . . . . . . . . 84

4.8 Mean absolute error analysis at γ = 0.5% noise. . . . . . . . . . . . . . . . 84

5.1 Pairwise competitive trading results for AUD/USD Straddles . . . . . . . 103

5.2 Pairwise competitive trading results for EUR/USD Straddles . . . . . . . 103

5.3 Pairwise competitive trading results for GBP/USD Straddles . . . . . . . 104

5.4 Fair price trading results for AUD/USD Straddles . . . . . . . . . . . . . 104

5.5 Fair price trading results for EUR/USD Straddles . . . . . . . . . . . . . 105

5.6 Fair price trading results for GBP/USD Straddles . . . . . . . . . . . . . 105

5.7 AUD/USD Competitive pairwise trading results in full . . . . . . . . . . . 109

5.8 EUR/USD Competitive pairwise trading results in full . . . . . . . . . . . 109

5.9 GBP/USD Competitive pairwise trading results in full . . . . . . . . . . . 110

5.10 AUD/USD Fair price trading results in full . . . . . . . . . . . . . . . . . 110

5.11 EUR/USD Fair price trading results in full . . . . . . . . . . . . . . . . . 111

5.12 GBP/USD Fair price trading results in full . . . . . . . . . . . . . . . . . 111

ix

Abbreviations

All Ords All Ordinaries

ARMA AutoRegressive Moving Average

DST Discrete Sine Transform

DHT Discrete Hilbert Transform

EMD Empirical Mode Decomposition

EWMA Exponentially Weighted Moving Average

fBm fractional Brownian motion

FFT Fast Fourier Transform

fGn fractional Gaussian Noise

GARCH Generalised AutoRegressive Conditional Heteroskedasticity

GFC Global Financial Crisis

GWN Gaussian White Noise

HHT Hilbert-Huang Transform

IID Independent and Identically Distributed

IMF Intrinsic Mode Function

IV Integrated Volatility

MACD Moving Average Convergence Divergence

MAE Mean Absolute Error

MAPE Mean Absolute Percentage Error

ME Mean Error

MSE Mean Square Error

NASDAQ National Association of Securities Dealers Automated Quotations

PDS Power Density Spectra

PMF Proto Mode Function

RV Realized Volatility

SIRCA Securities Industry Research Centre of Asia-Pacific SNR Signal to Noise Ratio

TSRV Two Scale Realized Volatility

VIX CBOE Volatility Index

x

To My Parents

xi

Chapter 1

Introduction

Our lives are governed by random processes, whether it’s the stock market, weather pat-

terns or quantum behaviour we are surrounded by uncertainty at all times. This thesis

seeks to shed some light on the nature of randomness, with a focus on financial time

series volatility, through the lens of a powerful new tool, the Hilbert-Huang Transform

(HHT) developed by Huang et al. [1].

A system contains randomness if not all parts of it are fully deterministic, i.e. ex-

actly how a random process will behave can not be fully described in advance. Not fully

described —luckily this still leaves much room for some particularly useful descriptions.

The descriptive which most concerns the topic of this thesis is the volatility, best thought

of as the magnitude of the uncertainty. At this point it is important to distinguish be-

tween uncertainty and risk. While risk may be more concerned with a negative outcome,

i.e. a large loss made on an investment, the volatility is akin to the uncertainty and is

more closely related to the magnitude of the movement, not its direction. A measure

of risk that focuses more closely on negative returns was proposed by Markowitz [2]

however it has not been widely adopted. Although conceptually distinct, volatility and

risk are inextricably linked, thus to manage risk one must also understand volatility.

Following the stock market crash of 1987 the first Basel Accord made understanding

volatility a matter of policy for many financial institutions which were henceforth re-

quired to manage their exposure to risk in terms of volatility. Unfortunately, recent

1

Chapter 1. Introduction 2

events in stock markets around the world show that even the experts still have a thing

or two to learn about the nature of volatility.

One reason why there is so much interest in understanding volatility is because the

volatility of a stochastic system impedes on our ability to give accurate forecasts. The

more volatile a system, the more rapidly a forecast becomes addled by uncertainty. It

quickly becomes apparent that the confidence placed in any such forecast is closely linked

to an accurate measure of volatility. This is true for all manner of stochastic systems,

it is why long term weather forecasts are notoriously unreliable and economists’ predic-

tions are often taken as opinions rather than facts.

The type of stochastic system that is investigated over the course of this thesis is a

time series, this is simply a sequence of points or data which come in some order. Ex-

amples of a time series include the temperature of your city measured daily, the position

of a seismograph needle during an earthquake or the current and voltage output of your

computer’s power supply over the course of a few seconds. Thus, any useful insights

into the nature of time series volatility will not be restricted to financial mathematics,

but to a broad swath of scientific disciplines.

To examine volatility in a practical setting, a focus is placed on financial time series.

This is partly because there is a wealth of available data in the area, but also because it is

a place where forecasts on stochastic systems are tested and refined on a daily basis and

any contributions to this field are likely to have a maximal impact. Volatility estimates

are also very important in option pricing, where the price of the financial instrument

is directly affected by the perceived variability of an underlying stock. Nowhere is this

more evident than the VIX Index, which trades in perceived uncertainty itself.

Methods for determining local volatility fall into two categories, time series meth-

ods which base their estimate on the underlying historical data and implied volatility

methods, which calculate the theoretical volatility needed to account for the price of an


option under a particular choice of pricing model. The time series approach to volatility

modelling assumes that some important characteristics of the volatility can be extracted

from the underlying historical data and as Poon and Granger [3] put it, these charac-

teristics can provide some insight into forward or future volatility. Thus any accurate

means of measuring past volatility also has a place in volatility forecasting.

1.1 Defining and Measuring Volatility

Firstly, it is important to clarify that our objective is to measure the volatility of the

log returns of a financial time series. It is standard practice in much of the financial

literature to use the log returns and then treat them as normally distributed. The log

return at time t is given by:

rt = log(Pt)− log(Pt−1), (1.1)

where Pt is the price or value of some underlying asset at time t. For notational con-

venience, log(Pt) is denoted pt henceforth, so the log returns can also be expressed as:

rt = pt − pt−1.

If one is to assume that the volatility is constant over time, estimating the volatility

from N return observations would be a simple matter of finding the Sample Variance

SV σ2, which is given by:

SV σ2 =1

N − 1

N∑t=1

(rt − r)2 , (1.2)

where r is the arithmetic mean of the returns. Note that r is assumed to be zero for many

financial time series sampled daily or at higher frequencies. However, the assumption

of constant volatility is rarely justified and there has been much work towards finding

an accurate and meaningful measure of a time varying variance σ2t . It is worth noting

at this point that σ2 only represents the second moment of the data and that it is of


limited usefulness without considering the probability density function that it is asso-

ciated with. The assumption of a log-normal return distribution is also questioned in

much of the literature as the log returns are often observed to be fat tailed or leptokurtic.

To understand the continuous time analogue of Equation (1.2) we let the log of the

frictionless price follow

dpt = σtdW(1)t , (1.3)

where W(1)t is a standard Brownian motion, the superscript is merely to distinguish this

Brownian motion from those used to drive a stochastic volatility model described in

Chapter 4. The instantaneous or spot volatility, denoted σt, has a continuous sample

path. It is assumed that the σt and W(1)t processes are uncorrelated. One period (often

one trading day for convenience) of Integrated Volatility (IV) is defined as:

IVt ≡∫ t

t−1σ2τdτ. (1.4)

This definition of IVt is sometimes referred to as the quadratic return variation. If

the discretely sampled returns are serially uncorrelated and the sample path for σt is

continuous, it follows from the theory of quadratic variation described by Merton [4] &

Karatzas and Shreve [5], that

lim1/h→∞

∫ t

t−1σ2τdτ −

∑i=1,...,1/h

r2t−1+ih(h)

= 0, (1.5)

where h is the increment size between samples, i.e. a period of length 1 has 1/h many

return observations. The key result from this is that a variable volatility is theoretically

observable at a time t provided that the sampling frequency is high enough. Andersen

et al. [6, 7] have shown that even though the spot volatility is not a directly observable

variable, estimations of the integrated volatility can be closely estimated by the easily

calculable Realized Volatility (RV), which is covered in more detail in Chapter 2. Thus,

the goal of much of this thesis and much literature is to reconstruct the path of σ2t for a

better understanding of market dynamics, to improve volatility forecasts, manage risk


more effectively and price options with higher accuracy.

1.2 Low and High Frequency Volatility Dynamics and Mea-

surement

This thesis can roughly be split into two sections, volatility estimation in the low fre-

quency domain and volatility estimation in the high frequency domain. The reason for

this division is that the problems faced at each time scale differ, as do the approaches

needed to overcome them. The low frequency domain is largely restricted to data col-

lected on the daily scale, like the closing price and opening price of an index, stock or

currency etc. The high frequency realm on the other hand involves orders of magnitude

more data and usually includes samples anywhere from one every fifteen minutes to only

milliseconds apart.

1.2.1 Low Frequency Volatility Dynamics

There is an escalating struggle between those that discover interesting new market dy-

namics and those that try to model them. Like lepidopterists collecting butterflies,

financial practitioners delight in naming new market phenomena and challenging aca-

demics to accurately describe them before pinning them down in a useful model. The

first to understand any new market dynamic will often have an edge over their compe-

tition, this has led to considerable research into all manner of market dynamics and an

arms race between algorithmic trading strategies that is evolving daily.

As previously stated, volatility appears to change over time in financial markets,

rendering the standard deviation of past returns of limited use. One of the simplest

tools to overcome this is simply to take the standard deviation over a shorter period of

time and trade off an increase in sample error (due to a smaller sample size) against an


increase in contemporariness. This approach yields the simple moving average estimate

of volatility described in Chapter 2.

Many financial returns series also appear to have the property of long memories which

results in the autocorrelations of variances staying positive for a long time, i.e. shocks

in the volatility process tend to persist. Methods to deal with this phenomenon include

the popular Exponentially Weighted Moving Average (EWMA) model (see Chou et al.

[8], Ding and Meade [9], Wilmott [10]) and the more general form of Auto Regressive

Conditional Heteroskedasticity (ARCH) models, first described by Engle [11].

The phenomenon of volatility clustering, described by Mandelbrot [12] as “large

changes tend to be followed by large changes, of either sign, and small changes tend to

be followed by small changes” can be described by ARCH models with large lags but

is more parsimoniously described by the Generalized Auto Regressive Conditional Het-

eroskedasticity or GARCH model, see Bollerslev [13] for more on the origins of GARCH.

Another advantage of the GARCH model is that it easily extends to accommodate even

more interesting phenomena.

The leverage effect is a phenomenon described by Black [14] as the negative corre-

lation that exists between recent returns and volatility. Typically, increases in asset

prices lead to decreasing levels of volatility and vice versa. The effect is also usually

asymmetric, i.e. a drop in an asset price would lead to larger change in volatility than

a corresponding increase in asset price. Models built to take this into account include

the GJR-GARCH model by Glosten et al. [15] and the EGARCH model by Nelson [16].

There is also the ARCH-in-Mean type modification by Engle et al. [17] to take into

account a similar phenomenon known as the volatility feedback effect. Note that the

volatility feedback effect leads to the same observed correlation between asset prices and

volatility as the leverage effect, however as Ait-Sahalia et al. [18] put it, “[the] effect is

consistent with the same correlation but reverses the causality: increases in volatility

lead to future negative returns”. Indeed the catalogue of volatility dynamics and their


corresponding ARCH / GARCH formulations seem to be endless, for more information

see two excellent review articles on the subject by Poon and Granger [3] & Andersen

et al. [19].

For financial forecasters, one aspect that must be considered is not just the accuracy

of a volatility estimate but also its time scale, what might be an accurate account of daily

volatility levels might be of less use to a long term forecaster than a less accurate record

of monthly volatility estimates. This leads to an interesting property of financial time

series; what is considered noise and what is considered a genuine movement in volatility is

often just a matter of scale. This phenomenon is caused by volatility structures not being

preserved after intertemporal aggregation, i.e. volatility tends to lose its self similarity

when viewed at different time scales. This property’s effect on volatility forecasts using

daily vs monthly data was described by Figlewski [20] however it is a more dominant

phenomenon when dealing with high frequency returns and will be revisited in this

context during Chapters 4 & 5. This raises a perplexing question; at any particular

scale, which changes in volatility can be described as noise and which are legitimate

changes in variance?

While there are techniques available to determine volatility in the low frequency

realm, none of them are perfect. The main drawback of many existing and popular

techniques is that assumptions made on the underlying behaviour of the time series may

introduce artefacts which can artificially inflate or retard volatility estimates. Further-

more, many of the models described so far need large amounts of in-sample data for the

purposes of parameter fitting. There is room for more non-parametric methods which

require no lead in data for parameter fitting and make no assumptions on the underlying

process.

To address the points made above, one of the aims of this thesis is to construct a

parameter free volatility estimate which is less susceptible to the kinds of artefacts which

can result in misleading volatility estimates.


1.2.2 High Frequency Volatility Dynamics

High frequency volatility estimation is also an increasingly interesting topic for re-

searchers and of growing practical importance for market participants. Recent increases

in the availability of high frequency financial tick data has inexorably led to the high

frequency trading strategies that take advantage of it. There is an inherent risk in most

trading strategies, even if the exposure is for a matter of seconds. Some of the trading

strategies which expose themselves to short term risk include asset pricing, statistical ar-

bitrage, market making, trading signal generation and a growing plethora of autonomous

trading strategies. Of paramount importance to the successful implementation of these

strategies is a very recent and reliable measure of volatility.

Unfortunately in the high frequency realm, measuring volatility doesn’t necessar-

ily become easier as the frequency of the available data increases. For many kinds of

financial markets, e.g. stock indices and Foreign eXchange (FX) rates, as the sample fre-

quency increases the nature of the random process appears to change. This breakdown

of self similarity is due to price movements after long time intervals being driven by a

perceived change in the true price (or frictionless price) of the asset, while price move-

ments after short intervals are dominated by microstructure effects (market frictions).

These microstructure effects, or market frictions include:

• price discretisation; e.g. when asset prices are only being traded to the nearest

cent,

• bid ask bounce; anyone wanting to initiate a buy or sell order of an asset must

cross the ‘spread’ which is the difference between the lowest sell quote and the

highest buy quote,

• heterogeneous beliefs; traders acting on different, often opposite beliefs may cause

prices to swing rapidly,

• simultaneous quotations & trades occurring on different networks or markets; in-

formation collected on asset prices over different trading networks can show the

one asset with two prices simultaneously.


Typically, the whole collection of microstructure effects are accounted for in the literature

by adding the Independent and Identically Distributed (IID) noise term ut in (1.6), where

pt represents the unobservable frictionless log price and pt the noisy logarithmic price.

pt = pt + ut (1.6)

Up until this point no distinction has been made between the observable and the

frictionless price because at low frequencies market frictions contribute little towards

the observed returns and so pt ≈ pt under these circumstances. However, as the time

between observations gets smaller, market microstructure effects make up a larger pro-

portion of any observed change in price. Thus, the frictionless returns are increasingly

obfuscated under the observed returns at high frequencies. Hence, we are presented

with a dichotomy. Ideally, more observations would correspond to a lower stochastic

error within a volatility measurement; on the other hand, a higher sampling rate can

introduce biases due to microstructure effects.

Much progress has been made in the attempt to overcome the difficulties in high

frequency volatility estimation. Most significantly, quadratic variation theory allows us

to measure the integrated volatility using squared daily returns, see Andersen et al. [6].

While empirical analysis by Martens [21] has shown that this approach does yield better

estimates than that obtained by methods that use daily data, it does have a tendency

to break down when microstructure noise becomes significant, see Andersen et al. [22],

Bandi and Russell [23], Zhang et al. [24]. Corsi et al. [25] found that this breakdown in

the realized volatility measure at high frequencies can be explained by the autocorrela-

tion structure of the returns data. This phenomenon is most commonly observed in the

negative autocorrelation of high frequency returns which results in positive anomalous

scaling for volatility estimates in FX markets. Conversely, positive autocorrelation in

returns leads to negative anomalous scaling of volatility estimates in many stocks and

indices.


Several suggestions have been put forward to solve the problem of microstructure

noise, firstly Andersen et al. [6], [26] put forward the idea of sparse sampling, which ef-

fectively decreases the proportion of the return volatility which is due to noise. However,

it was shown by Zhang et al. [24] that this is not an adequate solution to the problem.

This led to the development of the Two Scale Realized Volatility (TSRV) estimator,

which was put forward by Zhang and later refined by Ait-Sahalia et al. [27]. There have

also been several proposals for consistent Kernel based estimators, most notable are the

contributions made by Barndorff-Nielsen et al. [28] whose kernel estimators were shown

to be robust to endogenous and dependent noise as well as endogenously spaced data.

The method of applying filters has also been applied to the quandary of microstructure

noise, in particular Hansen et al. [29] were able to produce a consistent estimator of the

integrated volatility using the Moving Average (MA(1)) structure outlined by Ander-

sen et al. [26] and Ebens [30]. Finally, a conceptually similar approach to the method

developed in this thesis was taken by Curci and Corsi [31] in which their Discrete Sine

Transform (DST) diagonalised a moving average process into an orthonormal basis, from

which the microstructure and actual price could be separated.

The HHT based volatility estimates proposed in this thesis are also useful in the

realm of high frequency volatility estimation. It is shown that the empirical mode de-

composition process which forms a key component of the HHT, is capable of filtering

out a proxy for the ‘true’ or frictionless price from the noisy observable price which

has been obfuscated by the effects of market microstructure. The proposed approach

is also capable of delivering spot volatility estimates as opposed to the estimations of

integrated volatility that RV based techniques provide, with further improvements this

feature may be of great interest to those with an interest in ultra high frequency volatility.


1.3 A New Tool in the Toolbox

The Hilbert-Huang transform is a powerful new tool for the analysis of non-stationary

and nonlinear time series. Its adaptive and data driven nature make it highly flexible,

this has driven its use in a growing number of fields. The HHT is comprised of two parts,

first the Empirical Mode Decomposition (EMD) breaks a signal down into simple wave

like components that are sorted by frequency and then the Hilbert transform is applied,

yielding instantaneous amplitudes and frequencies of the simple waves. One of the main

advantages of using this transform is the ease with which the information from the HHT

can be interpreted. Due to the nature of the frequency and amplitude information the

process extracts, it is natural to make a comparison to Fourier methods, as indeed many

have, Huang et al. [1], Donnelly [32] to name a few. While Fourier techniques are ex-

tremely useful and are perfect for transforming signals from the time-amplitude domain

into the amplitude-frequency domain, they suffer from the drawback that this transform

is done at the expense of the time domain. Thus, where the Fourier transform is lacking,

is in the identification of changes in frequency with respect to time.

While Fourier techniques transform a signal from the time domain into a frequency

domain signal they cannot transform back into the amplitude-time domain without as-

suming that the signal is periodic and stationary. Thus, the main advantage of the HHT

is that the time domain is preserved, allowing for not only the identification of different

frequencies in a signal, but also their temporal location. This property is particularly

attractive when analysing non-stationary time series which, by definition change their

properties over time. There are also Fourier methods which analyse only a small sliding

window of data at a time, this procedure can maintain the time domain to an extent,

however there is a costly trade off between frequency and temporal resolution which is

difficult to overcome.

Wavelet transforms are another tool which can yield frequency information without


compromising the time domain however they are described by Huang et al. [1] as suf-

fering from uniformly poor resolution at different time scales because it is restricted

by essentially the same dichotomy as windowed Fourier transforms, i.e. one may have

high temporal resolution or high frequency resolution but not both simultaneously. This

being said, the literature on wavelets is growing rapidly and new wavelet decomposition

techniques are challenging this unfortunate dichotomy regularly. Comparisons between

the HHT, or more strictly the EMD which underpins it and wavelet transforms seem

more apt, indeed Flandrin et al. [33, 34] has made a strong case for the the EMD be-

having like a diadic filter bank, resembling some wavelet decompositions.

One of the more appealing attributes of the HHT is its seemingly natural ability to

extract periodic signals even when they are hidden by high levels of noise, see Li and

Meng [35]. This ability to perform hidden feature extraction was given more weight by

the work of Wu and Huang [36] who were able to assign a statistical significance to the

these extracted trends.

Flandrin et al. [33, 34] & Wu and Huang [37] further explored the properties of the

EMD when applied to noisy and stochastic signals and much of their methodology is

employed over the course of this thesis to examine the spectral properties of the wave

like structures that are produced by the EMD process, Chapter 2 elaborates further on

these developments.

The HHT and the EMD process which underpins it, have found many and varied

applications in a number of fields. In mechanical engineering the HHT has been used by

Liu et al. [38] to identify faults in gearboxes by analysing the telltale changes in vibration

which may indicate a cracked tooth or a some other gearbox related malady. In electrical

engineering the HHT is becoming an increasingly popular method for monitoring power

quality, see Senroy et al. [39], Yalcin et al. [40]. Due to the adroit manner in which the

EMD process handles seasonal data it is has found use as an analytical tool in the area

of climate science, see Molla et al. [41], ocean waves Rao and Hsu [42], Hwang et al.


[43] and pathology Cummings et al. [44]. The EMD and HHT processes are also slowly

gaining in popularity as tools in the financial world due to the complex, occasionally

seasonal, and reliably chaotic nature of financial data. The first and most natural appli-

cation of these new tools is seasonality analysis, Zhang et al. [45], Crowley [46], Huang

et al. [47], all used these techniques to break down financial time series data into seasonal

components for easier analysis.

The HHT has even been used as a financial volatility indicator in the past, in par-

ticular by Rao and Hsu [42], Huang et al. [47]. These approaches are often referred to

as variability or fluctuation measures as they do not give a volatility measure which

can be interpreted in terms of the variance. Thus the volatility estimates they produce,

while being useful volatility indicators, are harder to fit into existing financial models

and so there exists an enticing gap in the literature where a usable HHT based volatility

estimator should be.

The HHT specialises in analysing nonlinear and non-stationary time series, two prop-

erties which make it an ideal candidate for the backbone of a volatility estimator. In this

thesis the HHT is used as an actual measure of financial time series variance for the first

time. The only HHT based volatility measure conceptually similar to the one introduced

in this thesis was created by Wen and Gu [48] for earthquake simulation and modelling,

however its full potential as a volatility estimator was left largely unexplored. In this

thesis the HHT is shown to be capable of extracting the spot volatility σ2t which is easily

interpreted, rather than the less tractable volatility indicators it had hitherto produced.

Since the HHT procedure has not been used in this context before, the goal of this thesis

is not to necessarily create the best volatility estimator possible, but rather to evaluate

the viability of the HHT procedure as a spot volatility estimator and to examine the

nature of financial time series dynamics through a new and powerful analytical tool.


1.4 Thesis Outline

• Chapter 2

This chapter goes into more detail regarding a selection of popular volatility es-

timation procedures which were mentioned in the introduction. The essential

background information needed to understand the Hilbert-Huang transform as

well as the volatility estimator derived from it are also discussed in detail. The

components of the HHT procedure are also used to perform some analysis on de-

terministic and stochastic time series in order to familiarise the reader with some

of the properties of the HHT procedure. This chapter also covers some of the

testing and comparison procedures and some of the difficulties faced therein.

• Chapter 3

This chapter introduces an intuitive means of estimating the low frequency (daily)

historical volatility using the HHT. The HHT procedure is used to generate volatil-

ity estimates which are then compared against estimates given by a handful of con-

temporary spot volatility estimators as well as the realized volatility which is used

as a proxy for the true volatility. The proposed procedure is applied to ten years

worth of data from the All Ordinaries (All Ords) and the National Association of

Securities Dealers Automated Quotations (NASDAQ) Indices.

• Chapter 4

This chapter contributes to both low and high frequency volatility measurement

by first proposing and testing an improved HHT based method of measuring the

volatility of a time series. Firstly a test is proposed in order to examine and

quantify the trade off between temporal resolution long term accuracy. The HHT

based volatility estimate is then adapted to handle time series with microstructure

noise and its effectiveness is thoroughly tested using both simulated and data in

this chapter. Monte Carlo techniques are employed to simulate three different

kinds of time varying volatility for different levels of market microstructure noise.

• Chapter 5

This chapter tests a variant of the procedure developed in Chapter 4, using real


high frequency FX data. The inherent difficulty of assessing the estimation of

a latent variable is overcome by setting up a simulated options market, which

enabled an effective comparison based evaluation under real market conditions.

• Chapter 6

This chapter concludes the thesis. There is a summary of progress made and a

some thoughts are shared on the possible directions of future work.

Chapter 2

Background

This chapter is split up into three sections, the first section covers some of the most

widely practiced methods for calculating volatility at both low and high frequencies.

The second section describes the Hilbert-Huang transform and some analysis is into

its properties is provided. Finally, section three discusses some of the problems faced

when analysing a latent variable such as volatility and presents some of the measures

commonly used to evaluate model accuracy.

2.1 Volatility Measures

This section introduces some of the problems faced when measuring historical volatility

and clarifies which variants of each model have been used in this study. A descrip-

tion of the Moving Average (MA), Exponentially Weighted Moving Average (EWMA),

Generalized Autoregressive Conditional Heteroskedasticity (GARCH(1,1)) and Realized

Volatility (RV) models are given.

2.1.1 The Moving Average Volatility Model

The MA volatility estimate is calculated by sliding a window of interest along a time

series and calculating the sample variance of points within this moving window. When

this technique is applied a sliding window length must be chosen, this parameter can

16

Chapter 2. Background 17

greatly alter the volatility observed at any point. While the MA method may be suit-

able for describing the volatility of some time series, it suffers from the drawback that

one may only increase long term accuracy with a decrease in temporal resolution. This

drawback gives rise to artefacts such as ghosting and plateauing, in which a short term

spike or trough in volatility will be equally represented along an entire window length.

Overly inflated or retarded volatility estimates persist until the short term disturbance

is no longer included within the sliding window. This plateauing effect is clearly visible

in Figure 2.1, where short term fluctuations in price are followed by a long period of

overly high volatility estimates. The issue with poor temporal resolution and plateauing

renders the MA method of limited use, Wilmott [10] suggests that its use is confined

to cases with slowly varying volatility. The MA variance of the log returns r, usually

calculated on market closing prices is given by:

MAσ2n(N) =

1

N − 1

N−1∑j=0

(rn−j − r)2, (2.1)

where r is the expected (or mean) return which is usually assumed to be zero when the

sampling rate is daily or shorter and MAσ2n(N) should be understood as the variance

on the nth day where N is the sliding window length . Note that the ∼ symbol above

a character is often used to denote estimates throughout this thesis and that the MA

standard deviation is simply the square root of (2.1).

2.1.2 The Exponentially Weighted Moving Average Model

EWMA is a popular and effective technique for calculating daily variances, it largely

overcomes the plateauing effects of the MA model by weighting the importance of returns

based on their time of arrival, with the most recent returns given higher weights. The


2000 2001 2002 2003 2004 2005

2000

3000

4000

5000

All

Ord

s P

rice

Date

All Ords Price and MA Volatility

0

0.1

0.2

0.3

MA

Vol

atili

ty σ

n

Figure 2.1: All Ords Index prices from 1st Jan 2000 to 31st Dec 2004 (bold) and asimple MA volatility estimate with a window length of 50 days (dashed).

EWMA procedure for estimating volatility is given by:

EWMAσ2n(λ) = (1− λ)

∞∑j=0

λjr2n−j ,

or, written recursively:

EWMAσ2n(λ) = λEWMAσ2

n−1(λ) + (1− λ)r2n. (2.2)

where EWMAσ2n−1(λ) is the last variance estimate, r2

n the most recent squared return and

λ the decay rate. For daily returns it is common to let λ = 0.94 and for monthly returns

it is common practice to let λ = 0.97 because these values were found to minimise mean

square errors when performing one step ahead volatility forecasts, this is often referred

to as the RiskMetrics model for volatility modelling, see Longerstaey and Spencer [49].

The EWMA approach does indeed decrease the persistence of volatility spikes how-

ever they are still observed to decay exponentially at a rate proportional to λ. This

brings one to the inevitable question, what is a suitable choice for λ? Whatever our

choice for the decay rate it is still a balancing act between a noisy volatility estimate


with high temporal resolution and a smooth estimate with better long term accuracy.

This in turn begs the question of just what is noise and what is a legitimate short term

change in volatility. Suppose you had a time series with rapidly changing volatility fluc-

tuating around a long term mean; when determining parameters such as the decay rate,

a choice must be made as to which property of the time series your volatility estimate is

designed to extract, i.e. the long term average or the short term fluctuations. Figure 2.2

shows how two different choices for λ result in one noisy but highly responsive measure

and one that is smooth and better suited to picking up long term trends. The latter of

these two parameter choices is more susceptible to a plateauing like effect, in which the

volatility gradually decays down to its new level, rather than producing the tabletop

like artefact the MA procedure generates. This effect is particularly evident after sharp

declines in volatility as seen around mid 2000 and late 2001 in Figure 2.2. The impact

of different choices for λ are explored further in Chapter 4.

2000 2001 2002 2003 2004 20050

0.05

0.1

0.15

0.2

0.25

0.3

0.35

0.4

0.45

0.5

Date

Vol

atili

ty σ

n

All Ords EWMA Volatility

λ=0.7λ=0.94

Figure 2.2: EWMA volatility estimates with decay rates λ = 0.94 & λ = 0.7 for theAll Ords data set covering 1st Jan 2000 - 31st Dec 2004.


2.1.3 The GARCH(1,1) Model

The GARCH(1,1) model given by Equation (2.5) is widely used to model volatility

because of its relative simplicity and flexibility. It was designed to capture the volatil-

ity clustering behaviour, a phenomenon involving periods of high and low volatility that

have a tendency to group together, see Ruppert [50] for a more in depth look at volatility

clustering in financial markets. The GARCH(1,1) model is similar to the EWMA model

because it too depends on the last variance estimate the model returned (GARCH σ2n−1)

and the most recent squared return r2n, however in addition to this there is also the long

term average term average variance form, represented in its weighted form as ω below.

The disadvantage involved with using a GARCH(1,1) model is that it can require

a large amount of data in order to determine its three parameters, also like EWMA

methods GARCH(1,1) still shows some evidence of plateauing which can be observed

most prominently after a sharp decrease in volatility. The GARCH(1,1) model is usually

written in the following recursive form when used as a variance estimate:

rn = r + εn (2.3)

εn | Φn−1 ∼ N(0,GARCH σ2n) (2.4)

GARCH σ2n = ω + αε2n + βGARCH σ2

n−1, (2.5)

where rn is the daily log return, r is the mean return which is assumed to be zero for

sufficiently high frequency, Φn−1 contains all information up to day n-1, and the residu-

als εn follow a conditional normal distribution with mean zero and variance GARCH σ2n.

Also ω, α and β are parameters to be determined, with ω > 0; α ≥ 0; and for long

term stability it is required that α+β < 1 so that the long term average volatility stays

positive. More specifically, α is the weighting of the dependence on the most recent

innovation, β is the weight assigned to the last volatility estimate and ω is a constant.

Note that the unconditional variance of is given by σ2 = ω1−(α+β) .


The GARCH(1,1) model behaves similarly to the EWMA model, with β taking

the place of λ and α that of (1 − λ). The more general GARCH model is given as

GARCH(p,q) which uses the p most recent values of the returns r and q most recent

estimates of GARCH σ2. For more information on the GARCH procedure and its imple-

mentation see Bollerslev [13], Posedel [51] and Reinhard Hansen and Lunde [52].

2.1.4 Basic Realized Volatility

The realized volatility measure is a non-parametric, model free indicator and is generally

considered to be a very accurate means of extracting the volatility of a time series for

which intraday data is available. This section formalises and expands upon the realized

volatility discussion given in the introduction.

For the purposes of the RV measures we assume a price model like that described in

Andersen et al. [53] and Barucci et al. [54], i.e. the log of the frictionless price follows

Equation (1.3). The goal of the RV method is to measure the Integrated Volatility (IV)

over a certain period, for convenience, the unit for the time interval is usually set as one

day. Recall that one period of integrated volatility is defined by Equation (1.4). This

definition of IVt is sometimes referred to as the quadratic return variation. Although the

integrated volatility is not a directly observable variable it has been shown by Andersen

et al. [7] and Barndorff-Nielsen and Shephard [55] to be closely estimated by the realized

volatility, which is defined in Equation (2.7). First, we recall that the log return at time

t for step size h is defined as:

rt(h) = pt − pt−h. (2.6)

Then the realized volatility can be defined as:

BasicRV t(h) ≡1/h∑i=1

(rt−1+ih(h))2, (2.7)

which should be understood as the RV on day t for sample interval h, where 1/h is a

positive integer and corresponds to the daily sampling frequency. Note that when h is

at the smallest increment available, i.e. every sample point is used, the resulting RV


measure is denoted in this thesis as AllRV t.

Theoretically, without the presence of microstructure noise the realized volatility

estimate given by Equation (2.7) converges towards the integrated volatility given by

Equation (1.4) as h → 0 because the sample error approaches zero. In practice, when

microstructure noise is included into the observed log price, as in Equation (1.6), the

RV estimate can be greatly distorted as the the time between samples decreases. For

this reason, RV estimates usually use returns separated by intervals of between 5 and

15 minutes.

2.1.5 Overnight Returns

While many FX markets are open and trading frequently for 24 hours a day, it is typical

of most stock markets to only operate for a fraction of the day, usually between the

hours of 10am and 4pm local time. This cessation of trading usually results in a jump

(up or down) in asset prices when the market is reopened which leads to something

of a disconnect between the volatility measures that use strictly intraday trading data

and those which use daily data. Since this overnight change can be quite large, usually

following some important event, any volatility estimate that uses strictly intraday data

risks becoming a poor proxy to the daily volatility. To take into account this change

in overnight prices, a number of approaches are considered here, see Hol and Koopman

[56] or Tsay [57] for alternative approaches.

Since the more formal definitions of the RV model defined in Section 2.1.4 do not

easily lend themselves to the discussion of overnight returns, some simple notation is

introduced. Let the log prices on day t be defined as pt(k), with intraday values denoted

by k = 0 . . . 1/h. A value of k = 0 gives the opening price on day t and k = 1/h gives

the closing price on day t. Hence, a simple “close to open” log return is defined as

cort = pt(0)− pt−1(1/h).


One simple approach to to calculate the RV for a whole day is simply to add any

overnight return square to the RV estimated for that day, i.e.:

WDRV t(h) =co r2t +Basic RV t(h), (2.8)

where the superscript WD simply denotes that the estimate covers a whole day. The

problem with this approach is the noisy nature of cort which has a variance far greater

than that of BasicRV t(h), leading to an estimate of volatility for the whole day that is

relatively noisy.

When overnight return data is available, even if it is sparsely sampled then the

overnight volatility can be calculated by including the overnight returns at intervals h

and combined with the intraday RV to give:

ContRV2t (h, h) ≈

1/h∑j=1

(rt−1+jh(h))2 +

1/h∑i=1

(rt−1+ih(h))2, (2.9)

where the superscript Cont denotes continuous trading, 1/h is an integer, the summation∑1/hj=1 covers the sparsely sampled overnight (close to open) period and the summation∑1/hi=1 covers the (open to close) trading hours. Naturally, this reduces to Equation (2.8)

when the overnight sample period is very low. Conversely, when the sampling rate is

consistently high, this approach is equivalent to Equation (2.7) with a 24 hour period.

An alternative approach was developed by Martens [21] & Koopman et al. [58], it is

defined as follows:

WDRV2t (h) = (1 + c)BasicRV t(h), (2.10)

where WD simply denotes that the estimate covers a whole day and c is a positive

constant, i.e. taking into account overnight returns will only increase a variance estimate

that has only used data collected while the market was open. The scaling factor (1 + c)

is defined as:

1 + c =σ2oc + σ2

co

σ2oc

, (2.11)


where σ2oc and σ2

oc are variance estimates of the “open to close” and “close to open”

data, i.e.:

σ2oc =

1

N

N∑t=1

(ocrt)2, (2.12)

σ2co =

1

N

N∑t=1

(cort)2, (2.13)

where a simple “open to close” log return is defined as ocrt = pt(1/h) − pt(0) and cort

was defied earlier.

The observation that σ2oc is inherently noisier than RV oc led Hansen and Lunde

[59, 60] to develop their own whole day volatility estimate. Their approach proposes

that the adjustment factor of Equation (2.10) follows:

1 + c =

∑Nt=1(ccrt − r)2∑N

t=1BasicRV t(h)

, (2.14)

where ccrt denotes the “close to close” return on day t, i.e. ccrt = rt(1/h) − rt−1(1/h)

and r = 1N

∑Nt=1

ccrt. Equations (2.10) & (2.14) are used to calculate the daily returns

for the data examined in Chapter 3 of this thesis.

2.1.6 Sparse Realized Volatility and the Averaged or Subsampled Re-

alized Volatility

Consider a model with microstructure noise as given by Equation (1.6) with ut being

IID noise and the observed log returns given by:

rt(h) ≡ pt − pt−h, (2.15)

the noise is linked to the contaminated (noisy) observable returns rt(h) by:

rt(h) = rt(h) + et(h), (2.16)


where

et(h) = ut − ut−h. (2.17)

The sparse RV estimate is essentially the same as the basic RV measure however it

is not sampled at the highest available frequency, but rather at spacings h which are

some multiple (denoted nh) of the minimum sample interval, e.g. h = nhh where 1/h is

a positive integer. The sparse volatility estimator sparseRV is then defined as:

sparseRV t(h) ≡1/h∑i=1

(rt−1+ih(h))2. (2.18)

The advantage of this method is that it is less susceptible to the microstructure noise

that plagues higher frequency measurements. This is where an important compromise

must be made, one must choose between a low sample error and high microstructure

interference or high sample error and less microstructure effects.

Motivated by the desire to keep sample error at a minimum, while still retaining the

more robust nature of sparse sampling, the method of averaged or subsampled RV was

born. Firstly, an offset version of the subsampled method described above is given as:

OffsetRV t(h, k) ≡Nk∑i=1

(rt−1+ih+kh(h))2, (2.19)

with k = 0, . . . , nh − 1 and Nk = 1/h.

The AverageRV volatility estimator is then produced by taking the average of many

offset sparse RV estimates, i.e.

AverageRV t(h) =1

nh

nh−1∑k=0

OffsetRV t(h, k) (2.20)

The result of this procedure is a volatility estimator that can be resistant to market

microstructure noise when sampled sparsely and has a smaller sample error than the

standard sparse RV estimate.


2.1.7 Two Sample Realized Volatility

Unfortunately the sparse estimator and the averaged sparse estimator can still suffer

from a bias, especially if the sample rate is not chosen carefully. Fortunately though,

the size of this bias can be estimated and taken into account. This is achieved by the

Two Scale Realized Volatility (TSRV) procedure, which first uses an average sparse

estimator and then subtracts the theoretical bias.

˜TSRV t(h) =Average RV t(h)− (h/h)AllRV (h). (2.21)

A simple adjustment can be made to the above formula to account for the finite-sample

size of the two summands in Equation (2.21), this adjusted variant is given by:

Adjusted ˜TSRV = (1− h/h)−1 ˜TSRV t(h) (2.22)

While the family of RV methods are extremely useful, they may not be perfect for

every scenario. For example, Figlewski [20] showed that using daily volatility estimates

can lead to poorer long term forecasts than those produced from lower frequency data.

Realized volatility also requires a large amount of high frequency intraday data, this is

somewhat of a drawback since such high frequency data may not be available or may be

costly. Still, when high frequency data is available the Adjusted-TSRV method forms a

good benchmark and is widely used.

2.2 The Hilbert-Huang Transform

The HHT algorithm includes two separate procedures, firstly the EMD process breaks up

a signal into different frequency components known as Intrinsic Mode Functions (IMFs)

denoted by Ψ and secondly the Hilbert transform is used to calculate the instantaneous

amplitude and frequency for each separate IMF component. The IMFs are naturally

sorted from the highest to the lowest frequency, i.e. the first IMF is a time series con-

taining the rapidly changing components of a signal while later IMFs extract features


that vary more slowly in time.

The reader should be aware that there is also a slight change in notation at this

point. To be in keeping with the standard notation of the HHT, time will no longer be

denoted by a subscript as those positions are now reserved for IMF numbers. Also, a

resource that was of great help in understanding and performing the EMD process is

provided by Flandrin [61].

2.2.1 Empirical Mode Decomposition

For the HHT process to give meaningful information about a signal, the EMD process

will decompose a signal S(t) into IMFs which have the following properties

(i) Each IMF must have exactly one zero between any two consecutive local extrema.

(ii) At any point the mean value of the upper envelope (consisting of splines connecting

all of the maxima) and the lower envelope (consisting of splines connecting all of

the minima) is zero (within a tolerance).

The following is an outline of the steps needed to complete the EMD process, it

follows an outline described by Senroy et al. [39]:

1. Identify local maxima and minima of the signal S(t).

2. Create a cubic spline going through all of the maxima Cmax(t) and another going

through all the minima Cmin(t). This step is illustrated in Figure 2.3 where the

signal S(t) is clearly seen to be encased by the maximum and minimum splines.

3. Calculate the mean of the two splines.

Cmean(t) =Cmax(t) + Cmin(t)

2. (2.23)


4. Calculate the first potential IMF known as a Proto-Mode Function (PMF),

PMF1(t) = S(t)− Cmean(t). (2.24)

5. If PMF1(t) satisfies the conditions to be an IMF then Ψ1(t) = PMF1(t). If not

repeat steps 1 - 4 on PMF1 until it becomes an IMF.

6. Calculate the first residue Ω1(t),

Ω1(t) = S(t)−Ψ1(t). (2.25)

7. If the maximum amplitude of the residue is below a threshold or there are three

or fewer local maxima or minima, terminate the EMD process, otherwise repeat

steps 1 - 6 on the residue Ω1(t).

When the procedure has terminated, the result is a set of wave like functions as seen

in Figure 2.4 plus a leftover which can be recombined to give the original signal i.e.

S(t) =

n∑j=1

Ψj(t) + Ωn(t), (2.26)

where Ψj(t) are the wave like IMFs and Ωn is the leftover or residue. Also, as observed

in Wu and Huang [36], Equation (2.26) may be rewritten as:

S(t) =n∑j=1

Ψj(t) + Ωn(t) =m∑j=1

Ψj(t) + Ωm(t) =m+1∑j=1

Ψj(t) + Ωm+1(t), (2.27)

where m < n and Ωm(t) is the remainder of S(t) after m number of IMFs are extracted.

Also, it can be shown that each residue, with the exception of the last, can be written

as an IMF plus a lower frequency residue, i.e.

Ωj(t) = Ψj+1(t) + Ωj+1(t). (2.28)

As Wu and Huang [36] observe, Ωj+1(t) is a local mean of Ωj(t) which is valid for a

time approximately equal to the local period of Ψj+1(t), i.e. each successively lower


0 0.2 0.4 0.6 0.8 1−6

−4

−2

0

2

4

Time

S(t

)

Signal within Envelope

SignalMinMax

Figure 2.3: Super imposed sinusoids (solid blue), the local maxima and connectingspline (red circle and dot-dash line respectively) and the local minima and connecting

spline (green circle and dashed line respectively).

frequency residue forms a local mean to its higher frequency predecessor. This concept

of extracting a local mean in terms of residues and IMFs is used to construct a proxy to

the return series, as described in Section 3.1.1.

A simple example of the procedure follows. Consider a sinusoidal signal that contains

two waveforms, one at 5 Hz at amplitude 3 arbitrary units (arb.) and another at 15

Hz at an amplitude of 1 arb. which persists until t = 0.5, after which point the higher

frequency component increases from 15 to 25 Hz and persists until t = 1. This signal of

superimposed sinusoids can be seen in Figure 2.3 along with local maxima and minima

indicated as well as the splines joining those extrema points. Figure 2.4 shows the

IMFs extracted from the sinusoidal signal, note how the first IMF contains the two

high frequency components consecutively while the second IMF only contains the lower

frequency component. This demonstrates, in a simple manner, the frequency sorting

nature of the EMD process.


0 0.2 0.4 0.6 0.8 1−1.5

−1

−0.5

0

0.5

1

1.5First IMF

Time0 0.2 0.4 0.6 0.8 1

−4

−3

−2

−1

0

1

2

3

4

Time

Second IMF

Figure 2.4: The first two IMFs of S(t).

2.2.2 The HHT as a Variability Measure

The EMD process has been used by Rao and Hsu [42], Huang et al. [47] to estimate

historical volatility, the estimate is described by:

V ariabilityHHT (t) =|∑n

j=1 Ψj(t)|S(t)

, (2.29)

where n is the number of IMFs in the summation. In practice this means that n is a

parameter that determines the time scale to which the variability measure applies. In

effect this procedure gives the distance between the current value of the time series and

a variable local mean which is set by changing n. Perhaps the biggest disadvantage of

this method is the fact that the results it yields aren’t in terms of the variance and

are therefore not easily compatible with existing models. While this variability measure

does have several drawbacks, many described by Rao and Hsu [42], Huang et al. [47], it

does at least serve as a starting point from which more practical HHT based volatility

estimators may be built.

2.2.3 Monte Carlo Analysis of the EMD Procedure

In this section a quick study is carried out on the properties of the EMD procedure

when applied to both Gaussian White Noise (GWN) and the more complicated process


0 1 2 3 4 5 6 7 8−11

−10

−9

−8

−7

−6

−5

−4

−3

−2Semi−log plot of IMF Number vs Period

IMF Number

Log 2 P

erio

d

Figure 2.5: IMF number vs log2 IMF period for GWN.

of fractional Gaussian noise (fGn).

Some interesting properties the EMD procedure are revealed when it is applied to

simple Gaussian White Noise. In this simulated study a simple GWN is generated for

2048 points and repeated 1000 times. Applying the EMD procedure to this data set and

taking the average over the 1000 simulations it is possible to extract some interesting

features, namely the log-linear relationship between the IMF number and the log of the

average period of each IMF, as shown in Figure 2.5. From this figure we can also observe

that the period almost exactly doubles with each successive IMF, this is in line with the

observations made by Flandrin et al. [33] and Wu and Huang [37]. This period doubling

of successive IMFs is similar to the behaviour of dyadic wavelet filters, see Flandrin et al.

[33, 34].

Research done by Huang et al. [62] suggests that for white noise, the spread of energy

(or variance in our case, as seen in the top of Figure 2.6) within each IMF should follow a

χ2 distribution. Also, Huang proposes that deviations from this spread are evidence for

statistically significant deviations from random white noise. This finding has important


implications for seasonality analysis as it may allow for the identification of seasonal

activity, or trend extraction and provide a means for statistical significance testing.

Figure 2.7 depicts the Fourier power spectra of the first 10 IMFs, it illustrates another

important feature of the EMD process observed by Flandrin et al. [33] & Rilling et al.

[63], in that the first IMF acts as a high pass filter, i.e. only components of a signal

that are above some frequency threshold are admitted. This behaviour can be observed

in the right most part of the figure where the spectrum which corresponds to the first

IMF maintains a high power spectral density right through to the limit of observable

frequencies (the Nyquist frequency). The last IMF, visible on the far left of the figure

behaves in quite the opposite way, acting as a low pass filter. Contrary to the first and

last IMFs, every other IMF behave more like a band-pass filter, i.e. the frequencies

found in the IMFs are bound above and below.

The abscissa (horizontal position) of the IMF power spectra can also be shifted to

demonstrate that IMFs produced by the EMD procedure are highly self similar at dif-

ferent time scales. This property is depicted in Figure 2.8, note that with the exception

of the first and last IMF which have been removed from the plot, the remaining IMFs

largely lay on top of one another. This result, as with many others in this section con-

firm the results found by Flandrin et al. [33] & Rilling et al. [63], which support the

proposition that the EMD procedure has a dyadic filter-bank structure.

An interesting linear relationship exists between the log period of an IMF and the

log of the variance within that IMF, a similar relationship also exists between the IMF

number and the log variance. Both of these relationships can be seen in Figure 2.6. From

Figure 2.6 it is evident that there is a rough doubling of variance as the IMF number

increases and that a similar behaviour is being followed by the variance in relation to the

IMF number. These two properties, or rather the breakdown of these two properties,

is of practical importance during the self similarity analysis which is to follow. The

research carried out by Flandrin et al. [33] & Rilling et al. [63] also shows that the EMD

procedure can be used to investigate the self similarity of a time series at different scales


−10 −9 −8 −7 −6 −5 −4 −3 −2−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

Log2 Mean Period

Log 2 V

aria

nce

IMF Period vs IMF Variance for Gaussian White Noise

0 1 2 3 4 5 6 7−10

−9

−8

−7

−6

−5

−4

−3

−2

−1

0

IMF Number

Log 2 V

aria

nce

IMF Number vs IMF Variance for Gaussian White Noise

Figure 2.6: Log mean period vs log2 variance for GWN (top) & IMF number vs log2IMF variance for GWN.


0 2 4 6 8 10−30

−25

−20

−15

−10

−5

log Frequency

log

PS

D (

dB)

Fourier Power Spectra of IMFs

Figure 2.7: Power Spectra of 10 IMFs on GWN.

by examining the change in relationship between the variance and the IMF number or

log period. This is of particular interest since it may provide a useful tool when exam-

ining the autocorrelation structure of market microstructure noise .

Fractional Gaussian noise (fGn) can be used to demonstrate different autocorrelation

structures by altering the Hurst exponent H. We now turn to the more complex model

of fGn to generate our data which is of length 2048 and has 1000 repeats. This simu-

lation model is more flexible and allows for changes in self similarity which will aid our

later examination of market microstructure noise. The Hurst exponent is of particular

interest because it yields information on the self similarity of the process at different

scales and the parameter is directly related to the fractal dimension of the series. This

is of interest because when the Hurst estimate is 0 < H < 0.5, there is evidence of a

negative autocorrelation which is a property that also leads to the anomalous scaling

caused by market microstructure. This negative autocorrelation means that the time


5 6 7 8 9 10−16

−15

−14

−13

−12

−11

−10

−9

−8

Shifted Frequency

log

PS

D (

dB)

Shifted Fourier Power Spectra of IMFs

Figure 2.8: Rescaled Power Spectra of 10 IMFs on GWN.

series will have a tendency to regress towards the mean quickly, i.e. any large jumps in

one direction are likely to be followed by large jumps in the opposite direction. Con-

versely if 0.5 < H < 1 this implies a positive autocorrelation, when this is the case any

movements of a time series in one particular direction are more likely to be followed my

more movements in the same direction, also known as clustering. Evidence of positive

autocorrelations in some financial time series returns have led to the rise of momentum

trading.

The change in behaviour of fGn with different Hurst exponents is well demonstrated

by its Fourier power spectra as see in Figure 2.9. Note that in the top plot when H = 0.25

the spectral peaks are increasing in height from left to right meaning that the short term

fluctuations have more spectral power. Conversely when H = 0.75 as depicted in the

bottom figure, the height of the peaks are decreasing from left to right which means that

lower frequency components are more powerful. Finally in the case when H = 0.5 the


0 1 2 3 4 5 6 7 8 9 10−35

−30

−25

−20

−15

−10

−5

log

PS

D (

dB)

H=0.25

0 1 2 3 4 5 6 7 8 9 10−35

−30

−25

−20

−15

−10

−5

log

PS

D (

dB)

H=0.5

0 1 2 3 4 5 6 7 8 9 10−30

−25

−20

−15

−10

−5

0

log

PS

D (

dB)

H=0.75

log Frequency

Figure 2.9: Power Spectra of 10 IMFs on fGn for three values of H.

spectral peaks are the same for each IMF which is to be expected since fGn is actually

GWN for this value of H and the spectra is identical to that in Figure 2.7.

Following the procedures for examining self similarity outlined by Flandrin et al.

[33, 34] and Rilling et al. [63], the Hurst exponent was calculated using slight changes

in the slope of the linear relationship between the log of the mean period of an IMF and


the log variance within each IMF. Alternatively the IMF number and log variance can

be used since the same linear relationship is observed, as shown in Figure 2.10. As the

figure shows, the slope of the linear regression between the log IMF period and the log

IMF variance is observed to vary in with changes in the Hurst exponent and it is this

property which allows for the approximation of H using the EMD procedure and the

simple formula H = 1 + slope2 .

While the actual values of H in Figure 2.10 were 0.250, 0.500 and 0.750 the estimates

yielded from this procedure were 0.343, 0.535, 0.773 using the change in gradient of the

IMF number vs variance, as seen in the right hand side of Figure 2.10. However, using

the change in gradient in the period vs variance as seen in the left hand side of Figure

2.10 yielded the estimates 0.361, 0.534, 0.765. Both of these estimates were made using

the least squares merit function to maximise the fit of a line on the mean values of the

log period or IMF number with the log variance. Due to the usage of the means in

each case, only one H estimate was obtained for each method over the whole simulation,

making some forms of error analysis difficult.

In an effort to give some means to assess the variability of these H estimates the

process was repeated without first taking the mean values of the data resulting in a H

estimate for every simulation so that its variability could be investigated. This process

yielded slightly different H estimates of 0.413, 0.531 & 0.712 with standard deviations

of 0.052, 0.054 & 0.059 respectively. The EMD procedure to calculate the Hurst expo-

nent is significantly biased for smaller values of H and moderately biased in the case

when H > 0.5, this was also found to be the case by Rilling et al. [63]. That be-

ing said, the Hurst exponent is still a useful indicator of long or short term dependence

in a time series and so having another procedure to estimate it may provide a useful tool.


−10 −9 −8 −7 −6 −5 −4 −3 −2 −1−12

−10

−8

−6

−4

−2

0

Log2 Mean Period

Log 2 V

aria

nce

IMF Period vs IMF Variance for Fractional Gaussian Noise

1 2 3 4 5 6 7 8−12

−10

−8

−6

−4

−2

0

IMF Number

Log 2 V

aria

nce

IMF Number vs IMF Variance for Fractional Gaussian Noise

H=0.75

H=0.50

H=0.50

H=0.25

H=0.25

H=0.75

Figure 2.10: Log period vs log variance for 3 values of H (top). IMF number vs logvariance for 3 values of H (bottom).


2.2.4 The Hilbert Transform

Now we move on to the Hilbert part of the Hilbert-Huang transform. Essentially the

Hilbert transform (denoted Hilb) is applied to each IMF to create what is known as an

analytic signal, which is like the IMF it was applied to, but in the complex plane and

out of phase with the original signal by π/2. From this, the instantaneous amplitude

and frequency can be calculated for each IMF.

More formally, the Hilbert transform Hilb(t) of a function S(t) of the continuous

variable t is defined as:

Hilb(t) =1

πP

∫ ∞−∞

S(η)

η − tdη, (2.30)

where P is the Cauchy Principal Value integral. Huang et al. [1] goes on to say that S(t)

and Hilb(t) form a complex conjugate pair, so the analytic signal Z(t) can be expressed

as:

Z(t) = S(t) + iHilb(t) = A(t)eiϕ(t) (2.31)

where i =√−1 with instantaneous phase ϕ and amplitude A(t) given by:

• Instantaneous Amplitude:

A(t) =

√Hilb2(t) + S2(t). (2.32)

• Instantaneous Phase:

ϕ(t) = arctan

(Hilb(t)

S(t)

). (2.33)

• The Instantaneous frequency f(t) is found using:

ϕ(t) = ω(t) = 2πf(t), (2.34)

or, as described by Deering and Kaiser [64] as:

ω(t) =d

dtarctan

Hilb(t)

S(t)=S(t) Hilb′(t)−Hilb(t)S′(t)

S2(t) + Hilb2(t). (2.35)


The Discrete Hilbert Transform (DHT) Hilb(k) of a function S(k) may be written in

the form:

Hilb(k) =

N−1∑m=0

h(k −m)S(m), (2.36)

where N is even and

h(k) =2

Nsin2(

πk

2) cot(

πk

N). (2.37)

As in Deering and Kaiser [64], a centered difference approximation was used on the

derivatives in Equation (2.35):

f(k) =S(k)(Hilb(k + 1)−Hilb(k − 1))−Hilb(k)(S(k + 1)− S(k − 1))

2ts(S2(k) + Hilb2(k)), (2.38)

where ts is the sampling time.

Finally, the frequency can be corrected by f(k) = fs2π arcsin( f(k)

fs), where fs is the

sampling rate. Therefore, by using the DHT on a signal S(t), a meaningful instanta-

neous magnitude, frequency and phase can be obtained. Figure 2.11 shows the result of

this Hilbert transform applied to the IMFs given in Figure 2.4. This gives a point by

point (instantaneous) amplitude and frequency for each part of the signal.

Returning for a moment to the simple example made by the superposition of sinusoids

described in Section 2.2.1. The time vs amplitude plot in Figure 2.11 clearly identifies

two distinct amplitudes at 3 and 1 arb. units. The time vs frequency plot in Figure

2.11 clearly shows that the dashed curve which represents the components of amplitude

1 arb. unit in the graph above clearly undergo a frequency change at the half way point,

changing from 15 to 25 Hz while the solid line of amplitude three in the graph above

stays at a constant frequency for the duration. Note that the edge effects are due to the

interpretation of the beginning and ends of the data as a local minima or maxima, these

edge effects are much less pronounced when there are multiple frequencies and only the

highest ones are of interest, the effect can also be mitigated with some prior or post

history which may be used as a lead in or out. The speed at which the HHT is able to

determine changes in frequency is the main advantage of using this method over other

techniques like the Fourier transform which lacks such fine temporal resolution, for a


comparison between the two methods see Donnelly [32].

The HHT hinges on the ability of the EMD process to decompose a signal into

separate components based on their frequencies, however the EMD process has been

found to have difficulties when multiple frequencies are close together which can result

in IMFs with mixed frequency and amplitude information. This phenomenon arises

because the two sinusoids superimposed on one another may be interpreted equivalently

as either the sum of two sinusoids or a single amplitude modulated sinusoid, this is

known as mode mixing and is the focus of much research, most notably by Deering and

Kaiser [64] and Rilling and Flandrin [65].

0 0.2 0.4 0.6 0.8 10.5

1

1.5

2

2.5

3

3.5

4

Time

Am

plitu

de

Amplitude Vs. Time

IMF1

IMF2

0 0.2 0.4 0.6 0.8 10

5

10

15

20

25

30

Time

F (

Hz)

Frequency Vs. Time

IMF1

IMF2

Figure 2.11: Instantaneous amplitude (left) and frequency (right) given by a Hilberttransform of the IMFs displayed in Figure 2.4.

2.3 Comparison and Evaluation Methods

There are many ways to compare the accuracy of volatility estimates. A few evaluation

measures that are used throughout this thesis are described by Equations (2.39)−(2.42):

• Mean Absolute Error (MAE):

MAE =1

N

N∑t=1

|σ2t − σ2

t |. (2.39)


• Mean Error (ME):

ME =1

N

N∑t=1

(σ2t − σ2

t). (2.40)

• Mean Square Error (MSE):

MSE =1

N

N∑t=1

(σ2t − σ2

t)2. (2.41)

• Mean Absolute Percent Error (MAPE):

MAPE =100

N

N∑t=1

|σ2t − σ2

t|σ2t

. (2.42)

Where σ2t is the ‘true’ volatility and σ2

t is the estimate under comparison.

Another useful test to compare the distribution of the return estimate to that of the

‘true’ volatility is the two-sample Kolmogorov-Smirnov (K-S) test which is a measure of

the maximum distance between the two cumulative distributions, for more on the K-S

test see Massey [66].

Measuring and comparing volatility estimates isn’t always straightforward. The prob-

lems of computing an accurate volatility estimate are compounded by the fact that

volatility is a latent or unobservable variable. In reality, the ‘true’ volatility is only

known during simulated studies, while simulated studies have provided many insights

into the nature of how microstructure noise affects high frequency volatility estimates,

the advantage of knowing the model volatility is obviously lost when real financial data

is considered, thus making any estimates of real market volatility difficult to accurately

assess.

When real market data is considered the advantage of being able to compare against

this ideal volatility is lost, as is the case in Chapters 3 & 5. Luckily though, there are

some workarounds. In Chapter 3 the problem of not knowing the true daily volatility was


overcome by using the RV which is theoretically much more accurate and has the advan-

tage of orders of magnitude more data. However, due to microstructure effects, one can

not simply increase the sampling frequency to yield better estimates ad infinitum. Thus,

another problem must be overcome when comparing between different high frequency

volatility estimates. A novel solution to this problem based on a virtual options trading

market was proposed by Engle et al. [67] and explored in the context of high frequency

financial forecasts by Bandi et al. [68]. This is the approach that has been adopted in

Chapter 5 for the evaluation of our proposed volatility estimates on high frequency FX

data obtained from the SIRCA database. In this approach, competing volatility esti-

mates are used to give one step ahead forecasts, these are then used to price short term

straddle options which are bought and sold among virtual traders, each with their own

short term forecast of future volatility. See Chapter 5 for more details on this procedure.

Chapter 3

Low Frequency Volatility

Estimation & the HHT

This chapter introduces a method to extract volatility from a time series of daily index

data using the HHT. The proposed method is suitable for non-stationary processes with

volatility changing over time, like the behaviour observed in most financial time series.

Firstly, a proxy to the log returns is constructed using the HHT process and a statistical

measure of fit. The second step is to calculate the variance of the wave like components

that make up the proxy. Volatility estimates produced by this method are then compared

against the EWMA and GARCH(1,1) procedures on NASDAQ and All Ords Index data

covering a ten year period.

3.1 Volatility Estimation using the HHT

This section explains in detail the procedure used to estimate volatility from a time

series using the HHT.

3.1.1 Generating a Proxy of the Returns

The first step towards using the HHT to extract a measure of volatility is to generate a

proxy to the log return series of S(t) using IMFs. Once a proxy is generated which is

44

Chapter 3. Low Frequency Volatility 45

statistically similar to the log returns the simple sinusoidal like nature of the IMFs can

be exploited to extract the volatility.

The proxy to the returns implemented in this thesis is based on fitting the distribution

of the log returns to the distribution of the proxy by the implementation of a cutoff

frequency and a statistical measure of fit. A more detailed explanation of the procedure

follows:

• Apply the HHT process on the log of the price data to obtain the IMFs Ψj(t) and

the instantaneous amplitudes and frequencies Aj(t) & fj(t), where j is the IMF

number. Recall that the log returns on the frictionless price under this notation

are given by:

r(t, h) = p(t)− p(t− h), (3.1)

where h is some time increment, considered to be one day in the context of this

chapter.

• We wish to construct a proxy to these log returns which is easy to analyse and has

a similar volatility. Our proxy to the returns P r(t) is of the form:

P r(t) = p(t)−M(t). (3.2)

where M(t) represents a local mean of the log price at time t. Conceptually it

helps to view the log returns as a relative difference between consecutive prices

while the proxy can be viewed as a measure of short term fluctuations away from

a “local mean”. Although they are different measures they can behave similarly

when an appropriate choice is made for the local mean. A simple measure for the

local mean which could be defined over a variable time scale is given by the residues

of Equation 2.28. However, one drawback with this approach is that filtering by

whole IMFs at a time acts as a rather coarse filter. Thus, a finer filter which is

able to take fractions of an IMF would allow for grater flexibility. Creating a local

mean that can use partial IMFs was achieved by implementing a cutoff frequency

which is chosen in such a way as to which maximise a measure of fit between the


proxy and the log returns. The statistical measure of fit used in this chapter was

the K statistic of a two sample Kolmogorov-Smirnov (K-S) test.

• The log price can be written in terms of high frequency fluctuations and low

frequency long terms trends as::

p(t) =n∑j=1

Ψj(t) + Ωn(t) =m∑j=1

Ψj(t)︸︷︷︸RapidF luctuations

+n∑

j=m+1

Ψj(t) + Ωn(t)︸︷︷︸LongTermTrend

, (3.3)

where j is the IMF number and m is an integer smaller than n representing the

divide between high and low frequencies. Similarly a specific frequency may be

viewed as a cutoff between high and low frequency, i.e.:

p(t) =n∑j=1

H(fj(t)− F )Ψj(t) +n∑j=1

(1−H (fj(t)− F )) Ψj(t) + Ωn(t), (3.4)

where F is a frequency cutoff separating high frequency rapid fluctuations from

low frequency long term trends and H is the unit step function defined below:

H(x) =

0, if x < 0.

1, otherwise.

(3.5)

• Applying the idea that low frequency IMFs can express a local mean allows Equa-

tion (3.2) to be rewritten in terms of the original signal minus the low frequency

components, i.e.

P r(t) =n∑j=1

Ψj(t)−M(t), (3.6)

where

M(t) =

n∑j=1

H(1− (fj(t)− F ))Ψj(t). (3.7)

It follows that a convenient approximation to the proxy is given simply by the

IMFs containing only the high frequency components, i.e.:

P r(t) =n∑j=1

H(fj(t)− F )Ψj(t). (3.8)


The frequency cutoff F is the only parameter needed for this model, it may be searched

for efficiently using one of many gradient descent methods that are capable of dealing

with noise however for this thesis a simple grid search in the vicinity of the average

frequency of the fourth IMF was sufficient for a proof of concept. Figure 3.1 shows a

typical search for the optimal cutoff frequency using the K-S test. Note how there is a

clear optimal frequency which minimises the K statistic of a K-S test.

If too many IMFs are included in the proxy, then the volatility is over estimated as

the “local mean” covers too wide a swath of time. Conversely if too few IMFs are chosen

for the proxy then the volatility is under estimated as some short term fluctuations are

interpreted as long term trends and disregarded because of an overly tight interpreta-

tion of the local mean. Currently K-S fitting is used to decide on a suitable frequency

cutoff and hence a frequency value for the local mean, however there are many possible

ways to construct a proxy to the returns using the information given by the HHT process.

Figure 3.2 demonstrates how closely the proxy actually does behave like the returns.

Note how visually similar the returns and the proxy is in the upper two plots of Figure

3.2 and how closely the distribution matches as shown by the Quantile-Quantile (QQ)

plot in the lower left and Cumulative Distribution Function (CDF) plot in the lower

right. The proxy selection process outlined in this chapter is merely one possibility and

is not purported to be optimal. All of the approaches to fitting the proxy explored in the

course of the research for this chapter share one common feature, they are all trying to

optimise some fitting criteria between a selection of IMFs and the return series. While

the procedure outlined above happens to optimise the fit between the distributions of the

returns and the proxy to the returns, others could be made to optimise other measures

of fit such as the mean error, mean absolute error, squared returns etc. Indeed some of

the methods could be entirely parameter free which may save on computation time.


0 5 10 15 20 250.02

0.04

0.06

0.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22Proxy to Returns KS Fitting

Frequency Cutoff

K V

alue

s

Figure 3.1: Frequency vs K statistic of a K-S test during a typical search for a returnproxy.

3.1.2 Extracting Volatility from the Returns Proxy

In this section the wave like nature of the proxy is exploited to yield a simple estimate

of the volatility. The Hilbert spectral representation of the proxy to the log returns is

given by:

P r(t) = Re

n∑j=1

H(fj(t)− F )Aj(t)ei(ϕj(t))

. (3.9)

where i =√−1. At this point a random phase is introduced in order to extract the

volatility of the system at a point. This is akin to considering each IMF at each point as

just one possibility within a ensemble consisting of waves with similar amplitudes and

frequency but differing phases. A similar approach was taken by Wen and Gu [48] in

order to simulate earthquake data. The uniformly distributed random phase is given by

φ, so we have:

P r(t) = Re

n∑j=1

H(fj(t)− F )Aj(t)ei(ϕj(t)+φ)

. (3.10)

The instantaneous variance of P r(t) is given by:

HHT σ2(t) =1

∆tEφ

((P r(t)− Eφ

(P r(t)

))2), (3.11)


2000 2001 2002 2003 2004 2005−0.1

−0.05

0

0.05

0.1Log Returns for NASDAQ Index

Date

Log

Ret

urn

2000 2002 2004 2000 2002 2004−0.1

−0.05

0

0.05

0.1Proxy Returns for NASDAQ Index

Date

Log

Ret

urn

−0.2 −0.1 0 0.1 0.2−0.15

−0.1

−0.05

0

0.05

0.1

0.15

0.2

X Quantiles

Y Q

uant

iles

QQ plot of Proxy against Log Returns

−0.2 −0.1 0 0.1 0.20

0.2

0.4

0.6

0.8

1

x

F(x

)

CDF plots of Proxy and Log Returns

Log ReturnsProxy

Figure 3.2: Actual returns (top left). Proxy to the log returns P (t) (top right). QQplot of the the proxy P (t) against the actual returns (bottom left). CDF plot of the

proxy P (t) against the actual returns (bottom right).

where ∆t is the time between discrete samples. Since Eφ(P r(t)

)= 0, we have:

HHT σ2(t) =1

∆tEφ

Re n∑j=1

H(fj(t)− F )Aj(t)ei(ϕj(t)+φ)

2=

1

∆tEφ

n∑j=1

H(fj(t)− F )Aj(t) cos (ϕj(t) + φ)

2 (3.12)

Note that the IMFs have been described as “orthogonal for all practical purposes” by

Huang Huang et al. [1] due to the frequency sorting nature of the EMD process, so any


cross products in the above equation are zero, resulting in the simpler expression:

HHT σ2(t) =1

∆tEφ

n∑j=1

H(fj(t)− F )A2j (t) cos2 (ϕj(t) + φ)

. (3.13)

Recall that the nth central moment is defined by E[(X−E(X))n] =∫ ba (x−µ)nf(x)dx,

where f(x) is the probability density function of the random variable x. For the uniform

random variable φ, f(φ) = 1b−a and since the function is periodic, we can let a = 0 &

b = 2π. Now, taking the expectation of the function given by Equation (3.13), with

respect to the random phase gives:

HHT σ2(t) =1

∆t

∫ 2π

0

n∑j=1

H(fj(t)− F )A2j (t) cos2 (ϕj(t) + φ)

1

2πdφ

=

1

∆t

n∑j=1

H(fj(t)− F )A2j (t)

(ϕj(t) + φ

2+

sin (2(ϕj(t) + φ))

4

)1

2π

2π

0

=1

2∆t

n∑j=1

H(fj(t)− F )A2j (t), (3.14)

which is our local volatility estimate of the log returns.

3.2 Results and Discussion

3.2.1 Data and Testing Methodology

The intraday data for the NASDAQ and All Ords indices was obtained from the SIRCA

database, covering the period from 1st January 2000 to 31st December 2009. The All

Ords Index consisted of 2,958,287 lines of high frequency data and the NASDAQ Index

contained 13,333,424 lines. The text files containing this data were 168 Mb and 745 Mb

respectively. To put the size of these arrays into context, Microsoft Excel is known to

crash and run out of resources after 65,536 lines. When text files grow to this length,

efficient data handling and cleaning the data becomes a necessity.


Matlab was used for much of the analysis carried out in this thesis and this program

requires that arrays of data are stored into contiguous RAM memory. Unfortunately

though, even a computer with 16 Gb of RAM memory will usually only have 512 Mb

to 1 Gb of contiguous memory at the best of times, depending on hardware. Worse

still, once the computer has been operational for a short while the memory will begin to

fragment and the size of the largest available contiguous memory reduces quickly. Thus,

care must be taken when handling data that stretches into the millions of lines and it

must be cleaned section by section and split into more manageable formats, i.e. date

strings formatted into date numbers which use less memory, etc.

Once considerations were made for the size of the data, it was cleaned by removing

simultaneous quotations and extraneous data that occasionally filled a cell where price

or date information should be. The data was also sorted into periods of high frequency

and low frequency, such delineations also marked the boundaries of when the market

was considered opened or closed. From these high frequency data sets the daily realized

volatility was extracted and the last value from each trading day was then used as the

daily closing price. Both ten year sets were then split in half and each half analysed

separately because of different market conditions in the first half of the decade as com-

pared to the the latter half, namely the Global Financial Crisis (GFC).

The volatility estimates obtained via the HHT volatility estimate as well as the

EWMA and GARCH(1,1) models are compared against the more accurate but also

more data intensive realized volatility method, allowing error measurements to be ob-

tained and comparisons drawn. The GARCH parameter estimation was done using the

garchfit procedure in Matlab 2010b for each five year period. For the EWMA method,

the exponential decay rate of 0.94 was used because it is popularly referred to in the

literature as the value J.P. Morgan use in their RiskMetrics EWMA volatility model,

see Bauwens et al. [69].


Analysis is split up into two parts, firstly the NASDAQ Index covering the period 1st

Jan 2000 - 31st Dec 2004 is analysed qualitatively via the graphical results and discus-

sion. Secondly, Section 3.2.4 provides some numerical error analysis for the four effective

data sets mentioned above.

The comparison methods used to evaluate the validity of the HHT procedure as a

volatility estimate are given by Equations (2.39)-(2.42). Note that the realized volatil-

ity estimate, obtained using intraday log returns, will be used as the true volatility θ2t ,

while the volatility estimate under comparison θt2, uses only daily returns. The stan-

dard K-S test was also used to determine how close the volatility distributions are to

one another. It is important to note that all error measures are only as reliable as the

realized volatility, which is used as the basis for comparison. The RV is often regarded

as a very accurate measure to use when intraday data is available, in this case Equations

(2.10) & (2.14) were used as the RV estimate with 15 minute returns with over night

effects taken into account.

3.2.2 Graphical Volatility Analysis

One apparent feature of the HHT volatility estimate is its high degree of temporal

resolution compared to the other methods examined, i.e. the proposed method adapts

quickly to large changes in volatility. As Figure 3.3 clearly indicates the HHT based

volatility estimate is capturing much of the the same volatility behaviour as the realized

volatility method, all be it with less noise and working with orders of magnitude less

data. This is most evident around mid to late 2000 and early 2001 in Figure 3.3, where

the realized volatility as well as the HHT measure both drop sharply, while the GARCH

and EWMA methods are observed to decay exponentially towards the new volatility

level, resulting in inflated GARCH and EWMA volatility estimates for this period.

The EWMA procedure exhibits similar artefacts to GARCH when the decay factor is

set at the commonly used value of λ = 0.94. This slowly decaying over estimation

can be largely reduced by placing more weight on the most recent returns, however


2000 2001 2002 2003 2004 20050

0.2

0.4

0.6

0.8

1

Date

Vol

atili

ty

HHT vs Realized: NASDAQ 1st Jan 2000 − 31st Dec 2004

RealizedHHT

2000 2001 2002 2003 2004 20050

0.2

0.4

0.6

0.8

1

Date

Vol

atili

ty

GARCH vs Realized: NASDAQ 1st Jan 2000 − 31st Dec 2004

RealizedGARCH(1,1)

2000 2001 2002 2003 2004 20050

0.2

0.4

0.6

0.8

1

Date

Vol

atili

ty

EWMA vs Realized: NASDAQ 1st Jan 2000 − 31st Dec 2004

RealizedEWMA

Figure 3.3: HHT based and realized volatility estimates for the NASDAQ data set,1st Jan 2000 - 31st Dec 2004 (top). Corresponding GARCH(1,1) and realized volatilityestimates (middle). Corresponding EWMA and realized volatility estimates (bottom).

this increase in temporal resolution comes at the cost of potentially yielding a noisier

volatility estimate. This relationship between noise and temporal resolution is examined

in more detail in Section 4.2.


3.2.3 Further Analysis of the HHT Procedure

The procedure used to generate the returns proxy also has the added benefit of pro-

ducing an interesting local mean as a by-product, this can be seen in Figure 3.4. This

local mean is made up of the information with a frequency too low to be included in the

proxy. Since the EMD procedure was applied to the log price, the exponential of the

running mean eM(t) can be taken to allow for comparison with the original price series,

such a comparison is given in Figure 3.4. While the high frequency part of the signal

makes up our proxy which is needed for our volatility estimates, the lower frequency

elements of the signal may also be of interest to some analysts as they represents a

centered mean that is scalable with time and which differs from that used in regular

Moving Average Convergence/Divergence (MACD) style analysis. Furthermore if the

local mean is defined in terms of the IMF number alone, then MACD style analysis could

be carried out using the EMD procedure. This would generate means that capture more

of inherent dynamics of the underlying system, because those means are made up of

intrinsic modes. Another advantage of the HHT as a local mean estimator is the fact

that it doesn’t appear to suffer from the plateauing effects similar to those described in

Section 2.1.1.

The proposed K-S based procedure for calculating the returns proxy usually resulted

in an optimal cutoff frequency somewhere between three and four IMFs. These IMFs

capture enough stochastic behaviour of the system to yield a reasonable estimate of

the local variance. Figure 3.5 is an alternative representation of the NASDAQ data

displayed in 3.4. Figure 3.5 shows the distinction between the components considered

high frequency (top) and those considered low frequency (bottom). The high frequency

component is effectively the proxy, converted into units of $ and is equivalent to the

difference between the two lines in Figure 3.4. The low frequency component is identical

to the dashed line if Figure 3.4, and forms a measure of the local mean to the original

index price. In this context the HHT is acting like a high pass filter and those combined

high frequency components are effectively the difference from the longer term trend at

that point to the current value thus they can be thought of as an alternative to the


2000 2001 20021000

1500

2000

2500

3000

3500

4000

4500

5000

Date

NASDAQ Index Price and HHT Based Mean

Inde

x P

rice

($)

Index PriceHHT Mean

Figure 3.4: NASDAQ Index price for 1st Jan 2000 - 31st Dec 2001 and a measure ofthe local average eM(t).

return series.

The scaling of the highest frequency IMF components by the data at that point was

first done by Huang et al. [47], however they then rectified this signal and used that

as their volatility or ‘variability’ measure described by Equation (2.29). Applying this

method to the NASDAQ data produces Figure 3.6. The problem with this approach is

that it is not only noisy but at each point when the combined IMF components, which

are visible in the upper plot of Figure 3.5, cross the x axis, the volatility measure drops

to zero. This property is undesirable in a volatility estimator since the variance hasn’t

become any smaller at this point, the random walk was just crossing a local average.

Thus its use is limited to that of a volatility indicator as it would be difficult to use as

a practical measure.

The key difference between Huang’s variability measure and the variance measure

proposed in this chapter is the usage of the amplitude information given by the Hilbert

transform on the IMFs. It is these amplitudes that are of value when measuring volatil-

ity since they capture the behaviour of the system without going to zero when an IMF


2000 2001 2002−400

−200

0

200

400High Frequency Components: NASDAQ Index

HF

com

pone

nt (

$)

2000 2001 20021000

2000

3000

4000

5000

Date

Low Frequency Components: NASDAQ Index

LF c

ompo

nent

($)

Figure 3.5: The high frequency components which form the proxy (top). The remain-ing components which are a form of local mean (bottom).

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 20100

0.05

0.1

0.15

0.2Huang Variability measure

Date

Var

iabi

lity

Figure 3.6: The ‘variability’ measure proposed by Huang applied to the NASDAQdata set.

crosses a local mean and they form the backbone of the proposed HHT volatility esti-

mator. Figure 3.7 depicts the amplitudes of the first 5 IMFs of the NASDAQ data set,

scaled by the data. It can be seen from the figure that the lower frequencies are more

prone to edge effects, like the one visible towards the end of the fifth IMF. These edge

effects are due to the splining procedure used in the EMD process not having an extrema

point near the edge of the now filtered data and could be remedied by the inclusion of

ex post data. Furthermore, these edge effects are rarely of practical importance because


2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 20100

0.05

0.1HHT Amp Scaled by Data

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 20100

0.05

0.1

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 20100

0.05

0.1

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 20100

0.05

0.1

2000 2001 2002 2003 2004 2005 2006 2007 2008 2009 20100

0.1

0.2

Figure 3.7: Amplitudes of the first 5 IMFs scaled by the data for 10 years of theNASDAQ Index data set.

they have a tendency to effect the lower frequency IMF components which are discarded

in the proxy fitting procedure.

The HHT also yields the frequencies of the IMF components which allows us to look

for any underlying periodic dynamics like weekly, fortnightly and monthly trends. Figure

3.8 displays the frequencies extracted from the same five IMFs covering two years worth

of NASDAQ Index data. The mean frequency in cycles per year of the IMFs in Figure

3.8 were found to be 83.0, 43.5, 24.6, 14.7, 8.7 and 4.3 cycles per year respectively.

The EMD procedure was applied to two years of the NASDAQ Index data from 1st

Jan 2000 - 31st Dec 2001 resulting in nine IMFs and one residue, the first six IMFs are

given in Figure 3.9. From the figure it is evident that the EMD procedure is working

as intended for the financial data, separating it into intrinsic modes with periods of

increasing length as the IMF number increases. Note that the period is increasing by

just under a factor of 2, deviations from this trend or the appearance of a mode with an

unusually high amplitude may indicate the presence of seasonality in the modes. The

fifth IMF in Figure 3.9 may show some signs of seasonality as it has a comparatively


5

148HHT Frequency for NASDAQ Jan 2000 − Jan 2002

4

105

3

81

2

49

Fre

quen

cy

1

25

2000 2001 20021

10

Date

Figure 3.8: Frequency information given by the Hilbert transform of the first 6 IMFsfor two years of the NASDAQ Index data set.

high amplitude and a fairly steady quarterly period. Research carried out by Wu and

Huang [36] has put a quantitative measure on the statistical significant of IMFs gener-

ated from Gaussian white noise so that effects attributable to seasonal trends may be

distinguishable from stochastic noise with some confidence. Such research could give an

indication of the statistical significance of any apparent seasonality in financial systems,

such the apparent trends visible in Figure 3.9, however the application of such a measure

is out of the scope of this thesis.

The HHT volatility estimate also has a tendency to capture the distribution of the

realized volatility using only daily data more closely than the alternatives tested. As the

top Quartile-Quartile plots and cumulative frequency diagrams in Figure 3.10 demon-

strate, the distribution of the HHT volatility matches that of the realized more closely

than the EWMA and GARCH procedures. All three estimators shown are light or skinny

tailed when compared to the distribution of the realized volatility, this is expected as


−0.07

0.07 Mean Period=4.5 Days

IMFs for NASDAQ Jan 2000 − Jan 2002

−0.05


−0.06


−0.06


−0.06


2000 2001 2002−0.11


IMF

Date

Figure 3.9: First six IMFs for the log of the NASDAQ Index from 1st Jan 2000 -31st Dec 2001. The average period of each IMF is given in calendar days as opposed to

trading days.

the lower frequency methods under comparison appear as less noisy than the RV method.

This apparent noise can be deceptive, the RV procedure is by all accounts more accu-

rate than the lower frequency procedures however this high accuracy on one time scale

can appear as noise on another. Therefore, the term ‘noise’ should be used with care

and one must be careful not to confuse apparent noise with a lack of smoothing. Note

that due to issues of readability, the log of the volatility was taken for Figure 3.10. The

cumulative frequency diagrams given in Figure 3.11 also indicate how the GARCH and

EWMA estimates deviate from the realized distribution and the close fit of the HHT

procedure for this data set. Note that the EWMA procedure can be tuned to capture

the distribution more accurately by decreasing λ which gives more weight to recent data

and less weight to past volatility estimates.


−6 −5 −4 −3 −2 −1 0 1−6

−4

−2

0

2

X Quantiles

Y Q

uant

iles

Log Volatility QQ plot for HHT vs Realized

−4.5 −4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0 0.5−6

−4

−2

0

2

X Quantiles

Y Q

uant

iles

Log Volatility QQ plot for EWMA vs Realized

−4 −3.5 −3 −2.5 −2 −1.5 −1 −0.5 0−6

−4

−2

0

2

X Quantiles

Y Q

uant

iles

Log Volatility QQ plot for GARCH vs Realized

Figure 3.10: QQ plots of the volatility distribution for the HHT method (top),EWMA (middle) and GARCH (bottom) for the NASDAQ Index covering the period

1st Jan 2000 - 31st Dec 2004.


0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

x

F(x

)

CDF plots of HHT and Realized

HHTRealized

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

x

F(x

)

CDF plots of EWMA and Realized

EWMA λ=0.94Realized

0 0.1 0.2 0.3 0.4 0.5 0.6 0.7 0.8 0.9 10

0.2

0.4

0.6

0.8

1

x

F(x

)

CDF plots of GARCH and Realized

GARCH(1,1)Realized

Figure 3.11: CDF plots of volatility distribution for the HHT method (top), EWMA(middle) and GARCH (bottom) for the NASDAQ Index covering 1st Jan 2000 - 31st

Dec 2004.


3.2.4 Numerical Results

The volatility estimators under comparison were evaluated using a number of error mea-

sures which were described in Chapter 2. Since each measure is itself imperfect, each

measure only indicates the performance of a volatility estimator. The error measures

for several volatility measures are summarised in the Tables 3.1-3.4. For the data con-

sidered in Tables 3.1 -3.4 it appears as though the HHT based volatility estimates are

competitive with the EWMA and GARCH(1,1) volatility estimators. The 95% con-

fidence intervals are also given for most error measures in the tables. Note that the

method labelled Fast-HHT is a fully parameter free version of the HHT volatility esti-

mator which is described in Section 4.1. Its results have been included here for later

comparison.

The high accuracy of the RV method at one scale appears as noise against the more

slowly varying methods which use only daily closing prices. This apparent noise in the

more accurate measure which was used as the basis for comparison has led to high

MAPE errors and is a source of noise in the ME, MSE and MAE measures.

The numerical evidence provided by the K-S fit results in Tables 3.2, 3.3 support a

previous claim that for some data sets, the HHT procedure was capable of fitting the

realized volatility distribution more accurately than the GARCH(1,1) or popularly used

EWMA procedure in two of the four data sets that were examined.

3.3 Chapter Conclusion

This chapter has introduced a new and novel method for calculating local historical

volatility which is highly adaptable and likely to seed a variety of new HHT based

volatility measures. Previous attempts at using the HHT for a volatility estimator have

resulted in either volatility indicators like that developed by Huang et al. [47] which are


Table3.1:

Err

ors

for

vari

ous

vola

tili

tyes

tim

ato

rsfo

rth

eA

llO

rds

Ind

exco

veri

ng

the

per

iod

1st

Jan

2000

-31s

tD

ec2004

Met

hod

MA

EM

EM

SE

MA

PE

K-S

Fit

EW

MAλ

=0.

94.0

107±

.001

6.0

054±

.0016

9.0

6E

-04±

4.7

5E

-04

64.6981±

5.0042

0.0906

GA

RC

H.0105±

.0015

.0053±

.0016

8.56E-04±

4.38E-04

68.9

140±

5.2

138

0.1

950

HH

T.0

122±

.001

6.0

078±

.0017

9.8

7E

-04±

4.9

6E

-04

69.6

290±

4.1

322

0.1

904

Fas

t-H

HT

*.0

121±

.001

4.0035±

.0015

7.48E-04±

2.65E-04

86.1

083±

7.3

922

0.0432

Table3.2:

Err

ors

for

vari

ous

vola

tili

tyes

tim

ato

rsfo

rth

eN

AS

DA

QIn

dex

cove

rin

gth

e1st

Jan

2000

-31st

Dec

2004

Met

hod

MA

EM

EM

SE

MA

PE

K-S

Fit

EW

MAλ

=0.

94.1

053±

.0129

-.0034±

.0141

.0644±

.0469

66.5

741±

4.9

277

0.1

185

GA

RC

H.1

058±

.0129

-.0042±

.0142

.0650±

.0478

68.9

055±

5.1

517

0.1

257

HH

T.0978±

.0125

.0319±

.0135

.0601±

.0465

58.1922±

3.9149

0.0454

Fas

t-H

HT

*.1

198±

.0128

-.0265±

.0144

.0675±

.0335

75.7

504±

5.2

459

0.0

515

Table3.3:

Err

ors

for

vari

ous

vola

tili

tyes

tim

ato

rsfo

rth

eA

llO

rds

Ind

exco

veri

ng

the

1st

Jan

2005

-31s

tD

ec2009

Met

hod

MA

EM

EM

SE

MA

PE

K-S

Fit

EW

MAλ

=0.

94.0212±

.0025

-.0053±

.0027

2.41E-03±

8.77E-04

78.7745±

5.2724

0.1

449

GA

RC

H.0

220±

.002

6-.

0069±

.0028

2.6

5E

-03±

9.1

6E

-04

87.6

885±

5.8

322

0.2

070

HH

T.0

267±

.002

9-.0032±

.0033

3.5

1E

-03±

1.0

9E

-03

98.0

982±

8.2

663

0.0657

Fas

t-H

HT

*.0

287±

.003

9-.

0097±

.0042

5.7

0E

-03±

2.2

8E

-03

93.4

691±

7.1

633

0.1

250

Table3.4:

Err

ors

for

vari

ous

vola

tili

tyes

tim

ato

rsfo

rth

eN

AS

DA

QIn

dex

cove

rin

gth

e1st

Jan

2005

-31st

Dec

2009

Met

hod

MA

EM

EM

SE

MA

PE

K-S

Fit

EW

MAλ

=0.

94.0

357±

.004

4.0012±

.0049

7.6

6E

-03±

3.5

0E

-03

67.0

557±

8.3

383

0.0948

GA

RC

H.0337±

.0043

.0030±

.0047

7.15E-03±

3.52E-03

67.8

501±

7.8

068

0.1

695

HH

T.0

391±

.004

9.0

215±

.0052

9.1

8E

-03±

3.9

7E

-03

64.6630±

6.2877

0.1

645

Fas

t-H

HT

*.0

424±

.005

9-.

0063±

.0063

1.2

9E

-02±

6.1

9E

-03

75.3

533±

7.1

483

0.0409


difficult to interpret as they are not in terms of variance or in the case of Wen and Gu

[48] were not fully explored.

For the first time the HHT has been used to give volatility estimates in terms of

the local variance which is easily understood and, as the the tables in Section 3.2.4

show, the degree of accuracy is favorably comparable to the popularly used EWMA and

GARCH(1,1) methods.

As Section 3.2.2 shows, for the data set considered, the HHT volatility estimate does

not appear to suffer from the same kinds of artefacts which can render GARCH and

EWMA unreliable after sharp increases or decreases in volatility. The reason for this

apparent advantage of the HHT volatility estimate is the high temporal resolution of

the HHT procedure which underpins it.

From the numerical evidence provided here there is no clear winner in terms of

overall performance. All three methods are fitting the RV to a similar degree of success.

What makes an assessment of performance difficult is the absence of a “true” account of

volatility, with only the RV standing in as a proxy. In comparison to methods which rely

on daily data, the RV volatility measure can be perceived as noisy, this brings one back

to a question posed in the introduction; which changes in volatility can be described as

noise and which can be attributed to legitimate changes in variance? There is no one

answer, what is considered the best volatility estimate appears to be dependent on the

the intended use of the estimator. This apparent noise of the RV method has led to

relatively large standard errors and consequently quite broad 95% confidence intervals.

Thus, care must be exercised when judging the performance of volatility estimators that

have access to only daily data against the those measure which uses intraday data. This

is again a matter of scale, while the former estimators might be suitable for weekly

forecasts, the latter is better suited to daily forecasts

.


Note that while there are many different possible implementations of the GARCH,

EWMA or RV procedures, there is also a growing number of ways to implement the

HHT process. One such variant of the HHT process proposed by Hu [70] forces IMFs

into being completely orthogonal by a process known as the Orthogonal Hilbert-Huang

Transform (OHHT). This OHHT process was implemented and tested over the course

of this chapter but results varied so little from those obtained by the standard HHT

procedure that they were omitted from the results and further discussion. This close

resemblance between the HHT and the OHHT are not surprising since, as previously

described the IMFs may be considered orthogonal for all practical purposes.

The volatility estimator proposed in this chapter also has much scope for development

as it is based on the highly flexible HHT procedure. Applications of this method also

include the analysis of unevenly spaced time series data as the HHT has no requirement

for evenly spaced data. As briefly explored in Section 3.2.3, the EMD procedure also

gives information which may be used for seasonality analysis and MACD style technical

analysis. Further work has also been done towards making the HHT volatility procedure

completely parameter free for greater simplicity and computational efficiency. The first

such method which was seeded by the work first undertaken in this chapter is denoted

Fast-HHT in Tables 3.1-3.4, this method is parameter free and therefore much faster, it

forms a core component in the analysis which is to follow.

Chapter 4

High Frequency Simulated Data

and Microstructure Noise

In this chapter an improved version of the HHT volatility estimator is proposed and

evaluated. Its simpler and parameter free nature make it computationally more efficient

than the variant proposed in Chapter 3. This faster, simpler approach has enabled a

Monte Carlo study and allowed for further refinements which can even handle market

microstructure noise. In this chapter, two simulated studies are carried out, the first

compares the improved HHT volatility estimator against the EWMA procedure and the

second tests a variant of the HHT procedure in the presence of market microstructure.

4.1 Improved HHT Volatility Estimator

In this section the wave like nature of the IMFs is exploited to yield a simple estimate of

the volatility. Its main advantage over the procedure which was outlined in Chapter 3,

is that it requires no parameter fitting at all, making it substantially faster and able to

be investigated more thoroughly through detailed Monte Carlo simulations. The main

difference is that there is no need to create a proxy to the returns as the procedure is

applied directly to the log returns.

The HHT procedure can be used to give the Hilbert spectral representation of the

66

Chapter 4. HF Simulated Data and Microstructure 67

log returns as:

r(t,∆t) = Re

n∑j=1

Aj(t,∆t)eiϕj(t,∆t)

, (4.1)

where ∆t is the period between returns. Note that IMFs Ψj(t) are actually continuous

because of the splining nature of the EMD algorithm and that discrete returns can be

seen as a subset of these continuous functions. Also, the zero centered nature of the

IMFs implies that all of them satisfy Et (Ψj(t)) = 0, however this does not hold true for

the final residue Ωn(t).

Now the concept a random phase is introduced in order to extract the volatility of

the system at any point. This is akin to considering each IMF at each point as just

one possibility within a virtual ensemble consisting of waves with similar amplitudes

and frequency but differing phases. A similar approach was taken by Wen and Gu [48]

in order to simulate earthquake data, however the focus was on simulating different

volatility levels rather than estimating them. The uniformly distributed random phase

is given by φ, so we have:

r(t,∆t, φ) = Re

n∑j=1

Aj(t,∆t)ei(ϕj(t,∆t)+φ)

, (4.2)

where r(t,∆t, φ) represents a whole ensemble of waves. The instantaneous variance of

r(t,∆t, φ) is given by:

σ2(t,∆t) =1

∆tEφ

((r(t,∆t)− Eφ (r(t,∆t)))2

). (4.3)

Note that when IMFs are constructed using log return data, the IMF volatility drops

off exponentially with the IMF number, so any long term trends contained in the last

residue can be safely ignored for the purposes of this volatility analysis. Now we have:

σ2(t,∆t) =1

∆tEφ

Re n∑j=1

Aj(t,∆t)ei(ϕj(t,∆t)+φ)

2=

1

∆tEφ

n∑j=1

Aj(t,∆t) cos (ϕj(t,∆t) + φ)

2 . (4.4)


Note that the IMFs have been described as “orthogonal for all practical purposes” by

Huang et al. [1] due to the frequency sorting nature of the EMD process, so any cross

products in the above equation are zero, resulting in the simpler expression:

σ2(t,∆t) =1

∆tEφ

n∑j=1

A2j (t,∆t) cos2 (ϕj(t,∆t) + φ)

. (4.5)

Now, taking the expectation with respect to the uniform random phase between 0 and

2π gives:

σ2(t,∆t) =1

∆t

∫ 2π

0

n∑j=1

A2j (t,∆t) cos2 (ϕj(t,∆t) + φ)

1

2πdφ

=

1

2∆t

n∑j=1

A2j (t,∆t), (4.6)

which is our instantaneous and parameter free volatility estimate for the log returns.

Note that in a similar fashion to the way in which sparse and average RV estimates

were described in Chapter 2, the sparse HHT estimator is described as Equation (4.6)

with sample spacing being some integer multiple of the smallest available step size, i.e.

∆t = n∆t and for ease of notation let 1∆t be a positive integer representing the number

of sparse subsamples in one period. The average HHT can then be described by a mean

of sparse estimates with different offsets, i.e. each offset is given by:

offsetHHTσ2(t,∆t, k) =1

2∆t

n∑j=1

A2j (t+ k∆t,∆t). (4.7)

where k = 0, . . . , n − 1. Then the average HHT based estimator is given by a sum of

these sparse estimations as:

AverageHHTσ2(t,∆t) =1

n

n−1∑k=0

offsetHHTσ2(t,∆t, k) (4.8)

The estimate described by Equation (4.6) was used to give the values described as “Fast-

HHT” in Section 3.2.4. The numerical results given in Chapter 3 support the claim that

the algorithm described in this section is as accurate, if not more accurate than the


procedure described in Chapter 3. The main advantage is the the method described by

Equation (4.6) requires no fitting of a proxy to the returns and is therefore completely

parameter free and therefore much faster to run. This advantage of speed has enabled

more specialised versions of the HHT estimator to be developed which are comprised

of multiple estimates such as that described by Equation (4.8) and later the Filtered-

HHT procedure which can deal with market microstructure. It should also be noted

that, while the Fast-HHT procedure is much faster than the proxy dependent variant

described in Chapter 3, it is still substantially more complex, and thus slower than a

simple realized volatility measure.

4.2 Volatility Comparison Test

The results given in Chapter 3, particularly the graphical volatility analysis of Figure

3.3 appear to show that the HHT is capable of maintaining low levels of noise while also

adapting to changes in volatility more rapidly than the popular EWMA and GARCH

volatility methods. In the absence of any test to verify these claims in a straightforward

manner, one has been developed specifically to quantify this kind of behaviour. As

discussed in Section 2.1.2, there is often a compromise that must be made between the

ability of a method to capture short term or long term trends. This test seeks to quantify

the previous statement by examining this tradeoff in more detail.

4.2.1 Test Design

The test consists of two main parts, the calculation of the Signal to Noise Ratio (SNR)

and the response time.

I Calculate the Signal to Noise Ratio

• A Geometric Brownian Motion (GBM) time series of length m is simulated

with constant volatility, i.e. σ2(t) = c for t = 1, . . . ,m.


• Competing instantaneous volatility methods are used to approximate the volatil-

ity over the length of the time series. The estimates are denoted σ2(t) for

t = 1, . . . ,m.

• The SNR is then calculated using:

SNR =σ2(t)

1m

∑mt=1 (|σ2(t)− σ2(t)|)

(4.9)

This should then be repeated many times for each variable parameter within

the estimation method, i.e. λ in the case of EWMA.

II Determine the Response Time

• Secondly, a GBM time series is simulated with a distinct step in volatility mid

way through the data, i.e. σ2(t) = c for t = 1, . . . ,m/2 which then switches to

σ2(t) = c/2 for t = m/2 + 1, . . . ,m. The time it takes for competing volatility

estimate to approach to within a threshold (within ±20% of c/2 was used for

this research) of the new volatility level is then recorded. As before, this step

should be repeated for different values of any parameters in the estimation

method.

III Compare the Signal to Noise Ratio to the Response Time.

• A useful comparison can be made by graphing the SNR vs response time results

across across each variable parameter, this procedure produced Figure 4.3.

Note that the EWMA estimator gives results which can vary over λ and so the

EWMA estimator yields a curve as its properties change with this parameter.

On the other hand, the HHT procedure is parameter free and so the SNR vs

response plot for this method yields only a single point for comparison.

4.2.2 Test Results

Figure 4.1 shows how the EWMA procedure adjusts to the sharp drop in volatility levels

for several values of the parameter λ. The estimator can be seen to approach the new

volatility level more slowly as the decay rate gets closer to one. Figure 4.2 selects one


0 50 100 1500.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Time

Vol

atili

ty

Response Test for the EWMA Procedure

λ=0.1λ=0.86λ=0.98

Figure 4.1: EWMA Response profiles for three values of λ.

of the EWMA estimates, corresponding to λ = 0.67 and compares it against the HHT

volatility estimator. The value of λ = 0.67 was chosen because it has a similar SNR

to the HHT method. The HHT volatility estimate descends towards the new volatility

level faster than the EWMA estimate. Note that the HHT volatility estimate actually

dips before the actual volatility drops, this is because the EMD process which underpins

the HHT procedure uses cubic splines and ex-post data, it is in fact an artefact and does

not indicate a hidden predictive ability.

Importantly, Figure 4.3 shows that this single point produced by the HHT procedure

is below the line formed by considering all of the parameter choices for the EWMA pro-

cedure. This implies that the HHT volatility estimator (4.6) is adapting to new volatility

levels faster, while maintaining a lower SNR than the EWMA procedure under test con-

ditions, which is the claim that this test was designed to investigate.


0 50 100 1500.08

0.1

0.12

0.14

0.16

0.18

0.2

0.22

Time

Vol

atili

ty

Response Test for the EWMA & HHT Methods

HHT VolEWMA λ=0.67Actual

Figure 4.2: Response profiles for HHT estimator & EWMA with λ = 0.67

4.3 The Filtered-HHT Volatility Estimator for Microstruc-

ture Noise

The HHT volatility estimate can also provide consistent estimates even in the presence

of market microstructure noise. This is achieved by effectively filtering off the higher

frequency components which constitute the unwanted noise. The main advantage of

the HHT procedure over any of the RV techniques is that it provides an instantaneous

volatility measure so that finer scale volatility characteristics are revealed. Note that

this method of filtering off excess noise is only suitable for the kind of microstructure

which results in positive anomalous scaling of volatility, such as the kind of noise found

in high frequency FX markets.


2 4 6 8 10 12 14 16 180

10

20

30

40

50

60

70

80

90

100

Signal−Noise Ratio

Res

pons

e T

ime

EWMA and HHT Signal−Noise vs Response Time Comparison

EWMAHHT

Figure 4.3: SNR vs response time for EWMA with 0.1 ≤ λ ≤ 0.98 (curve) and theHHT volatility estimate (circle).

4.3.1 Adapting the HHT Estimator to the High Frequency Realm

The procedure to calculate a consistent high frequency volatility estimator based on the

HHT method is as follows:

1. EMD Step:

Apply the EMD procedure to the observed log price series p, note that the pro-

cedure works equally well when applied to the regular price series. The IMFs are

then passed to the next step. Also, recall that in Chapter 3 the observed price

was given by p, however this was the low frequency case where market frictions

are negligible and the frictionless price is effectively observable. In this chapter

the observed log price which includes market microstructure is denoted as p, with

the frictionless p being unobservable.

2. Filtration Step:

If this is the first iteration of the filtration step, skip it and pass all IMFs through

to the next step. Otherwise, pass on the residue and all but the highest frequency


IMF to the next step. This step is essentially using the EMD process a low pass

filter.

3. Volatility and Averaging Step:

Now, using the IMFs passed on from the previous step, the HHT volatility mea-

sure described by Equation (4.8) is used at various sampling rates. For exam-

ple, in this chapter the simulated time series was sub-sampled at ∆t(1, . . . , 12) =

[0.4, 0.6, 0.8, 1, 2, 3, 4, 5, 6, 7, 8, 9] minute intervals and for convenience, the time

scale for one period is 24 hours, or 1440 minutes so that the daily sample rate

is given by: fs(j) = 1440∆t(j) , j = 1, . . . , 12. The sample rates should be chosen so

that the they capture both high frequency inflationary noise as well as some sparse

samples which are relatively unaffected by market microstructure. Note that the

method proposed above gives the spot volatility at each sample point in time and

for each sample rate, which results in 12 volatility estimates spanning the length

of the simulation for each filtration. However in the comparisons to follow, we

wish to compare this instantaneous or spot volatility to a measure of integrated

volatility and so the mean of the spot volatility should be taken over a length

comparable to that in the integrated volatility procedure (one day). Let this mean

or integrated volatility at sample rate fs(j), day t and filtration step i be denoted

by HHTIV (t, fs(j), i).

4. Iteration Step:

Now, iterate the previous two steps until only the residual of the signal is left.

5. Optimisation Search Step:

• Calculate the mean volatility with respect to time at every sample rate and

filtration step. This mean volatility is given by:

meanHHTIV (fs(j), i) =1

∆t(j)

1/∆t(j)∑t=1

HHTIV (t, fs(j), i). (4.10)

This provides a good indication of overall inflation when compared to means

at other sample rates and filtration steps.


• While one approach would be to simply determine which filtration step yields

the most consistent volatility estimates over multiple sample rates, experience

from Chapter 3 has taught that filtering by whole IMFs can be too coarse a

filter. One alternative to this coarse filter approach is a finer frequency filter,

such as the step function approach taken by chapter 3. An alternative, which

is used during this chapter, is to use a linear combination of under filtered

and over filtered volatility estimates to yield the final result.

More specifically, mean volatility measures at neighboring filtration steps

(i − 1 & i) are linearly combined in such a way as to maximise the flatness

of the combined volatility profile over all sample rates. Note that depar-

tures from flatness in the volatility profile are quantified by the gradient of

meanHHTIV (fs(j), i) with respect to changes in fs. Let the gradient at each

sample rate fs(j), j = 1, . . . , k and filtration step i, be defined using simple

numerical differentiation as:

Γ(fs(j), i) =

meanHHTIV (fs(j+1),i)−meanHHTIV (fs(j),i)

fs(j+1)−fs(j) for j = 1,

meanHHTIV (fs(j+1),i)−meanHHTIV (fs(j−1),i)fs(j+1)−fs(j−1) for j = 2, . . . , k − 1,

meanHHTIV (fs(j−1),i)−meanHHTIV (fs(j),i)fs(j)−fs(j−1) for j = k.

(4.11)

• The idea now is to search for a linear combination of neighboring filtration

steps which give the minimum absolute gradient sum, i.e. the cost function

f(C) is minimised by searching for an optimal C (for a proof of concept a

simple grid search was used).

f(C) =k∑j=1

|(C)Γ(fs(j), i) + (1− C)Γ(fs(j), i− 1)| , (4.12)

where 0 ≤ C ≤ 1 and k is the number of sample rates.

6. Calculation of Volatility Step:

Finally, once the pair of neighboring IMFs which lead to the flattest volatility

profile have been found, the filtered volatility estimate is given by a linear combi-

nation of these over filtered and under filtered estimates at various sampling rates


as:

σ2(t, fs(j)) = (C)HHTIV (t, fs(j), i) + (1− C)HHTIV (t, fs(j), i− 1). (4.13)

This procedure is best understood with the aid of Figure 4.4, depicting the character-

istic volatility profile or volatility signature plot for the Two-Factor Affine model at

the γ = 0.1 microstructure level. This figure demonstrates how changes in sample rate

can effect volatility estimates. In this figure the flattening of the blue lines indicate

clear that successive applications of the filtration process are resulting in a reduction of

upwards bias at shorter sampling intervals. The relative flatness of the Filtered-HHT

volatility estimation is also evident in Figure 4.4. This demonstrates that the goal of

a consistent estimator across different sample rates is also being achieved. Since the

proposed estimate yields volatility estimates across several different sampling rates, the

best result is obtained by averaging across several of these estimates. Further, the figure

also shows why this procedure can be considered as a combination of under and over

filtered components.

Ideally, taking the average over every sample rate would be optimal, however due to

occasional over filtering at the fastest sample rates it was found that averaging over the

flattest region of the volatility signature plot provided the most consistent results. This

was found to be around the last 5 sampling rates and this is the procedure that was

used in Section 4.4.2.

4.4 Monte Carlo Analysis

In this section, Monte Carlo analysis is used to test the performance of the Filtered-HHT

volatility estimation procedure proposed above. Following the models used by Andersen

et al. [53] and Barucci et al. [54] in similar investigations of market microstructure, a

log price with frictions is considered. Specifically, the underlying price model follows

Equation (1.3): dpt = σtdW(1)t with additive microstructure noise given by Equation


0 1 2 3 4 5 6 7 8 90

0.2

0.4

0.6

0.8

1

1.2

1.4

1.6

1.8Characteristic Volatility Profile

Sample interval in minutes (simulated)

Dai

ly V

olat

ility

Successive filtrationsOptimised

Figure 4.4: Volatility estimates for various sample rates across different levels offiltration. The ‘o’ symbol denotes the estimate provided by Equation (4.13).

(1.6): pt = pt + ut, where ut is IID noise with mean zero and the variance is propor-

tional to the expected volatility for the frictionless log price, i.e. V ar[u(t)] = γE[IVt]

with γ = 0%, 0.1%, 0.5%. The observable price is then given by Equation (2.15):

rt(h) ≡ pt − pt−h.

The volatility, σt in Equation (1.3) is then set to one of three models:

• Model I: GARCH

dσ2t = k(θ − σ2

t )dt+ cGσ2t dW

(2)t , (4.14)

with k = 0.6, θ = 0.636 and cG = 0.1439, noting that E[IVt] = θ. Note that this

model has a strong mean reversion for this choice of parameters.


• Model II: Two-Factor Affine

σ2t = σ2

1,t + σ22,t (4.15)

dσ2i,t = ki(θi − σ2

i,t)dt+ ciσ2i,tdW

(i,2)t , (4.16)

with k1 = 0.5708, k2 = 0.0757, θ1 = 0.3257, θ2 = 0.1786, c1 = 0.2286, c2 = 0.1096,

noting that E[IVt] = θ1 + θ2. This model has a very volatile first factor with

strong mean reversion, and a less volatile component which is only weakly mean

reverting.

• Model III: Log-Normal Diffusion

d log σ2t = k(θ − log σ2

t )dt+ cLNdW(2)t , (4.17)

with k = 0.0136, θ = −0.8382 and cLN = 0.1148, noting that E[IVt] = eθ+cLN/4k.

All three of these models are Ornstein-Uhlenbeck process, they are widely used in the

literature to simulate market microstructure noise.

4.4.1 Spectral Analysis and Hurst Parameter Estimation

Before we delve into volatility analysis, some of the tools that were outlined in Chapter

2 are revisited, specifically, the spectral structures and Hurst exponents are examined

for the three stochastic volatility models described above.

One of the most powerful tools available in spectral analysis is the Fast Fourier Trans-

form (FFT). Using the FFT it is possible to transform a signal from the time-amplitude

domain into the frequency-power domain. When the EMD procedure is performed first,

a FFT can be applied to each of the IMFs separately, this procedure was undertaken for

the three models described above. The simulation ran for 250 virtual trading days with

one trade every 10 seconds for six hours giving 2160 trades each day and was repeated

at three microstructure noise level, namely 0%, 0.1% and 0.5%. The resulting spectra


are shown in Figure 4.5. The first thing to notice is that with zero noise, the spectra are

flat. This is consistent with a Gaussian white noise process with stochastic volatility.

When a small amount of microstructure noise of γ = 0.1% is introduced, it becomes

evident that there is an increase in the power of the IMFs with higher frequencies. This

effect becomes more pronounced as the noise is increased to γ = 0.5%, where the high

frequency IMFs are of considerably higher power in all three models.

One can not help but notice that the IMF power spectra plots of Figure 4.5 bear a

striking similarity to those produced by the examination of a fractional Gaussian noise

process as seen in Figure 2.9, namely the ones withH < 0.5. This link has been examined

in more detail by applying the EMD procedure for calculating the Hurst exponent for

all three models at the same noise levels as above, the results are given in Table 4.1.

As hinted by the spectral information in Figure 4.5, the Hurst exponents do indeed fall

below H = 0.5 and as the noise parameter γ increases the H estimates decrease.

Table 4.1: Hurst exponent estimates from the EMD procedure and their associatedstandard deviations for the three models at three noise levels.

γ = 0% Std γ = 0.1% Std γ = 0.5% Std

H GARCH 0.5186 0.0518 0.3428 0.0473 0.2980 0.0477

H Two-Factor Affine 0.5242 0.0505 0.4011 0.0508 0.3217 0.0517

H Log Normal Diffusion 0.5210 0.0543 0.3497 0.0499 0.3096 0.0498

4.4.2 Results and Comparisons

In this section the Monte Carlo models described above are used to generate 250 sim-

ulated trading days, each of duration 6 hours with one trade every second. Firstly, to

show the effects of even low levels of market microstructure on a volatility estimate,

the RV was calculated at different levels of daily sampling rates 1/h (available in Table

4.2) and the result is displayed in the volatility signature profile Figure 4.6. From this

one can observe that even a small amount of market microstructure noise can dominate

volatility estimates at high sampling rates. This emphasises the importance of using a

volatility estimator that is specifically designed to handle such inflationary noise.


24

68

1012

−30

−28

−26

−24

−22

−20

Log 2(F

requ

ency

)

Log PSD (dB)G

AR

CH

Spe

ctra

, 0%

Noi

se

24

68

1012

−30

−28

−26

−24

−22

−20

Affi

ne S

pect

ra, 0

% N

oise

Log 2(F

requ

ency

)2

46

810

12−

30

−28

−26

−24

−22

−20

Log 2(F

requ

ency

)

Log

Spe

ctra

, 0%

Noi

se

24

68

1012

−30

−28

−26

−24

−22

−20

−18

Log 2(F

requ

ency

)

Log PSD (dB)

GA

RC

H S

pect

ra, 0

.1%

Noi

se

24

68

1012

−30

−28

−26

−24

−22

−20

−18

Affi

ne S

pect

ra, 0

.1%

Noi

se

Log 2(F

requ

ency

)2

46

810

12−

30

−28

−26

−24

−22

−20

−18

Log 2(F

requ

ency

)

Log

Spe

ctra

, 0.1

% N

oise

24

68

1012

−30

−25

−20

−15

Log 2(F

requ

ency

)

Log PSD (dB)

GA

RC

H S

pect

ra, 0

.5%

Noi

se

24

68

1012

−30

−25

−20

−15

Affi

ne S

pect

ra, 0

.5%

Noi

se

Log 2(F

requ

ency

)2

46

810

12−

30

−25

−20

−15

Log 2(F

requ

ency

)

Log

Spe

ctra

, 0.5

% N

oise

Figure 4.5: Spectral analysis of IMFs from GARCH, Affine and Log volatility models.Top row: no microstructure noise. Middle row: 0.1% noise. Bottom row: 0.5% noise.

Note that the vertical axis is in terms of the Power Density Spectra (PDS)


Table 4.2: Sampling rate and equivalent sampling intervals in seconds.

1/h 2160 1440 720 360 288 96 48 24 12 6Seconds 10 15 30 60 75 225 450 900 1800 3600

2 3 4 5 6 7 8 90

0.5

1

1.5

2

2.5

3

3.5

4RV Volatility Signature Plot 0.1% Noise

log Sample Interval (seconds)

Vol

atili

ty

M1M2M3

Figure 4.6: Volatility signature plot at 0.1% noise. M1 M2 M3 refers to the threevolatility models, GARCH, Two-Factor Affine and Log-Normal diffusion respectively.

One of the benefits of using simulated data is that the actual frictionless volatility

of each process is known throughout the simulation and so a simple error analysis using

the procedures outlined by Equations 2.39- 2.41 in Chapter 2 is all that is needed to

determine the accuracy of each volatility measure. One year of simulated data at the

high frequency rate of one point per second is generated for three models at two noise

levels, γ = 0.1% and γ = 0.5%. From this the mean error, mean square error and mean

absolute errors are calculated for each volatility measure over the course of the simulated

year and the results are given in Tables 4.3 − 4.8. Several RV-based volatility estimation

methods were included in this study to give a good basis of comparison to the proposed

HHT method. As a guide to understand what variant of each method was implemented,

a short description of each follows:

• RV all

This will serve as a baseline for a poor measure in the presence of microstructure


noise. It is defined as the RV given by Equation (2.7) when the highest available

frequency of data is used. This procedure is not designed to be used under the

conditions of market microstructure.

• RV averaged

As described by Equation (2.18), this volatility estimate is simply a sparsely sam-

pled RV measure which has been averaged over many offsets. The sparse sampling

was done at a rate of one sample every 5 minutes. This procedure should have

some resistance to market microstructure.

• TSRV

This is the first real measure designed specifically to deal with market microstruc-

ture, defined by Equation (2.21)

• TSRV adj

Short for Adjusted−TSRV . A slight improvement on TSRV that adjusts for sam-

ple size difference which make up the two components in the summands, defined

by Equation (2.21)

• HHT averaged

This is the HHT equivalent of RV averaged and is the average of many offset HHT

volatility estimates sampled 5 minutes apart, defined by Equation (4.8)

• HHT filtered

This is the volatility estimate proposed in Section 4.3.1 and is defined therein.

The mean error results given in Tables 4.3 & 4.6 show that the Filtered-HHT method

is yielding similar results to the Adjusted-TSRV procedure, with each achieving the low-

est mean error 3 of 6 times. The good performance of the Filtered-HHT procedure in

this test is most likely due to the averaging process over several sample rates, since

the average of several unbiased estimators is likely to have a lower bias than each es-

timate individually. A variant of the TSRV procedure that does use an average over

many sample rates is known as the MSRV and was proposed by Zhang [71], this proce-

dure is likely to have a very low bias for the kind of simulated data tested in this chapter.


Table4.3:

Mea

ner

ror

an

aly

sis

atγ

=0.

1%

nois

e.

Mod

elRVall

RVaverage

TS

RV

TSRVadj

HHTaverage

HHTfiltered

GA

RC

H-1

0.9

6±

.0493

-0.3

552±

.011

90.

0320±

.011

90.0106±

.0123

-0.3

744±

.026

0-0

.026

4±

.018

4

AF

F-2

.385±

.0125

-0.0

641±

.009

40.

0334±

.009

40.

0160±

.009

6-0

.056

1±

.013

50.0110±

.0112

LO

GN

-9.5

20±

.0429

-0.3

061±

.005

60.

0213±

.005

70.

0116±

.005

8-0

.309

2±

.015

3-0.0073±

.0082

Table4.4:

Mea

nsq

uare

erro

ran

aly

sis

atγ

=0.

1%

nois

e.

Mod

elRVall

RVaverage

TS

RV

TSRVadj

HHTaverage

HHTfiltered

GA

RC

H120

.3±

1.0

873

0.13

53±

.008

90.

0102±

.001

80.0099±

.0018

0.18

36±

.023

10.

0225±

.003

8

AF

F5.

697±

.059

60.

0097±

.001

70.

0068±

.001

10.0062±

.0011

0.01

49±

.003

20.

0082±

.001

8

LO

GN

90.7

5±

.817

90.

0958±

.003

50.

0025±

.000

50.0023±

.0005

0.11

07±

.010

60.

0044±

.000

9

Table4.5:

Mea

nab

solu

teer

ror

an

aly

sis

atγ

=0.

1%

nois

e.

Mod

elRVall

RVaverage

TS

RV

TSRVadj

HHTaverage

HHTfiltered

GA

RC

H10

.96±

.049

30.

3552±

.011

90.

0794±

.007

70.0786±

.0076

0.37

51±

.025

80.

1223±

.010

8

AF

F2.

385±

.012

50.

0784±

.007

50.

0654±

.006

20.0632±

.0059

0.09

27±

.009

90.

0706±

.007

0

LO

GN

9.52

0±

.042

90.

3061±

.005

60.

0382±

.004

10.0364±

.0039

0.30

94±

.015

30.

0505±

.005

4


Table4.6:

Mea

ner

ror

an

aly

sis

atγ

=0.

5%

nois

e.

Mod

elRVall

RVaverage

TS

RV

TSRVadj

HHTaverage

HHTfiltered

GA

RC

H-5

5.1

1±

.2485

-1.7

944±

.014

10.

0629±

.012

90.0441±

.0133

-1.8

555±

.062

5-0

.059

9±

.014

4

AF

F-1

1.9

0±

.0540

-0.3

764±

.008

30.

0352±

.008

30.

0209±

.008

5-0

.364

4±

.019

5-0.0041±

.0122

LO

GN

-47.6

5±

.2129

-1.5

427±

.015

90.

0701±

.014

70.0471±

.0152

-1.5

703±

.057

8-0

.050

2±

.018

8

Table4.7:

Mea

nsq

uare

erro

ran

aly

sis

atγ

=0.

5%

nois

e.

Mod

elRVall

RVaverage

TS

RV

TSRVadj

HHTaverage

HHTfiltered

GA

RC

H30

41±

27.4

010

3.23

3±

.050

80.

0146±

.002

30.0133±

.0022

3.69

35±

.247

20.

0169±

.002

9

AF

F14

1.7±

1.28

300.

1461±

.006

50.

0057±

.000

90.0051±

.0008

0.15

72±

.016

00.

0096±

.001

9

LO

GN

2273±

20.3

422

2.39

6±

.049

50.

0188±

.003

10.0171±

.0030

2.68

01±

.202

10.

0252±

.006

0

Table4.8:

Mea

nab

solu

teer

ror

an

aly

sis

atγ

=0.

5%

nois

e.

Mod

elRVall

RVaverage

TS

RV

TSRVadj

HHTaverage

HHTfiltered

GA

RC

H55

.11±

.248

51.

794±

.014

10.

0982±

.008

80.0935±

.0084

1.85

55±

.062

50.

1039±

.009

8

AF

F11

.90±

.054

00.

3764±

.008

30.

0625±

.005

30.0590±

.0051

0.36

53±

.019

20.

0777±

.007

4

LO

GN

47.6

5±

.212

91.

543±

.015

90.

1104±

.010

10.1035±

.0100

1.57

03±

.057

80.

1204±

.012

9


The mean square error and mean absolute error information given in Tables 4.4, 4.5,

4.7 & 4.8 show that the Filtered-HHT procedure is a great improvement over the RV and

unfiltered HHT processes, however it appears to slightly under performing in this area

compared to the TSRV and its adjusted variant. It is worth pointing out though, while

the errors are larger than the TSRV-based methods they are still relatively small, they

are of the same magnitude and usually only slightly larger. This suggests that while

the Filtered-HHT method is non biased, it does vary further from the true volatility

more often than its counterparts, although it does so almost equally both above and

below the true volatility. This can be seen in Figure 4.7 and Figure 4.8 which show the

volatility estimates and the true model volatility for all three models and the two noise

levels of γ = 0.1% and γ = 0.5%. Also Figures 4.7 & 4.8 provide a clearer perspective of

how closely the Filtered-HHT procedure and the Adjusted-TSRV procedures are to one

another as well as the model’s true volatility. While there is a small difference between

the two estimates, both methods are capturing much of the model volatility with some

accuracy.

The errors in Tables 4.3 - 4.5 indicate the 95% confidence intervals for each measure.

A larger simulation could have been performed to reduce the uncertainty around these

errors, however due to the high computational load involved in repeatedly performing

the the EMD algorithm for each Filtered-HHT estimate, the simulation was kept to one

simulated year, with trades occurring every 10 seconds. The Filtered-HHT procedure

is not currently optimised for computational efficiency however in its current form it is

sufficient for a proof of concept, as was the size of the simulation carried out.

Such results are promising, as the TSRV and Adjusted-TSRV methods were designed

to deal with the specific kind of additive microstructure noise which was examined in

this section. The Filtered-HHT procedure on the other hand, made no assumptions on

the kind of microstructure which was present and the filtering algorithm successfully

yielded a consistent estimator. Not being tied to a specific kind of microstructure noise


0 50 100 150 200 2500.2

0.4

0.6

0.8

1

1.2GARCH Volatility γ=0.1

σ2

Model Vol

TSRVadj

HHTfiltered

0 50 100 150 200 2500.2

0.4

0.6

0.8

1

1.2Two−Factor Affine Volatility γ=0.1

σ2

Model Vol

TSRVadj

HHTfiltered

0 50 100 150 200 2500

0.2

0.4

0.6

0.8 Log−Normal Diffusion Volatility γ=0.1

Time (simulated days)

σ2

Model Vol

TSRVadj

HHTfiltered

Figure 4.7: Volatility vs Time plots for the GARCH (top), Two-Factor Affine (middle)and Log-Normal Diffusion (bottom) models at 0.1% noise. Each plot shows model

volatility, the Adjusted-TSRV estimate and the Filtered-HHT volatility estimate.


0 50 100 150 200 2500

0.5

1

1.5GARCH Volatility γ=0.5

σ2

Model Vol

TSRVadj

HHTfiltered

0 50 100 150 200 2500.2

0.4

0.6

0.8

1Two−Factor Affine Volatility γ=0.5

σ2

Model Vol

TSRVadj

HHTfiltered

0 50 100 150 200 2500

0.5

1

1.5

2 Log−Normal Diffusion Volatility γ=0.5

Time (simulated days)

σ2

Model Vol

TSRVadj

HHTfiltered

Figure 4.8: Volatility vs Time plots for the GARCH (top), Two-Factor Affine (middle)and Log-Normal Diffusion (bottom) models at 0.5% noise. Each plot shows model

volatility, the Adjusted-TSRV estimate and the Filtered-HHT volatility estimate.

may lead to the Filtered-HHT procedure being a more flexible estimator of HF volatil-

ity than those which are designed to account for a particular type of microstructure noise.



While no direct comparison is made between the improved HHT volatility measure pro-

posed in Section 4.1 with its predecessor proposed in Section 3.1, the improved estimator

effectively supplants the older one. It is theoretically more accurate since it’s using the

log returns directly and doesn’t have to create a proxy for them, furthermore it’s entirely

parameter free, making it much faster and enabling the Monte Carlo analysis carried

out in this chapter.

In Section 4.2 a test was designed specifically to test the claim that the HHT volatility

estimate was able to pick up on changes in volatility faster than EWMA methods while

maintaining a lower signal to noise ratio. The test results have shown that under certain

conditions this is indeed the case, namely the HHT estimates were able to respond to

changes volatility faster. The real price paid by the HHT based method is that it is best

suited to ex-post data, while no such requirement is placed on the EWMA procedure.

This early uptake of changes to volatility is likely due to the two sided ex-post nature

of the EMD process in which the cubic splining procedures at the heart of the EMD

procedure fitting data based on past and future information relative to each point. This

two sided nature of the EMD process can lead to a short lived artefact in which volatility

spikes can effect neighboring data on either side of the spike. An interesting investiga-

tion that may shed some light on how rapid changes or spikes propagate through the

EMD procedure was carried out by Rilling et al. [63].

Spectral analysis of the three simulated models performed in Section 4.4.1 showed

that when microstructure noise was introduced, there was an increase in the power level

of high frequency components and a simultaneous decrease in the Hurst exponents ex-

tracted from the data. This shares many parallels with the spectral analysis of the

fractional Gaussian noise conducted in Chapter 2. In effect, increasing the microstruc-

ture noise is leading to a rougher time series, with negative autocorrelations at the high

frequencies. Also, we have shown that the procedure proposed by Rilling et al. [63] to


estimate the Hurst exponent using the EMD process may be a useful means to analyse

inflationary market microstructure in high frequency data.

The Filtered-HHT method proposed in this chapter to deal with market microstruc-

ture noise appears to be functioning as intended. Figure 4.4 and Tables 4.3 − 4.8

clearly show that the EMD filtering approach taken clearly reduces the effect of market

microstructure on variance estimates. The low mean error results given by the tables

indicate that the Filtered-HHT approach has a comparably low bias with the TSRV-

based procedures which are also designed to handle market microstructure noise. The

Filtered-HHT procedure gave slightly higher mean square errors and mean absolute er-

rors than the TSRV and Adjusted-TSRV methods, however it is worth reiterating that

these procedures were specifically designed to handle the kind of uncorrelated additive

microstructure noise which was present in the simulation. Conversely, the Filtered-HHT

approach made no assumptions on the structure of the microstructure noise, other than

the fact that it was inflationary at higher frequencies. Although it is not made clear in

this chapter, the greater generality arising from a lack of assumptions on the underlying

structure can be a strength, just how much of an advantage it may be is more evident

in the following chapter.

Chapter 5

Simulated Options Market &

Real FX Data

In this chapter, a simulated options market in created in order to assess the performance

of several volatility estimators using real high frequency FX data for three currencies.

This approach of a simulated options market was developed by Engle et al. [67] to evalu-

ate the performance of volatility forecasting algorithms (ARMA, GARCH(1,1) etc.) for

low frequency financial data. More recently this approach was implemented by Bandi

et al. [68] to evaluate different volatility estimates (RV, TSRV, etc) at high frequencies.

The key feature of this approach is that it allows for the direct comparison of different

volatility estimates in the absence of the ‘true’ volatility.

ARMA(1,1) models are used to generate short term, one step ahead volatility fore-

casts for several different volatility estimating techniques. These different forecasts are

each represented by a virtual agent which buys and sells straddle options on $1 US of

the underlying (the currency) to other virtual agents which have differing volatility fore-

casts. These straddle options are then held to maturity and the profits and losses are

calculated. This chapter also presents a slightly modified version of the Filtered-HHT

procedure which reduces computational load.

90

Chapter 5. Simulated Options Market & Real FX Data 91

5.1 The Simulated Options Market and Clarification of

Methods

In this section the proposed volatility measure for dealing with HF microstructure noise

is tested with real FX data, namely the: AUD/USD, EUR/USD & GBP/USD exchange

rates. Volatility is a latent variable, therefore there is no true volatility measure one

can use as a basis for comparison as there would be in simulated data, thus we turn

to a competitive simulated options market and compare outcomes. The details of this

approach follow.

5.1.1 Simulated Option Market Construction

In this section we specify the simulated options market where one step ahead volatility

forecasts are pitted against one another. Every volatility estimator is considered to have

its own virtual trading agent that buys and sells straddle options to other agents on a

pairwise basis, based on whether or not they view the straddle as being over or under

priced by their pairwise partner. For simplicity, each option is priced on a $1 US share in

the underlying, or the exchange in our case with a strike price of $1 and a risk free rate

of zero. Note that over such short time intervals this zero expected return is entirely

plausible for stocks and even more so for FX markets. Under these assumptions the

Black-Scholes European call price (and also the put price) is given by:

Ct = 2Φ

(1

2σ(t)∆t0.5

)− 1, (5.1)

where Φ is the cumulative normal distribution and σ(t) is the specific volatility forecast

in terms of daily standard deviation at time t and ∆t = 124 since the option expires in

one hour. Note that a straddle is made up of a call and a put option purchased at the

same spot price and at the same strike price. A straddle will increase in value if the

underlying asset (or rate) either increases or decreases, thus trading straddles is akin to

trading in volatility.


The following procedure outlines the virtual market trading place.

1. Estimate past volatility

Historical volatility estimates (σ(t)) are calculated using a number of methods.

All volatility estimates are constructed from high frequency intra-day data in one

hour chunks.

2. Forecast future volatility

The one step ahead out of sample forecasts were computed using an ARMA(1,1)

model with parameters computed over a 500 hour sliding window on the log of the

historical volatility. Parameter fitting was performed on the log volatility because

the leptokurtic nature of the raw volatility distribution led to inflated volatility

forecasts, this in turn was due to the least squares parameter fitting procedure

that was used to find the ARMA(1,1) model parameters. The exponential of the

forecasted log volatility was then used to price the option.

3. Price a straddle option

For a given pair of volatility forecasts, each method computes their perceived fair

price for a one hour straddle option using Equation (5.1).

4. Trade with partner

Trades then take place between each pair at a price calculated using the midpoint

of the two volatility estimates. Note this differs from the literature slightly where

the trade is executed at the midpoint of the two perceived fair prices. The agent

with the higher volatility forecast of the two will perceive the option as being

under priced and take a long position in the straddle option (buy). Conversely,

the agent with the lower volatility forecast will perceive the option as being over

priced and take a short position (sell). If the two agents have the exact same

volatility estimate at any point then no trade is made.

5. Hedge

Finally, the two positions are hedged to minimise exposure to risk. Since a straddle

consists of a call and a put option, with corresponding hedge ratios of Φ(

12σt∆t

0.5)

short and 1−Φ(

12σt∆t

0.5)

long, the total hedge ratio is 1−2Φ(

12σt∆t

0.5). Hence,


the hourly profit for the agent who buys a straddle is given by:

|RFXratet |−2Ct +RFXratet

(1− Φ

(1

2σt∆t

0.5

))(5.2)

where RFXratet is the return on the $1 US spent on the currency at time t − 1.

Conversely, the seller of the straddle makes the hourly profit given by:

2Ct − |RFXratet |−RFXratet

(1− Φ

(1

2σt∆t

0.5

)). (5.3)

When it comes to assessing the performance of each forecaster, the most obvious, but

not the only useful metric, is the total profits or losses incurred after engaging in com-

petitive trading against virtual agents representing other volatility forecasts. This test

metric is described throughout this chapter as the pairwise competitive trading test.

There is also much to be learned by a parallel market in which each agent buys or

sells the straddle to the other party at what they consider to be their own fair price, i.e.

no midpoint is agreed on and each trader enters into the trade almost independently of

the other, with only their position determined by the others price. In this case the profit

from one will no longer be equal in magnitude to the corresponding loss of its trading

counterpart. Also, it is significant to note that the the most successful trader according

to this test, is not the one with the highest profit, but rather the one with the smallest

magnitude of profit or loss. This change in the assessment metric is due to the fact that

each agent is effectively trading at its own perceived fair price, thus only the agents with

the fairest price will result in a profit or loss that is small in magnitude. This test metric

is described throughout this chapter as the fair price trading test.

5.1.2 Data Specification and Clarification of Methods

The FX rate used for analysis was at the midpoint of the bid rate and the ask rate.

Actual time was used as opposed to tick or business time and the data was of suffi-

ciently high frequency that there were always points sufficiently close to the desired


sampling point. For the AUD/USD exchange rates there were 1565 total high frequency

trading hours considered between the 1st March 2014 and 31st May 2014, 1576 for the

EUR/USD exchange and 1552 for the GBP/USD exchange. Each of these hours was

considered high frequency if it had at least one trade very ten seconds, although most

hours had substantially more than this.

The FX quotes are provided around the clock throughout weekdays so there is no

consideration taken for overnight effects. Weekends were handled simply by closing all

trading positions before a weekend starts so that no options or stocks are held over the

weekend.

Nine distinct methods were used to provide historical volatility estimates as well as

three supplementary estimates of Min, Max and Average which are useful diagnostic

tools. The historical volatility estimation methods used were:

1. The RV method with a sample interval of one minute given by Equation (2.18)

and denoted here as ‘RV 1min’

2. The RV method with a sample interval of five minutes given by Equation (2.18)

and denoted here as ‘RV 5min’

3. The Averaged-RV method with a sample interval of one minute given by Equation

(2.20) and denoted here as ‘Av RV 1 min’

4. The Averaged-RV method with a sample interval of five minutes given by Equation

(2.20) and denoted here as ‘Av RV 1 min’

5. The Adjusted-TSRV method with a sample interval of one minute given by Equa-

tion (2.22) and denoted here as ‘Adj TSRV 1 min’

6. The Adjusted-TSRV method with a sample interval of five minutes given by Equa-

tion (2.22) and denoted here as ‘Adj TSRV 5 min’

7. The Average-HHT method with a sample interval of one minute given by Equation

(4.8) and denoted here as ‘HHT 1 min’


8. The Average-HHT method with a sample interval of five minutes given by Equation

(4.8) and denoted here as ‘HHT 5 min’

9. The Filtered-HHT method, described in Section 5.1.3 and denoted here as ‘HHT

Filtered’.

Finally, the minimum, maximum and mean of all of the above methods was taken at

each point as a diagnostic tool. These supplementary estimates are given as:

10. Average: Constructed by taking the arithmetic mean of the above forecasts on a

point-by-point basis

11. Max: Constructed by taking the highest volatility forecast from the above methods

on a point-by-point basis

12. Min: Constructed by taking the lowest volatility forecast from the above methods

on a point-by-point basis.

5.1.3 Modified HHT Estimator With Market Microstructure

This version of the HHT volatility estimator differs slightly from that outlined in Chap-

ter 4. Due to differences in the data and overall computational load, a slightly more

efficient procedure was used. The main difference is a change in the stopping criteria for

the iterative filtration process. The subsample times were also changed to capture the

dynamics of the underlying FX market more succinctly. As a guide, the range of sam-

ple intervals should include enough high frequency points to capture some inflationary

behaviour, as well as some relatively sparse samples which have not been significantly

effected by market microstucture. A step by step outline of the modified volatility

estimation procedure follows.

1. EMD Step:

Apply the EMD procedure to the observed raw price series P (t,∆t), where ∆t is

the highest available sample rate and the observed raw price with market frictions


is given by P . Note that this differs from the low frequency realm, where the ob-

served price was essentially the same as the frictionless price, which was denoted

P . Also, recall that the observed price can be reconstructed by a summation of

all IMFs, i.e. P (t,∆t) =∑n

l=1 Ψl(t,∆t) + Ωn(t,∆t).

2. Filtration Step:

Next, filter the observed price using the EMD procedure to give the filtered price

Fi(t,∆t), defined by:

Fi(t,∆t) =

n∑l=1+i

Ψl(t,∆t) + Ωn(t,∆t), (5.4)

for i = 0, . . . , n, where i is the number of IMFs filtered off, so that when i = 0,

F0(t,∆t) is simply the observed price. Note that the ith filtered signal is actually

equivalent to the ith residual, i.e.

Ωi(t,∆t) = P (t,∆t)−i∑l=0

Ψl(t,∆t) =

n∑l=i+1

Ψl(t,∆t) + Ωn(t,∆t). (5.5)

This observation is consistent provided that Ω0 is the considered original observed

price and Ψ0 is empty.

3. Subsampling Step:

Subsample Fi(t,∆t) at various time intervals to give Fi(t,∆t), where ∆t = n∆t.

For example, the numerical results given in this chapter used ∆t(1, . . . , 6) =

[0.25, 0.5, 1, 1.5, 2, 3] minute intervals. The daily sample rate is then given as

fs(j) = 1440∆t(j) , for j = 1, . . . , 6.

4. Volatility and Averaging Step:

Now, starting at the 0th filtration step, the procedure outlined in Section 4.1 is

used to estimate the volatility of Fi(t,∆t) for each ∆t. The the level of filtration

is increased with each successive iteration until the stopping criteria is met. Note

that this procedure gives the spot volatility, and for the purposes of comparison,

we wish to convert this into integrated volatility to make hourly estimates. The


mean of each of the resulting volatility measures is equivalent to the integrated

volatility over the period the mean was taken, which is one hour in this case. Let

this mean or integrated volatility at sample rate fs, hour t and filtration step i be

given by HHTIV (t, fs(j), i), defined by Equation (4.10).

5. Iteration and Stopping Criteria Step:

Iterate the previous step until the characteristic volatility curve changes the sign

of its gradient at the highest sample rates (fs(1)), i.e. the characteristic volatility

curve changes from decreasing as the sample interval increases, to increasing as

the sample interval increases. This can also be defined in terms of the gradient

changing with respect to changes in sample rate, i.e. Γ(fs(1), i− 1)) > 0 changes

to Γ(fs(1), i) < 0 (note that fs(j) decreases as j increases) where Γ is defined in

Section 4.11, with k = 6.

6. Optimisation Search & Calculation of Volatility Steps:

Once the stopping criteria is met, the last two filtrations of the volatility measure

are linearly combined in such a way as to maximise the flatness of the combined

volatility profile over all sample rates. This is achieved by minimising the cost

function described by Equation (4.12) in Chapter 4. Finally, Equation (4.13) is

used to calculate the Filtered-HHT volatility estimate.

The procedure is best illustrated through Figure 5.1, where it evident that successive

filtrations have removed much of the market microstructure. In Figure 5.1 it is also

possible to see that a change of gradient has occurred at the highest sampling rates in

the last filtration. This is the condition which triggered the stopping criteria, as the

change in gradient signals the point at which the filtered signal has changed from being

under filtered to over filtered.

The end result of this is a volatility measure which should be roughly the same

whether it was sampled at 0.5 minutes or 3 minute intervals, thus any of the values can

be chosen or better still, a mean over the different sample rates can be chosen as the

final integrated volatility estimate. A mean of all sample rates what the procedure used


0 0.5 1 1.5 2 2.5 3 3.5 42

4

6

8

10

12

14x 10

−5

Sample interval in minutes

Dai

ly V

olat

ility

Characteristic Volatility Profile

Successive filtrationsOptimised

Figure 5.1: Volatility estimates with sample intervals of ∆t(1, . . . , 8) =[0.2, 0.25, 0.5, 1, 1.5, 2, 3, 4], calculated across different levels of filtration using FX data.The ‘o’ symbol denotes the estimate provided by the procedure outlined in this chapter.

in this chapter.

Importantly, once the optimal choice for the parameter C is known, the volatility

estimates which used the highest frequency data can be used to describe the instan-

taneous volatility down to the smallest time scales. Figure 5.2 shows this in practice.

This spot volatility estimate makes it possible to see how volatility is evolving over the

course of a minute, making daily volatility estimates seem low frequency by comparison.

Unfortunately, the results from the previous chapter also suggest that the Filtered-HHT

procedure can be quite noisy, rendering volatility estimates at the tick scale of limited

use at the current time, however with further refinements this approach could lead to

better short term forecasts.


0 6 12 18 24−12

−11

−10

−9

−8

−7

−6

−5Average Daily Volatility Cycle

Hours

Log

Vol

atili

ty

Figure 5.2: Instantaneous or spot volatility estimates for 24 hours of high frequencyAUD/USD data.

Furthermore, by effectively de-noising a signal in order to create a consistent volatility

measure there is an added bonus of being able to either examine the frictionless price or

the noise separately. Figure 5.3 depicts AUD/USD FX data riddled with microstructure

noise as well as the frictionless price, which can easily be extracted once the optimal

multiplier C is known. It was also observed during the course of this research that once

this frictionless price is extracted, inconsistent integrated volatility estimates such as RV

also become consistent when applied to the filtered price. Figure 5.3 also shows how the

EMD process has been used as a smoother and a low pass filter, as the jagged move-

ments and jumps which are apparent at the highest frequencies have been smoothed out.

5.1.4 Results

The aim of this section is to see how the volatility estimates proposed above compare

against the commonly used RV and Adjusted-TSRV methods. Note that the return


Figure 5.3: The ‘frictionless’ price (red), as extracted from 24 hours of high frequencyAUD/USD data (blue).

profits were annualised assuming 252 trading days in a year, each day consisting of 24

trading hours.

For the competitive option market in which the highest profits is the metric of suc-

cess, Tables 5.1-5.3 show that the proposed Filtered-HHT method provides the best

volatility estimates for the EUR/USD and GBP/USD exchanges and achieves a rank

of second place for the AUD/USD exchange. Conversely, the realized volatility formed

by averaging multiple five minute estimates earns itself first place in the AUD/USD

exchange and second in both the EUR/USD and GBP/USD exchanges. Surprisingly

the Adjusted-TSRV method, which is designed specifically to handle high frequency

noise appears to be over adjusting for the inflationary noise and under estimating the

volatility. Unsurprisingly all of the volatility estimates which make no adjustment for

the presence of high frequency noise perform very poorly at higher sampling rates when

the effect of high frequency noise is at its greatest.

In the second performance test, straddle options were valued and bought or sold at


each virtual trader’s opinion of a fair price. Thus, the object under this measure is not

to have the largest profit, but the smallest magnitude profit or loss, since it implies the

fairest pricing. The results for this test are given in Tables 5.4-5.6 . Under this test the

rankings are dominated by the same two methods that performed well in the pairwise

trading test. The Filtered-HHT procedure achieving first place for the AUD/USD and

GBP/USD exchanges and second for the EUR/USD exchange. Conversely, the realized

volatility formed by averaging multiple five minute estimates earns itself first place in

the EUR/USD exchange and second in both the AUD/USD and GBP/USD exchanges.

As expected the methods that make no adjustment for noise are performing the worst

at the highest sampling rate when the inflationary bias is at its worst. Again, the

Adjusted-TSRV method appears to be biased below, resulting in poorer than expected

performance.

The annualised standard deviation for the percentage returns is also given in Tables

5.1-5.3 for the pairwise competitive test and in Tables 5.4-5.6 for the fair price trading

test. Similar annualised standard deviations are observed within each market because

of the way each estimator interacts with each other estimator, the profit from one is the

loss of the same magnitude for another. This observation of only a small change in the

standard deviation of profits within a market is consistent with observations made by

Bandi et al. [68].

The Sharpe ratios are also given for each method during the competitive trading

test in each currency in Tables 5.7-5.9. It should be noted that values given for this

measure are not particularly informative, as so much of the result is based on the choice

of methods present in the simulation. The values given in Tables 5.7-5.9 have to be

considered in the right context, these Sharpe ratios are entirely dependent on the mod-

els that were chosen to compete in the simulated market, some of which were not well

suited to market microstructure. Thus, the results given by this metric may not be

representative of performance in a real market because of its dependence on the models

chosen for the simulated study. As expected, the ranking results of the Sharpe ratios


follow the ranking of competitive trading test very closely because the standard devia-

tion of profits varies little across different methods. Sharpe ratios were omitted from the

test results in which each estimation procedure traded at its own perceived fair price be-

cause they could be misleading and are no more informative than the rankings provided.

Under both performance metrics it may be of some interest to examine the potential

profits if all trades were unhedged. However, the unhedged results were calculated but

are not displayed because the time interval was so short that there was no significant

difference between the hedged and unhedged results and no difference in the performance

rankings in this case. If the options were held for longer periods or were exposed to more

risk in some other way then the importance of hedging would become more apparent.

The estimates denoted as Min, Max and Average were used as diagnostic tools when

setting up the simulated market participants. Due to the inclusion of some volatility

estimation techniques which were not designed to handle market microstructure noise,

such as RV 1 min and HHT 1 min, it is reasonable to assume that there are going to

be some very inflated volatility estimates. The presence of these highly biased volatility

estimates has led to the Max volatility forecast being substantially worse than the Min

forecast. Such a result was expected as there is no mechanism present which was likely

to yield an extremely low volatility estimates, additionally the maximum volatility was

unbounded and thus had far greater room for error.

The presence of systematically biased estimates has also led to the Average forecaster

being biased above, however it is worth noting that if the competing set of forecasters

was made up of independent estimates without such a systematic bias, then the average

of these forecasts would be expected to perform quite well.

More detailed tables that describe how each method performed when trading against

every other method for both performance tests can be found in the appendix to this

chapter.


Table 5.1: Pairwise competitive trading results for AUD/USD Straddles

Method Av Annual Return % Std of Return% Return Rank

RV 1 min -31.06 4.69 9

Adj TSRV 17.12 4.68 6

Av RV 1 min -37.34 4.71 10

RV 5 min 27.35 4.69 3

Adj TSRV 5 min 10.03 4.53 7

Av RV 5 min 34.44 4.70 1

HHT 1 min -37.92 4.62 11

HHT 5 min 25.71 4.67 4

HHT Filtered 32.16 4.70 2

Max -69.75 4.48 12

Min 4.44 4.45 8

Average 24.82 4.70 5

Table 5.2: Pairwise competitive trading results for EUR/USD Straddles


RV 1 min -31.47 3.25 10

Adj TSRV 8.86 3.25 7

Av RV 1 min -15.57 3.28 9

RV 5 min 20.66 3.28 3

Adj TSRV 5 min 12.26 3.21 6

Av RV 5 min 22.45 3.29 2

HHT 1 min -34.75 3.26 11

HHT 5 min 17.48 3.27 5


Max -55.59 3.13 12

Min 6.19 3.13 8

Average 18.73 3.29 4


Assessing the performance of estimations on an unobservable variable is inherently a

difficult task. Without resorting to simulated models, which are highly specification

dependent, we were successfully able to use a virtual option market to compare a new

procedure for high frequency estimation with several popular methods.

The volatility estimate proposed in this chapter is designed to overcome the prob-

lem of inflated volatility estimates caused by microstructure noise. The Filtered-HHT


Table 5.3: Pairwise competitive trading results for GBP/USD Straddles


RV 1 min -9.79 3.13 10

Adj TSRV 11.18 3.14 7

Av RV 1 min -4.70 3.11 9

RV 5 min 18.65 3.13 3

Adj TSRV 5 min 13.02 3.05 6

Av RV 5 min 21.25 3.13 2

HHT 1 min -54.51 3.00 11

HHT 5 min 14.14 3.07 5


Max -62.00 2.96 12

Min 8.46 2.97 8

Average 16.81 3.13 4

Table 5.4: Fair price trading results for AUD/USD Straddles


RV 1 min -74.19 4.70 9

Adj TSRV -20.94 4.70 6

Av RV 1 min -81.29 4.72 10

RV 5 min -6.78 4.70 3

Adj TSRV 5 min -36.52 4.53 7

Av RV 5 min 0.71 4.70 2

HHT 1 min -82.70 4.63 11

HHT 5 min -13.99 4.68 5

HHT Filtered -0.50 4.71 1

Max -116.87 4.50 12

Min -45.62 4.46 8

Average -7.46 4.70 4

estimate proposed in this thesis achieved the best overall performance when considering

all three exchanges and both performance tests so there is strong evidence to suggest

that the aim of constructing a consistent volatility estimator using the HHT has been

achieved. The reduction in high frequency bias after successive filtrations as depicted

Figure 5.1 also provides some evidence supporting the argument that the proposed

method is acting as a consistent estimator. Also, the apparent dipping of the volatility

estimate with a sample interval of four minutes in Figure 5.1 is likely due to the higher

sample error of the more sparsely sampled estimates. This apparent dipping is specific to

this randomly selected profile and highlights why simply choosing a large sample width


Table 5.5: Fair price trading results for EUR/USD Straddles


RV 1 min -62.92 3.26 10

Adj TSRV -21.03 3.28 6

Av RV 1 min -45.40 3.28 9

RV 5 min -5.48 3.30 4

Adj TSRV 5 min -23.50 3.22 7

Av RV 5 min -3.32 3.29 1

HHT 1 min -66.30 3.25 11

HHT 5 min -11.95 3.29 5


Max -89.82 3.14 12

Min -33.63 3.15 8

Average -4.97 3.29 3

Table 5.6: Fair price trading results for GBP/USD Straddles


RV 1 min -37.90 3.12 10

Adj TSRV -16.76 3.15 5

Av RV 1 min -32.43 3.11 9

RV 5 min -6.86 3.13 4

Adj TSRV 5 min -20.77 3.06 7

Av RV 5 min -4.91 3.13 2

HHT 1 min -93.02 3.02 11

HHT 5 min -17.04 3.08 6


Max -101.52 2.98 12

Min -28.77 2.97 8

Average -6.29 3.12 3

is an inadequate solution to the problem of market microstructure.

Perhaps the main advantage of the Filtered-HHT approach is that there are no un-

derlying assumptions on the structure of the microstructure noise, only that the noise

is causing inflationary effects at higher frequencies. The lack of assumptions, combined

with an algorithmic approach to achieve consistency over different time scales, appears

to have made the HHT based estimator flexible enough to handle real data effectively.


Better volatility estimates aren’t the only useful outcome of the filtering procedure

outlined in Section 5.1.3. With filtering, the high frequency noise that usually obfus-

cates our view of the frictionless price can be stripped away one layer at a time until the

elusive true price is revealed. Furthermore, as Figure 5.2 demonstrates, the proposed

estimator is also capable of providing spot or instantaneous volatility estimates as op-

posed to just the integrated volatility over a period of time. This ultra high frequency

volatility estimate may led to better short term predictions as well as a more thorough

understanding of high frequency dynamics.

It is also worth noting that asymmetries in the Black-Scholes pricing formula would

also tend to penalise under estimates differently than over estimates, as the option

prices do not vary linearly with volatility due to the convexity of the Black-Scholes pric-

ing formula and the effect of Jensen’s Inequality. This raises an important point about

volatility assessment in general, using a more realistic and complicated cost function to

determine the superiority of one estimator over another may yield to completely different

conclusions than assessments of the ME, MSE, MAPE etc. This is due to asymmetries

in the cost function placing more weight on some characteristics than others, i.e. under

estimating volatility may be more costly than over estimating it or a low bias might be

more important to some cost functions than a low noise.

It should be noted that at such high frequencies, assumptions required for the Black-

Scholes equation start to break down, namely the requirements for log-normally dis-

tributed returns and continuous sample paths. At such high frequencies, market frictions

(microstructure effects) lead to a highly discretised return structure which may be more

suited to a binomial or trinomial option pricing model with variable volatility, such as

the one developed by Haahtela [72]. Although we can acknowledge that the B-S model

has its drawbacks, so long as results gained from it are put into their correct context,

there is still a lot of information that may be garnered from its use. Furthermore, the

wide adoption of the B-S model makes any new insights more accessible and relevant.

Alternative, perhaps more appropriate option pricing schemes may have a non-trivial


impact on the evaluation of volatility estimators, however this line of research falls out-

side of the scope of this thesis and is an area of future research.

Chapter Appendix

The tables contained in this appendix give a more comprehensive account of how each

volatility estimator and its associated virtual trader fared against its competitors on

a pairwise basis. Note that the tables for the competitive trading results should be

symmetric about the diagonal in magnitude, but opposite in sign. For the competitive

trading case in Tables 5.7 – 5.9, the result in cell (i,j) should be seen as the profit or loss

resulting in a trade with its partner in cell position (j,i). For the fair trading tests in

Tables 5.10 – 5.12, the tables no longer have any symmetry as different virtual traders

are no longer in direct competition. Although they are no longer in competition, the are

not completely independent of one another, this is why results still vary among different

trading partners during the fair pricing test. What is driving the difference in results is

simply the fact that its position to buy or sell a straddle (at its own perceived fair price)

is still dependent on its trading partner.

From the full tables, the performance of individual methods becomes more apparent.

Some of the features which stand out the most in all of the tables is the fact that the

composite volatility estimate obtained by taking the maximum volatility forecast on

a point by point basis is clearly grossly over estimating the volatility, resulting in the

poorest overall performance. This is closely followed by the three volatility estimators

which are not designed to handle market microstructure, namely the “RV 1 min”, “Av

RV 1 min” and “HHT 1 min” procedures.

The composite volatility estimate generated by taking the minimum volatility fore-

cast at each point also has very poor performance across the board. While not quite as


bad as the composite maximum or the three procedures which are particularly suscepti-

ble to market microstructure noise, the composite minimum was the next worst estimate

in every test. This should not come as a surprise, as the estimate should be biased below

the true volatility, unless every other forecaster is also biased above the true volatility.

The fact that all of the estimates which account for market microstructure in some way

are between the composite maximum and the composite minimum is a reassuring sign

for the validity of the test. It implies that the true volatility lies between these two

extreme estimates.

The composite “Average” volatility estimate is performing relatively well considering

that it is being biased above quite heavily by the three procedures which are known to

over estimate volatility in the presence of market microstructure. Note that this pro-

cedure is generated from the 9 different volatility estimates which does not include the

two composite methods of the minimum and maximum. With the removal of the three

procedures known to be heavily biased above, the Average estimate would actually be

expected to have the best performance, since an average of independent unbiased esti-

mates is often better than any individual estimate.

Surprisingly, the “Adj TSRV” and “Adj TSRV 5 min” procedures aren’t dominat-

ing the forecast performance tables, as the results from Chapter 4 would indicate they

should. The TSRV procedures appear to be under estimating the true volatility, as the

“RV 5 min” and “Av RV 5 min” have, by definition a higher volatility estimate than

the TSRV procedures and yet they have are performing better under this simulation.

What is left to discuss are the sparsely sampled RV and HHT estimates as well as

the Filtered-HHT procedure. The more sparsely sampled RV and HHT based estimates

are all performing relatively well in both the competitive trading and fair pricing tests.

Sampling at five minute intervals has largely reduced the effects of market microstruc-

ture, however not the sparse sampling didn’t quite meet the performance of the filtering

procedure which has the best overall performance.


Table5.7:

AU

D/U

SD

Com

pet

itiv

ep

air

wis

etr

ad

ing

resu

lts

infu

ll

Meth

od

RV

1min

AdjTSRV

Av

RV

1min

RV

5min

AdjTSRV

5min

Av

RV

5min

HHT

1min

HHT

5min

HHT

Filte

red

Max

Min

Avera

ge

RV

1min

08.59E-0

5-2

.73E-0

51.01E-0

47.29E-0

51.01E-0

4-2

.15E-0

58.39E-0

51.08E-0

4-1

.33E-0

46.72E-0

51.27E-0

4AdjTSRV

-8.59E-0

50

-9.14E-0

52.53E-0

5-1

.87E-0

53.80E-0

5-9

.34E-0

52.48E-0

52.93E-0

5-1

.04E-0

4-2

.66E-0

5-9

.07E-0

6Av

RV

1min

2.73E-0

59.14E-0

50

1.05E-0

47.73E-0

51.05E-0

42.20E-0

68.83E-0

51.13E-0

4-1

.34E-0

47.17E-0

51.32E-0

4RV

5min

-1.01E-0

4-2

.53E-0

5-1

.05E-0

40

-2.65E-0

52.17E-0

5-1

.09E-0

45.86E-0

61.60E-0

5-1

.16E-0

4-3

.36E-0

5-2

.50E-0

5AdjTSRV

5min

-7.29E-0

51.87E-0

5-7

.73E-0

52.65E-0

50

3.52E-0

5-7

.99E-0

55.16E-0

52.64E-0

5-8

.80E-0

5-2

.18E-0

5-9

.64E-0

7Av

RV

5min

-1.01E-0

4-3

.80E-0

5-1

.05E-0

4-2

.17E-0

5-3

.52E-0

50

-1.08E-0

4-2

.94E-0

5-7

.60E-0

6-1

.15E-0

4-3

.95E-0

5-2

.61E-0

5HHT

1min

2.15E-0

59.34E-0

5-2

.20E-0

61.09E-0

47.99E-0

51.08E-0

40

9.07E-0

51.15E-0

4-1

.34E-0

47.43E-0

51.34E-0

4HHT

5min

-8.39E-0

5-2

.48E-0

5-8

.83E-0

5-5

.86E-0

6-5

.16E-0

52.94E-0

5-9

.07E-0

50

1.49E-0

5-9

.87E-0

5-5

.71E-0

5-1

.09E-0

5HHT

Filte

red

-1.08E-0

4-2

.93E-0

5-1

.13E-0

4-1

.60E-0

5-2

.64E-0

57.60E-0

6-1

.15E-0

4-1

.49E-0

50

-1.23E-0

4-3

.09E-0

5-1

.60E-0

5M

ax

1.33E-0

41.04E-0

41.34E-0

41.16E-0

48.80E-0

51.15E-0

41.34E-0

49.87E-0

51.23E-0

40

8.24E-0

51.41E-0

4M

in-6

.72E-0

52.66E-0

5-7

.17E-0

53.36E-0

52.18E-0

53.95E-0

5-7

.43E-0

55.71E-0

53.09E-0

5-8

.24E-0

50

5.36E-0

6Avera

ge

-1.27E-0

49.07E-0

6-1

.32E-0

42.50E-0

59.64E-0

72.61E-0

5-1

.34E-0

41.09E-0

51.60E-0

5-1

.41E-0

4-5

.36E-0

60

Hourly

Retu

rn-5

.14E-0

52.83E-0

5-6

.17E-0

54.52E-0

51.66E-0

55.69E-0

5-6

.27E-0

54.25E-0

55.32E-0

5-1

.15E-0

47.35E-0

64.10E-0

5Hourly

Retu

rnStD

6.03E-0

46.02E-0

46.06E-0

46.03E-0

45.82E-0

46.04E-0

45.94E-0

46.01E-0

46.05E-0

45.76E-0

45.72E-0

46.05E-0

4Sim

ple

AnnualRetu

rn%

-31.06

17.12

-37.34

27.35

10.03

34.44

-37.92

25.71

32.16

-69.75

4.44

24.82

AnnualRetu

rnStd

%4.69

4.68

4.71

4.69

4.53

4.70

4.62

4.67

4.70

4.48

4.45

4.70

Hourly

Sharp

eRatio

-0.085

0.047

-0.102

0.075

0.029

0.094

-0.105

0.071

0.088

-0.200

0.013

0.068

AnnualSharp

eRatio

-6.62

3.66

-7.93

5.83

2.22

7.33

-8.20

5.50

6.84

-15.56

1.00

5.27

Retu

rnRankin

g9

610

37

111

42

12

85

Sharp

eRankin

g9

610

37

111

42

12

85

Table5.8:

EU

R/U

SD

Com

pet

itiv

ep

air

wis

etr

ad

ing

resu

lts

infu

ll

Meth

od

RV

1min

AdjTSRV

Av

RV

1min

RV

5min

AdjTSRV

5min

Av

RV

5min

HHT

1min

HHT

5min

HHT

Filte

red

Max

Min

Avera

ge

RV

1min

06.84E-0

56.11E-0

58.36E-0

56.32E-0

58.05E-0

5-7

.77E-0

67.04E-0

57.96E-0

5-8

.33E-0

55.70E-0

59.97E-0

5AdjTSRV

-6.84E-0

50

-6.03E-0

52.02E-0

57.01E-0

62.99E-0

5-7

.30E-0

51.69E-0

55.42E-0

5-8

.74E-0

5-5

.83E-0

65.60E-0

6Av

RV

1min

-6.11E-0

56.03E-0

50

8.13E-0

55.73E-0

57.51E-0

5-8

.50E-0

56.73E-0

57.65E-0

5-1

.35E-0

45.10E-0

59.53E-0

5RV

5min

-8.36E-0

5-2

.02E-0

5-8

.13E-0

50

-7.04E-0

69.24E-0

6-8

.39E-0

5-4

.82E-0

61.54E-0

5-9

.51E-0

5-1

.74E-0

5-7

.03E-0

6AdjTSRV

5min

-6.32E-0

5-7

.01E-0

6-5

.73E-0

57.04E-0

60

1.75E-0

5-6

.45E-0

51.97E-0

52.31E-0

5-7

.47E-0

5-1

.82E-0

5-5

.48E-0

6Av

RV

5min

-8.05E-0

5-2

.99E-0

5-7

.51E-0

5-9

.23E-0

6-1

.75E-0

50

-8.35E-0

5-2

.12E-0

53.78E-0

5-9

.41E-0

5-2

.34E-0

5-1

.17E-0

5HHT

1min

7.77E-0

67.30E-0

58.50E-0

58.39E-0

56.45E-0

58.35E-0

50

7.38E-0

58.28E-0

5-8

.47E-0

55.83E-0

51.04E-0

4HHT

5min

-7.04E-0

5-1

.69E-0

5-6

.73E-0

54.82E-0

6-1

.97E-0

52.12E-0

5-7

.38E-0

50

3.39E-0

5-8

.36E-0

5-3

.29E-0

5-1

.32E-0

5HHT

Filte

red

-7.96E-0

5-5

.42E-0

5-7

.65E-0

5-1

.54E-0

5-2

.31E-0

5-3

.78E-0

5-8

.28E-0

5-3

.39E-0

50

-9.33E-0

5-2

.56E-0

5-3

.72E-0

5M

ax

8.33E-0

58.74E-0

51.35E-0

49.51E-0

57.47E-0

59.41E-0

58.47E-0

58.36E-0

59.33E-0

50

6.82E-0

51.12E-0

4M

in-5

.70E-0

55.83E-0

6-5

.10E-0

51.74E-0

51.82E-0

52.34E-0

5-5

.83E-0

53.29E-0

52.56E-0

5-6

.82E-0

50

-1.51E-0

6Avera

ge

-9.97E-0

5-5

.60E-0

6-9

.53E-0

57.03E-0

65.48E-0

61.17E-0

5-1

.04E-0

41.32E-0

53.72E-0

5-1

.12E-0

41.52E-0

60

Hourly

Retu

rn-5

.20E-0

51.46E-0

5-2

.57E-0

53.42E-0

52.03E-0

53.71E-0

5-5

.75E-0

52.89E-0

55.09E-0

5-9

.19E-0

51.02E-0

53.10E-0

5Hourly

Retu

rnStD

4.18E-0

44.18E-0

44.22E-0

44.22E-0

44.13E-0

44.23E-0

44.19E-0

44.21E-0

44.20E-0

44.03E-0

44.02E-0

44.23E-0

4Sim

ple

AnnualRetu

rn%

-31.47

8.86

-15.57

20.66

12.26

22.45

-34.75

17.48

30.76

-55.59

6.19

18.73

AnnualRetu

rnStd

%3.25

3.25

3.28

3.28

3.21

3.29

3.26

3.27

3.27

3.13

3.13

3.29

Hourly

Sharp

eRatio

-0.125

0.035

-0.061

0.081

0.049

0.088

-0.137

0.069

0.121

-0.228

0.025

0.073

AnnualSharp

eRatio

-9.68

2.72

-4.75

6.29

3.82

6.83

-10.67

5.34

9.41

-17.75

1.98

5.69

Retu

rnRankin

g10

79

36

211

51

12

84

Sharp

eRankin

g10

79

36

211

51

12

84


Table5.9:

GB

P/U

SD

Com

pet

itiv

ep

air

wis

etr

ad

ing

resu

lts

infu

ll

Meth

od

RV

1min

AdjTSRV

Av

RV

1min

RV

5min

AdjTSRV

5min

Av

RV

5min

HHT

1min

HHT

5min

HHT

Filte

red

Max

Min

Avera

ge

RV

1min

02.73E-0

52.30E-0

55.61E-0

54.02E-0

55.73E-0

5-1

.04E-0

44.39E-0

56.26E-0

5-1

.31E-0

43.70E-0

56.59E-0

5AdjTSRV

-2.73E-0

50

-2.36E-0

5-3

.33E-0

62.56E-0

61.15E-0

5-9

.71E-0

59.44E-0

72.96E-0

5-1

.08E-0

4-6

.24E-0

71.16E-0

5Av

RV

1min

-2.30E-0

52.36E-0

50

5.42E-0

53.83E-0

55.77E-0

5-1

.32E-0

44.31E-0

56.20E-0

5-1

.42E-0

43.58E-0

56.76E-0

5RV

5min

-5.61E-0

53.33E-0

6-5

.42E-0

50

-8.48E-0

6-4

.27E-0

6-1

.05E-0

4-8

.15E-0

62.91E-0

5-1

.09E-0

4-1

.50E-0

5-1

.20E-0

5AdjTSRV

5min

-4.02E-0

5-2

.56E-0

6-3

.83E-0

58.48E-0

60

1.84E-0

5-8

.53E-0

5-2

.31E-0

61.71E-0

5-9

.06E-0

5-1

.29E-0

5-8

.58E-0

6Av

RV

5min

-5.73E-0

5-1

.15E-0

5-5

.77E-0

54.27E-0

6-1

.84E-0

50

-1.01E-0

4-2

.58E-0

52.97E-0

5-1

.05E-0

4-2

.31E-0

5-2

.10E-0

5HHT

1min

1.04E-0

49.71E-0

51.32E-0

41.05E-0

48.53E-0

51.01E-0

40

8.94E-0

51.07E-0

4-3

.02E-0

58.09E-0

51.20E-0

4HHT

5min

-4.39E-0

5-9

.44E-0

7-4

.31E-0

58.15E-0

62.31E-0

62.58E-0

5-8

.94E-0

50

1.19E-0

5-9

.33E-0

5-2

.12E-0

5-1

.36E-0

5HHT

Filte

red

-6.26E-0

5-2

.96E-0

5-6

.20E-0

5-2

.91E-0

5-1

.71E-0

5-2

.97E-0

5-1

.07E-0

4-1

.19E-0

50

-1.11E-0

4-1

.67E-0

5-2

.35E-0

5M

ax

1.31E-0

41.08E-0

41.42E-0

41.09E-0

49.06E-0

51.05E-0

43.02E-0

59.33E-0

51.11E-0

40

8.48E-0

51.24E-0

4M

in-3

.70E-0

56.24E-0

7-3

.58E-0

51.50E-0

51.29E-0

52.31E-0

5-8

.09E-0

52.12E-0

51.67E-0

5-8

.48E-0

50

-4.79E-0

6Avera

ge

-6.59E-0

5-1

.16E-0

5-6

.76E-0

51.20E-0

58.58E-0

62.10E-0

5-1

.20E-0

41.36E-0

52.35E-0

5-1

.24E-0

44.79E-0

60

Hourly

Retu

rn-1

.62E-0

51.85E-0

5-7

.77E-0

63.08E-0

52.15E-0

53.51E-0

5-9

.01E-0

52.34E-0

54.55E-0

5-1

.03E-0

41.40E-0

52.78E-0

5Hourly

Retu

rnStD

4.02E-0

44.03E-0

44.00E-0

44.02E-0

43.92E-0

44.02E-0

43.86E-0

43.94E-0

44.01E-0

43.81E-0

43.81E-0

44.02E-0

4Sim

ple

AnnualRetu

rn%

-9.79

11.18

-4.70

18.65

13.02

21.25

-54.51

14.14

27.51

-62.00

8.46

16.81

AnnualRetu

rnStd

%3.13

3.14

3.11

3.13

3.05

3.13

3.00

3.07

3.12

2.96

2.97

3.13

Hourly

Sharp

eRatio

-0.040

0.046

-0.019

0.077

0.055

0.087

-0.233

0.059

0.114

-0.269

0.037

0.069

AnnualSharp

eRatio

-3.13

3.57

-1.51

5.96

4.27

6.79

-18.15

4.61

8.83

-20.94

2.85

5.37

Retu

rnRankin

g10

79

36

211

51

12

84

Sharp

eRankin

g10

79

36

211

51

12

84

Table5.10:

AU

D/U

SD

Fair

pri

cetr

ad

ing

resu

lts

infu

ll

Meth

od

RV

1min

AdjTSRV

Av

RV

1min

RV

5min

AdjTSRV

5min

Av

RV

5min

HHT

1min

HHT

5min

HHT

Filte

red

Max

Min

Avera

ge

RV

1min

0-2

.71E-0

5-4

.09E-0

58.79E-0

7-6

.86E-0

57.75E-0

7-3

.73E-0

5-4

.00E-0

51.76E-0

5-1

.45E-0

4-8

.30E-0

56.20E-0

5AdjTSRV

-1.81E-0

40

-1.90E-0

4-1

.48E-0

6-5

.34E-0

51.10E-0

5-1

.94E-0

4-1

.87E-0

63.34E-0

6-2

.09E-0

4-6

.00E-0

5-5

.03E-0

5Av

RV

1min

1.35E-0

5-2

.61E-0

50

8.79E-0

7-6

.86E-0

57.75E-0

7-6

.63E-0

6-4

.00E-0

51.89E-0

5-1

.43E-0

4-8

.30E-0

56.20E-0

5RV

5min

-1.87E-0

4-5

.31E-0

5-1

.94E-0

40

-6.46E-0

56.32E-0

6-2

.00E-0

4-2

.03E-0

5-4

.65E-0

6-2

.13E-0

4-7

.79E-0

5-5

.54E-0

5AdjTSRV

5min

-1.87E-0

4-1

.27E-0

5-1

.95E-0

4-8

.26E-0

60

1.42E-0

6-1

.99E-0

43.20E-0

5-1

.60E-0

5-2

.13E-0

4-2

.92E-0

5-6

.20E-0

5Av

RV

5min

-1.87E-0

4-6

.59E-0

5-1

.95E-0

4-3

.70E-0

5-7

.17E-0

50

-1.99E-0

4-5

.11E-0

5-2

.63E-0

5-2

.13E-0

4-8

.34E-0

5-5

.59E-0

5HHT

1min

5.58E-0

6-2

.63E-0

5-1

.11E-0

52.10E-0

6-6

.86E-0

57.75E-0

70

-4.00E-0

51.89E-0

5-1

.40E-0

4-8

.30E-0

56.20E-0

5HHT

5min

-1.87E-0

4-5

.07E-0

5-1

.95E-0

4-3

.05E-0

5-7

.24E-0

58.81E-0

6-1

.99E-0

40

-1.52E-0

5-2

.13E-0

4-7

.95E-0

5-5

.95E-0

5HHT

Filte

red

-1.86E-0

4-5

.68E-0

5-1

.95E-0

4-3

.73E-0

5-7

.38E-0

5-1

.17E-0

5-1

.99E-0

4-4

.72E-0

50

-2.13E-0

4-8

.49E-0

5-3

.85E-0

5M

ax

1.20E-0

4-2

.31E-0

51.25E-0

41.31E-0

6-6

.86E-0

57.75E-0

71.27E-0

4-4

.00E-0

51.89E-0

50

-8.30E-0

56.20E-0

5M

in-1

.87E-0

4-3

.07E-0

6-1

.95E-0

4-6

.35E-0

61.49E-0

5-3

.64E-0

7-1

.99E-0

43.65E-0

5-1

.71E-0

5-2

.13E-0

40

-6.20E-0

5Avera

ge

-1.87E-0

4-3

.60E-0

5-1

.95E-0

4-7

.55E-0

6-6

.86E-0

5-5

.64E-0

6-1

.99E-0

4-4

.25E-0

5-7

.67E-0

6-2

.13E-0

4-8

.30E-0

50

Hourly

Retu

rn-1

.23E-0

4-3

.46E-0

5-1

.34E-0

4-1

.12E-0

5-6

.04E-0

51.17E-0

6-1

.37E-0

4-2

.31E-0

5-8

.30E-0

7-1

.93E-0

4-7

.54E-0

5-1

.23E-0

5Hourly

Retu

rnStD

6.04E-0

46.04E-0

46.07E-0

46.04E-0

45.82E-0

46.04E-0

45.96E-0

46.01E-0

46.05E-0

45.78E-0

45.73E-0

46.05E-0

4Sim

ple

AnnualRetu

rn%

-74.19

-20.94

-81.29

-6.78

-36.52

0.71

-82.70

-13.99

-0.50

-116.87

-45.62

-7.46

AnnualRetu

rnStd

%4.70

4.70

4.72

4.70

4.53

4.70

4.63

4.68

4.71

4.50

4.46

4.70

Retu

rnRankin

g9

610

37

211

51

12

84


Table5.11:

EU

R/U

SD

Fair

pri

cetr

ad

ing

resu

lts

infu

ll

Meth

od

RV

1min

AdjTSRV

Av

RV

1min

RV

5min

AdjTSRV

5min

Av

RV

5min

HHT

1min

HHT

5min

HHT

Filte

red

Max

Min

Avera

ge

RV

1min

0-1

.22E-0

54.91E-0

59.86E-0

6-4

.46E-0

54.87E-0

6-2

.17E-0

5-1

.97E-0

53.24E-0

6-9

.20E-0

5-6

.12E-0

55.08E-0

5AdjTSRV

-1.34E-0

40

-1.23E-0

4-1

.17E-0

5-3

.10E-0

5-1

.62E-0

6-1

.39E-0

4-1

.49E-0

52.30E-0

5-1

.59E-0

4-4

.18E-0

5-2

.58E-0

5Av

RV

1min

-7.30E-0

5-1

.53E-0

50

1.31E-0

5-4

.46E-0

55.46E-0

6-9

.45E-0

5-1

.69E-0

55.89E-0

6-1

.48E-0

4-6

.12E-0

55.17E-0

5RV

5min

-1.44E-0

4-5

.20E-0

5-1

.38E-0

40

-3.92E-0

5-4

.72E-0

6-1

.46E-0

4-2

.74E-0

5-5

.44E-0

6-1

.63E-0

4-5

.64E-0

5-2

.95E-0

5AdjTSRV

5min

-1.45E-0

4-3

.87E-0

5-1

.35E-0

4-2

.14E-0

50

-9.05E-0

6-1

.47E-0

4-2

.81E-0

7-3

.85E-0

6-1

.63E-0

4-2

.73E-0

5-4

.98E-0

5Av

RV

5min

-1.43E-0

4-6

.12E-0

5-1

.33E-0

4-2

.31E-0

5-4

.69E-0

50

-1.47E-0

4-3

.71E-0

52.16E-0

5-1

.63E-0

4-6

.06E-0

5-3

.48E-0

5HHT

1min

-6.25E-0

6-7

.77E-0

67.52E-0

58.72E-0

6-4

.45E-0

56.56E-0

60

-1.72E-0

55.57E-0

6-9

.25E-0

5-6

.12E-0

55.43E-0

5HHT

5min

-1.42E-0

4-4

.64E-0

5-1

.35E-0

4-1

.62E-0

5-4

.14E-0

56.32E-0

6-1

.46E-0

40

1.47E-0

5-1

.63E-0

4-5

.70E-0

5-4

.63E-0

5HHT

Filte

red

-1.42E-0

4-8

.48E-0

5-1

.35E-0

4-3

.62E-0

5-5

.37E-0

5-5

.43E-0

5-1

.46E-0

4-5

.46E-0

50

-1.63E-0

4-6

.27E-0

5-6

.11E-0

5M

ax

7.39E-0

5-1

.30E-0

61.21E-0

41.11E-0

5-4

.39E-0

58.49E-0

67.65E-0

5-1

.68E-0

57.05E-0

60

-6.12E-0

55.30E-0

5M

in-1

.45E-0

4-2

.26E-0

5-1

.36E-0

4-1

.61E-0

51.01E-0

5-8

.97E-0

6-1

.47E-0

41.18E-0

5-6

.07E-0

6-1

.63E-0

40

-5.30E-0

5Avera

ge

-1.43E-0

4-4

.02E-0

5-1

.35E-0

4-1

.77E-0

5-4

.78E-0

5-1

.35E-0

5-1

.49E-0

4-2

.43E-0

51.07E-0

5-1

.63E-0

4-6

.12E-0

50

Hourly

Retu

rn-1

.04E-0

4-3

.48E-0

5-7

.51E-0

5-9

.06E-0

6-3

.89E-0

5-5

.50E-0

6-1

.10E-0

4-1

.98E-0

56.94E-0

6-1

.49E-0

4-5

.56E-0

5-8

.22E-0

6Hourly

Retu

rnStD

4.19E-0

44.22E-0

44.21E-0

44.24E-0

44.15E-0

44.24E-0

44.18E-0

44.23E-0

44.19E-0

44.03E-0

44.05E-0

44.23E-0

4Sim

ple

AnnualRetu

rn%

-62.92

-21.03

-45.40

-5.48

-23.50

-3.32

-66.30

-11.95

4.20

-89.82

-33.63

-4.97

AnnualRetu

rnStd

%3.26

3.28

3.28

3.30

3.22

3.29

3.25

3.29

3.26

3.14

3.15

3.29

Retu

rnRankin

g10

69

47

111

52

12

83

Table5.12:

GB

P/U

SD

Fair

pri

cetr

ad

ing

resu

lts

infu

ll

Meth

od

RV

1min

AdjTSRV

Av

RV

1min

RV

5min

AdjTSRV

5min

Av

RV

5min

HHT

1min

HHT

5min

HHT

Filte

red

Max

Min

Avera

ge

RV

1min

0-3

.01E-0

51.12E-0

54.31E-0

6-4

.16E-0

55.05E-0

7-1

.39E-0

4-3

.14E-0

51.29E-0

5-1

.67E-0

4-5

.21E-0

53.41E-0

5AdjTSRV

-7.93E-0

50

-7.46E-0

5-3

.81E-0

5-4

.09E-0

5-2

.53E-0

5-1

.65E-0

4-3

.93E-0

5-4

.21E-0

6-1

.75E-0

4-4

.91E-0

5-1

.65E-0

5Av

RV

1min

-3.48E-0

5-3

.27E-0

50

3.15E-0

6-4

.25E-0

52.06E-0

6-1

.67E-0

4-3

.10E-0

51.36E-0

5-1

.78E-0

4-5

.23E-0

53.74E-0

5RV

5min

-1.01E-0

4-2

.79E-0

5-9

.80E-0

50

-3.82E-0

5-1

.85E-0

5-1

.78E-0

4-3

.35E-0

59.46E-0

6-1

.85E-0

4-4

.96E-0

5-3

.29E-0

5AdjTSRV

5min

-1.06E-0

4-3

.76E-0

5-1

.03E-0

4-1

.88E-0

50

-4.46E-0

6-1

.77E-0

4-1

.77E-0

5-1

.26E-0

5-1

.85E-0

4-2

.19E-0

5-5

.04E-0

5Av

RV

5min

-1.06E-0

4-4

.39E-0

5-1

.05E-0

4-9

.45E-0

6-4

.25E-0

50

-1.78E-0

4-4

.47E-0

51.31E-0

5-1

.85E-0

4-5

.20E-0

5-4

.53E-0

5HHT

1min

6.23E-0

51.50E-0

59.09E-0

51.24E-0

5-3

.76E-0

52.32E-0

60

-2.71E-0

51.82E-0

5-3

.33E-0

5-5

.23E-0

55.24E-0

5HHT

5min

-1.06E-0

4-3

.45E-0

5-1

.04E-0

4-1

.51E-0

5-1

.32E-0

58.06E-0

6-1

.78E-0

40

-1.22E-0

5-1

.85E-0

4-3

.46E-0

5-5

.08E-0

5HHT

Filte

red

-1.06E-0

4-6

.02E-0

5-1

.05E-0

4-4

.95E-0

5-5

.04E-0

5-4

.75E-0

5-1

.78E-0

4-3

.86E-0

50

-1.85E-0

4-5

.47E-0

5-4

.24E-0

5M

ax

8.93E-0

52.45E-0

59.93E-0

51.24E-0

5-3

.58E-0

52.31E-0

62.70E-0

5-2

.71E-0

51.82E-0

50

-5.23E-0

55.24E-0

5M

in-1

.07E-0

4-3

.71E-0

5-1

.05E-0

4-1

.51E-0

55.33E-0

6-2

.66E-0

6-1

.78E-0

49.32E-0

6-1

.58E-0

5-1

.85E-0

40

-5.24E-0

5Avera

ge

-9.55E-0

5-4

.03E-0

5-9

.56E-0

5-1

.11E-0

5-4

.04E-0

5-6

.18E-0

6-1

.78E-0

4-2

.91E-0

52.76E-0

6-1

.85E-0

4-5

.23E-0

50

Hourly

Retu

rn-6

.27E-0

5-2

.77E-0

5-5

.36E-0

5-1

.13E-0

5-3

.43E-0

5-8

.12E-0

6-1

.54E-0

4-2

.82E-0

53.96E-0

6-1

.68E-0

4-4

.76E-0

5-1

.04E-0

5Hourly

Retu

rnStD

4.02E-0

44.05E-0

44.00E-0

44.03E-0

43.93E-0

44.03E-0

43.88E-0

43.96E-0

44.00E-0

43.84E-0

43.82E-0

44.02E-0

4Sim

ple

AnnualRetu

rn%

-37.90

-16.76

-32.43

-6.86

-20.77

-4.91

-93.02

-17.04

2.39

-101.52

-28.77

-6.29

AnnualRetu

rnStd

%3.12

3.15

3.11

3.13

3.06

3.13

3.02

3.08

3.11

2.98

2.97

3.12

Retu

rnRankin

g10

59

47

211

61

12

83

Chapter 6

Discussion & Conclusion

6.1 Discussion

The world is full of complex systems and processes that change their stochastic behaviour

over time. By restricting our attention to financial time series the implications of this

research have not been fettered or narrowed, the financial setting is merely a focal point

for the methodology developed herein.

The goal of this thesis was essentially to explore low and high frequency volatility

estimation in the light of a new tool, the HHT, which had hitherto been largely neglected

in this context. To that end, clear contributions have been made in the realms of low and

high frequency volatility estimation using the HHT procedure, which has again proved

its worth and demonstrated that its true value lies in its flexibility.

The HHT was successfully used to break down complicated financial time series infor-

mation into simpler wave like structures which were more amenable to intuitive analysis.

This ability to decompose complicated signals is demonstrated by the spectral analysis

components of Sections 2.2.3 & 4.4.1. In these sections, the EMD process at the heart of

the HHT procedure can be seen to act as an effective band-pass filter and the spacings

between frequencies clearly indicate that it performs as a dyadic filter, a result which can

112

Chapter 6. Discussion & Conclusion 113

also be achieved with some wavelet transforms. This parallel with wavelets may lead

to further developments and a more complete mathematical description of the EMD

process which has so far proved an intractable problem for researchers due to the adap-

tive nature of the EMD algorithm. Furthermore, this filter bank structure allows for a

convenient method of estimating the Hurst exponent which can describe the autocorre-

lation structure of the data. In particular, a change in the Hurst exponent at different

sampling rates indicates that a change in autocorrelation structure has occurred and the

self similarity of the time series may not be preserved through intertemporal aggregation.

In Chapter 3, the filtering nature of the EMD process was also put to use as a means

to generate a proxy to the returns by subtracting a local mean (made up of low frequency

components) from the time series. Again, the filtration structure was used in Chapters

4 & 5 to effectively sift off high frequency noise from a return series. This generated an

approximation to the “true” or frictionless price as well as a volatility estimate which

was free from the inflationary effects of market microstructure.

A parametric HHT based procedure for estimating low frequency volatility is also

proposed and tested in Chapter 3. The procedure is further refined by the completely

non-parametric approach taken in Section 4.1. The later HHT based method needs

no in-sample data for parameter fitting and makes no assumptions on the stochastic

process driving the time series. Also, in the low frequency realm the HHT method is

able to extract key volatility features with a high temporal resolution while maintaining

relatively low levels of noise, a claim which is substantiated in Section 4.2. Furthermore,

perhaps the best feature of the methods developed over the course of this thesis is their

intuitive simplicity. This elegant simplicity comes from the ability of the HHT to rep-

resent seemingly chaotic signals in terms simpler wave like structures that combine to

form more complex ones.

In Chapter 4, a benchmark test was developed specifically to test this claim that

the HHT procedure could adapt quickly to changes in the level of volatility level, while


maintaining a low signal to noise ratio. Simulated data and this benchmark were used

to test the claim against the EWMA procedure for many values of the decay parameter

λ. To some satisfaction, the results can all be summarised in one simple plot given by

Figure 4.3, which does indeed show the claim to be true under test conditions.

As an introduction to market microstructure and as an exercise in spectral analysis,

the HHT procedure was then used to estimate the Hurst exponent at different levels

of market microstructure for the three kinds of stochastic volatility processes used in

Chapter 4. A drop in the Hurst exponent would indicate an increasingly rough time

series with a short term memory, just like the fGn studied in Chapter 2. This is exactly

what was observed, with (H < 0.5) indicating the presence of inflationary market mi-

crostructure noise. Markets with alternative kinds of market microstructure noise have

also been considered over the course of this research, some of which have a retarding or

dampening effect on volatility estimates at higher frequencies. These are often associ-

ated with long term memory processes (H > 0.5). Typically this deflationary behaviour

is observed in index data, such as the All Ords Index covering 1st Jan 2000- 31st Dec

2009. The ability of the HHT to estimate the Hurst exponent opens the possibility for

it to be used to indicate the presence and extent of market microstructure noise.

Perhaps the largest contribution the HHT procedure has to make to the study of

volatility estimation is its inherent flexibility. There is no requirement for evenly spaced

data and the approach described by Equation (4.6) is completely parameter free, requir-

ing no lead in data or parameter fitting. The improved speed this approach has over its

predecessor developed in Chapter 3 has enabled more complex methods to be developed

which can deal with the problems of high frequency volatility estimation.

The Filtered-HHT method was then proposed to deal with market microstructure

noise, comparisons with TSRV based methods in a simulated study followed. Results

indicated that the the proposed Filtered-HHT procedure was capable of adjusting for

high levels of market microstructure noise. While most of the error metrics were slightly


higher for the HHT based procedure, it is stipulated that this was largely due to the

fact that the TSRV methods were specifically designed to adjust for the kind of additive

noise used in the simulated data studied in Chapter 4, while the HHT based approach

remains more general.

In an effort to test the Filtered-HHT procedure with real high frequency data, a sim-

ulated options market was constructed in Chapter 5. This simulated market was based

on several competing volatility forecasts and used real high frequency exchange data.

This approach of using a simulated options market is a novel and effective way to assess

forecast performance on an unobservable variable. The purpose of this test was to see

how the different volatility estimates performed when real market microstructure was

encountered, microstructure which isn’t the simple additive noise used in the simulations

of Chapter 4. Another interesting aspect of this chapter is the more complex evaluation

criteria, while other chapters used measures like ME, MSE, MAPE, etc, this simulated

study involved a nonlinear pricing formula (the Black-Scholes equation), which added an

element of realism to the testing. The results for this chapter also indicate that the HHT

based approach combined with the frequency filtering capability of the EMD process

was able to accurately extract volatility levels in the presence of market microstructure.

Furthermore, this approach met with more success than its RV based competitors for

these simulations, perhaps owing to the lack of assumptions made by the HHT approach.

One possible drawback of the simulated options market approach is that the per-

formance indicators were entirely dependent on the methods that were chosen to be

included in the market. While the results obtained still provided an effective and infor-

mative rankings and pairwise comparisons, they are harder to quantify outside of the

context of this simulated market. For example, the average profits and losses for each

method are entirely dependent on the presence of the other eleven simulated market

participants, some of which (like the one denoted as RV 1 min) have no place in a high

frequency market and were only included to show how poorly they perform under such


conditions.

6.2 Future Work

The global financial markets are experiencing a paradigm shift in the way trading is

conducted. The trend is shifting away from speculators calling their brokers to make

long term investments and towards automated systems which trade against one another

in an environment where mere nanoseconds matter. In such a marketplace, having the

most up to date volatility estimate is a clear advantage.

One of the key features of the HHT procedure is that it can yield instantaneous,

rather than integrated volatility estimates. For clarification, the term ‘instantaneous

volatility’ isn’t referring to its calculation time, rather that volatility estimates are given

for every point of a time series. Furthermore, due to the cubic splining procedure used

in the EMD phase of the HHT process, IMFs produced and hence volatility estimates

are actually continuous functions. While the procedures outlined in this thesis are ca-

pable of producing such a high frequency estimate, as demonstrated by Figure 5.2, it is

a goal of future work to improve upon these instantaneous measurements and test their

performance as short term forecasters. There are two approaches currently undergoing

development. The first approach is to rescale IMFs so that the power spectra produced

are of equal power, the second involves making assumptions on the underlying auto-

correlation structure of the noise and taking it into account in a similar manner to the

TSRV method.

At the high frequencies considered for the simulated options market of Chapter 5,

many of the assumptions underlying the Black-Scholes model become invalid. In particu-

lar, the assumption that prices come from a continuous distribution becomes particularly

strained at scales when market microstructure starts to dominate and returns become

highly discretised. To account for this, an option pricing formula based on more discrete


stock movements such as a binomial or trinomial method could be used and the simu-

lated options market study could be redone under a more discretised framework which

may be better suited to the behaviour of markets at high frequency. The trinomial

model developed by Haahtela [72] would be particularly well suited to this as it allows

for changing volatility estimates over time.

There is also much interest in the relationship between volume, volatility and price.

A higher dimensional version of the EMD algorithm such as the one proposed by Sinclair

and Pegram [73] could be used to explore the relationship between volume, volatility

and price. This would allow for the creation of a three dimensional surface which, it

is hoped, will shed some light on how these variables interact with one another and

perhaps lead to better forecasting models.

Some research was also done into potential improvements in the way overnight re-

turns are handled with low frequency data. The research explored the idea of using the

opening and closing prices of a stock or index to split the information into two corre-

lated time series which could then be recombined to give a more accurate result. While

promising, further research into the idea was halted after the the work done by Bertram

[74] was found to cover the concept in sufficient depth.

Assessing the performance of the Filtered-HHT procedure on different kinds of mar-

ket microstructure noise is also an area of interest. Further work in this area would help

substantiate claims made on the flexibility of the methods developed herein. One line

of inquiry which was partially explored over the course of this thesis, are the random

processes generated by the Karhunen-Loeve transform. There are many similarities be-

tween the way this transform constructs random walks and the way in which the HHT

procedure breaks them back down again and so this could be the starting point in

an investigation into long memory processes with microstructure noise that attenuates

volatility estimates rather than inflates them.


6.3 Concluding Remarks

This thesis has provided an in depth look at the potential for the HHT procedure as a

volatility estimator. A procedure which has largely been ignored by financial practition-

ers has now been built up to the point where it could be considered an effective tool in

both the high and low frequency realms.

The results of several simulated experiments and two studies using real financial data

all indicate that the HHT based volatility estimators are competitively accurate with

popular alternatives and are worthy of further research. Furthermore, the interpretation

of volatility as a sum of wave amplitudes has a certain intuitive and elegant simplicity

which is appealing in its own right.

Bibliography

[1] Norden E. Huang, Zheng Shen, Steven R. Long, Manli C. Wu, Hsing H. Shih, Qua-

nan Zheng, Nai-Chyuan Yen, Chi Chao Tung, and Henry H. Liu. The empirical

mode decomposition and the Hilbert spectrum for nonlinear and non-stationary

time series analysis. Proceedings of the Royal Society of London. Series A: Mathe-

matical, Physical and Engineering Sciences, 454(1971):903–995, 1998.

[2] Harry M. Markowitz. Portfolio Selection. Blackwell Publishers, Malden, MA, 2nd

edition, 1991.

[3] Ser-Huang Poon and Clive W.J. Granger. Forecasting Volatility in Financial Mar-

kets: A Review. Journal of Economic Literature, 41:478–539, 2003.

[4] Robert C. Merton. On estimating the expected return on the market : An ex-

ploratory investigation. Journal of Financial Economics, 8(4):323–361, December

1980.

[5] Ioannis Karatzas and Steven E. Shreve. Brownian Motion and Stochastic Calculus.

Springer US, 1988.

[6] Torben G. Andersen, Tim Bollerslev, Francis X. Diebold, and Paul Labys. Modeling

and Forecasting Realized Volatility. Econometrica, 71(2):579–625, March 2003.

[7] Torben G. Andersen, Tim Bollerslev, Francis X. Diebold, and Paul Labys. (Under-

standing, Optimizing, Using and Forecasting) Realized Volatility and Correlation.

New York University, Leonard N. Stern school finance department working paper

seires, 1999.

119

Bibliography 120

[8] Shyan-Rong Chou, Andy Chien, Chang-Cheng Changchien, and Chien-Hui Wu. A

Comparison of the Forecasting Volatility Performance on EWMA Family Models.

International Research Journal of Finance and Economics, 54:19–28, 2010.

[9] Jie Ding and Nigel Meade. Forecasting accuracy of stochastic volatility, GARCH

and EWMA models under different volatility scenarios. Applied Financial Eco-

nomics, 20:771–783, 2010.

[10] P. Wilmott. Paul Wilmott on Quantitative Finance, volume 3. Springer, 2006.

[11] Robert F. Engle. Autoregressive Conditional Heteroskedasticity with Estimates of

the Variance of United Kingdom Inflation. Econometrica, 50:987–1007, 1982.

[12] Benoit Mandelbrot. The Variation of Certain Speculative Prices. The Journal of

Business, 36:394, 1963.

[13] Tim Bollerslev. Generalized Autoregressive Conditional Heteroskedasticity. Journal

of Econometrics, 31:307–327, 1986.

[14] F. Black. Studies of stock price volatility changes. In Proceedings of the 1976

Meeting of the Business and Economic Statistics Section. American Statistical As-

sociation, Washington, D.C, 1976.

[15] Lawrence R. Glosten, Ravi Jagannathan, and David E. Runkle. On the relation

between the expected value and the volatility of the nominal excess return on stocks.

Staff Report 157, Federal Reserve Bank of Minneapolis, 1993.

[16] Daniel B. Nelson. Conditional Heteroskedasticity in Asset Returns: A New Ap-

proach. Econometrica, 59(2):347–70, March 1991.

[17] Robert F. Engle, David M. Lilien, and Russell P. Robins. Estimating Time Varying

Risk Premia in the Term Structure: The ARCH-M Model. Econometrica, 55(2):

391–407, March 1987.

[18] Yacine Ait-Sahalia, Jianqing Fan, and Yingying Li. The Leverage Effect Puzzle:

Disentangling Sources of Bias at High Frequency. Working Paper 17592, National

Bureau of Economic Research, November 2011.

Bibliography 121

[19] Torben G. Andersen, Tim Bollerslev, and Francis X. Diebold. Parametric and

Nonparametric Volatility Measurement. Working Paper 279, National Bureau of

Economic Research, August 2002.

[20] Stephen Figlewski. Forecasting Volatility. Financial Markets, Institutions & In-

struments, 6(1):1–88, 1997.

[21] M. Martens. Measuring and Forecasting S&P 500 Index-Futures Volatility Using

High-Frequency Data. Journal of Futures Markets, 22:497–518, 2002.

[22] Torben G. Andersen, Tim Bollerslev, and Francis X. Diebold. Parametric and

Nonparametric Volatility Measurement. NBER Technical Working Papers 0279,

National Bureau of Economic Research, Inc, August 2002.

[23] F. M. Bandi and J. R. Russell. Microstructure Noise, Realized Variance, and Op-

timal Sampling. The Review of Economic Studies, 75(2):339–369, 2008.

[24] Lan Zhang, Per A. Mykland, and Yacine Ait-Sahalia. A Tale of Two Time Scales:

Determining Integrated Volatility With Noisy High-Frequency Data. Journal of the

American Statistical Association, 100:1394–1411, December 2005.

[25] Fulvio Corsi, Gilles Zumbach, Ulrich Muller, and Michel Dacorogna. Consistent

High-Precision Volatility from High-Frequency Data. Economic Notes, 30:183–204,

2001.

[26] Torben G. Andersen, Tim Bollerslev, Francis X. Diebold, and Paul Labys. The Dis-

tribution Of Realized Exchange Rate Volatility. Journal of the American Statistical

Association, 96(453):42–55, March 2001.

[27] Yacine Ait-Sahalia, Per A. Mykland, and Lan Zhang. Ultra high frequency volatility

estimation with dependent microstructure noise. Journal of Econometrics, 160(1):

160–175, January 2011.

[28] Ole E. Barndorff-Nielsen, Peter Reinhard Hansen, Asger Lunde, and Neil Shephard.

Designing Realized Kernels to Measure the ex post Variation of Equity Prices in

the Presence of Noise. Econometrica, 76(6):1481–1536, 2008.

Bibliography 122

[29] Peter R. Hansen, Jeremy Large, and Asger Lunde. Moving Average-Based Estima-

tors of Integrated Variance. Econometric Reviews, 27(1-3):79–111, 2008.

[30] H. Ebens. Realized stock index volatility. Working Paper 420, Department of

Economics Johns Hopkins University, Baltimore, 1999.

[31] G. Curci and F. Corsi. A discrete sine transform approach for realized volatility

measurement. Working paper, University of Southern Switzerland, 2004.

[32] D. Donnelly. The Fast Fourier and Hilbert-Huang Transforms: A Comparison.

In Proceedings of the Multiconference on Computational Engineering in Systems

Applications, pages 84–88, Beijing, China, 2006.

[33] Patrick Flandrin, Paulo Goncalves, and Gabriel Rilling. EMD equivalent filter

banks, from interpretation to applications. In Hilbert-Huang Transform and Its

Applications, pages 57–73. World Scientific Publishing Co. Pte. Ltd., 2005.

[34] P. Flandrin, G. Rilling, and P. Goncalves. Empirical mode decomposition as a filter

bank. Signal Processing Letters, IEEE , 11(2), 2004.

[35] H.G. Li and G. Meng. Detection of harmonic signals from chaotic interference by

empirical mode decomposition. Chaos, Solitons & Fractals , 30(4):930–935, 2006.

[36] Zhaohua Wu and Norden E. Huang. Statistical significance test of intrinsic mode

functions. In Hilbert-Huang Transform and Its Applications, pages 107–127. World

Scientific Publishing Co. Pte. Ltd., 2005.

[37] Zhaohua Wu and Norden E. Huang. A study of the characteristics of white noise

using the empirical mode decomposition method. Proceedings of the Royal Society

of London A: Mathematical, Physical and Engineering Sciences, 460(2046):1597–

1611, 2004.

[38] Bao Liu, S. Riemenschneider, and Y. Xu. Gearbox fault diagnosis using empir-

ical mode decomposition and Hilbert spectrum. Mechanical Systems and Signal

Processing, 20(3):718–734, 2006.

Bibliography 123

[39] N. Senroy, S. Suryanarayanan, and P.F. Ribeiro. An Improved Hilbert-Huang

Method for Analysis of Time-Varying Waveforms in Power Quality. IEEE Trans-

actions on Power Systems , 22:1843–1850, 2007.

[40] T. Yalcin, O. Ozgonenel, and U. Kurt. Feature vector extraction by using empirical

mode decomposition for power quality disturbances. In Proceedings of the 10th

International Conference on Environment and Electrical Engineering , pages 1–4,

May 2011.

[41] Md Khademul Islam Molla, Akimasa Sumi, and M. Sayedur Rahman. Analysis of

temperature change under global warming impact using empirical mode decompo-

sition. International Journal of Information Technology, 3(2):131–139, 2006.

[42] A.R. Rao and E.-C. Hsu. Hilbert-Huang Transform Analysis of Hydrological and

Environmental Time Series. Springer, 2008.

[43] Paul A. Hwang, Norden E. Huang, David W. Wang, and James M. Kaihatu. Hilbert

Spectra of Nonlinear Ocean Waves. In Hilbert-Huang Transform and Its Applica-

tions, pages 211–225. World Scientific Publishing Co. Pte. Ltd., 2005.

[44] Derek A.T. Cummings, Rafael A. Irizarry, Norden E. Huang, Timothy P. Endy,

Ananda Nisalak, Kumnuan Ungchusak, and Donald S. Burke. Travelling waves

in the occurrence of dengue haemorrhagic fever in Thailand. Nature, 427(6972):

344–347, 2004.

[45] Xun Zhang, KK Lai, and Shou-Yang Wang. A new approach for crude oil price

analysis based on Empirical Mode Decomposition. Energy economics, 30(3):905–

918, 2008.

[46] P. Crowley. How Do You Make A Time Series Sing Like a Choir? Extracting

Embedded Frequencies from Economic and Financial Time Series using Empirical

Mode Decomposition. Studies in Nonlinear Dynamics & Econometrics, 16(5), 2012.

[47] Norden E. Huang, Man-Li Wu, Wendong Qu, Steven R. Long, and Samuel S. P.

Shen. Applications of Hilbert-Huang transform to non-stationary financial time

series analysis. Applied Stochastic Models in Business and Industry, 19(3):245–268,

2003.

Bibliography 124

[48] Y. Kwei Wen and Ping Gu. The Hilbert-Huang Transform in Engineering. CRC

Press, 2005.

[49] Jacques Longerstaey and Martin Spencer. RiskMetrics Technical Document. Mor-

gan Guaranty Trust Company of New York: New York, 1996.

[50] D. Ruppert. Statistics and Data Analysis for Financial Engineering. Springer Texts

in Statistics. Springer, 2010.

[51] Petra Posedel. Properties and Estimation of GARCH(1,1) Model. metodoloski

zvezki, 2(2):243–257, 2005.

[52] P. Reinhard Hansen and A. Lunde. Does anything beat a GARCH(1,1)? A com-

parison based on test for superior predictive ability. In Proceedings of the IEEE

International Conference on Computational Intelligence for Financial Engineering,

pages 301–307, 2003.

[53] Torben G. Andersen, Tim Bollerslev, and Nour Meddahi. Realized volatility fore-

casting and market microstructure noise. Journal of Econometrics, 160(1):220–234,

January 2011.

[54] Emilio Barucci, Davide Magno, and Maria Elvira Mancino. Fourier volatility fore-

casting with high-frequency data and microstructure noise. Quantitative Finance,

12(2):281–293, September 2012.

[55] Ole E. Barndorff-Nielsen and Neil Shephard. Econometric analysis of realized

volatility and its use in estimating stochastic volatility models. Journal of the

Royal Statistical Society: Series B (Statistical Methodology), 64(2):253–280, 2002.

[56] Eugenie Hol and Siem J. Koopman. Stock Index Volatility Forecasting with High

Frequency Data. Tinbergen Institute, Tinbergen Institute Discussion Papers, 2002.

[57] Ruey S. Tsay. Analysis on Financial Time Series. Wiley, Hoboken, NJ, 3 edition,

2010.

[58] Siem J. Koopman, B. Jungbacker, and Eugenie Hol. Forecasting daily variability of

the S&P 100 stock index using historical, realised and implied volatility measure-

ments. Journal of Empirical Finance, 12:445, 2005.

Bibliography 125

[59] Peter Reinhard Hansen and Asger Lunde. A realized variance for the whole day

based on intermittent high-frequency data. Journal of Financial Econometrics, 3

(4):525–554, 2005.

[60] Peter R. Hansen and Asger Lunde. A forecast comparison of volatility models: does

anything beat a GARCH(1,1)? Journal of Applied Econometrics, 20(7):873–889,

2005.

[61] Patrick Flandrin. EMD code, 2007. URL http://perso.ens-lyon.fr/patrick.

flandrin/emd.html.

[62] Norden E. Huang, Man-Li C. Wu, Steven R. Long, Samuel S.P. Shen, Wendong

Qu, Per Gloersen, and Kuang L. Fan. A confidence limit for the empirical mode

decomposition and Hilbert spectral analysis. Proc. R. Soc. Lond., 459(2037), 2003.

[63] G. Rilling, P. Flandrin, and P. Goncalves. Empirical mode decomposition, frac-

tional Gaussian noise and Hurst exponent estimation. In Proceedings of the IEEE

International Conference on Acoustics, Speech, and Signal Processing , volume 4,

pages 489–492, March 2005.

[64] R. Deering and J.F. Kaiser. The use of a masking signal to improve empirical mode

decomposition. In Proceedings of the IEEE International Conference on Acoustics,

Speech, and Signal Processing , volume 4, pages 485–488, 2005.

[65] R. Rilling and P. Flandrin. One or Two Frequencies? The Empirical Mode De-

composition Answers. IEEE Transactions on Signal Processing, 56 Issue:1:85–95,

2008.

[66] Jr. Massey, Frank J. The Kolmogorov-Smirnov Test for Goodness of Fit. Journal

of the American Statistical Association, 46(253):68–78, 1951.

[67] Robert F. Engle, Che-Hsiung Hong, and Alex Kane. Valuation of Variance Forecast

with Simulated Option Markets. NBER Working Papers 3350, National Bureau of

Economic Research, Inc, 1990.

[68] Federico M. Bandi, Jeffrey R. Russell, and Chen Yang. Realized volatility forecast-

ing and option pricing. Journal of Econometrics, 147(1):34–46, 2008.

http://perso.ens-lyon.fr/patrick.flandrin/emd.html

http://perso.ens-lyon.fr/patrick.flandrin/emd.html

Bibliography 126

[69] Luc Bauwens, Sebastien Laurent, and Jeroen V. K. Rombouts. Multivariate

GARCH models: a survey. Journal of Applied Econometrics, 21(1):79–109, 2006.

[70] Canyang Hu. Simulation of nonstationary processes based on orthogonal Hilbert-

Huang transform. In Proceedings of the 3rd International Congress on Image and

Signal Processing, volume 7, pages 3111–3115, Oct 2010.

[71] Lan Zhang. Efficient estimation of stochastic volatility using noisy observations: a

multi-scale approach. Bernoulli, 12(6):1019–1043, December 2006.

[72] T. Haahtela. Recombining Trinomial Tree for Real Option Valuation with Changing

Volatility. In Proceedings of the 14th Annual International Conference on Real

Options, Rome, Italy, 2010.

[73] S. Sinclair and G. G. S. Pegram. Empirical Mode Decomposition in 2-D space and

time: a tool for space-time rainfall analysis and nowcasting. Hydrology and Earth

System Sciences, 9:127–137, 2005.

[74] William K. Bertram. An empirical investigation of Australian Stock Exchange data.

Physica A: Statistical Mechanics and its Applications, 341(C):533–546, 2004.

Documents

Volatility Analysis and the Hilbert-Huang Transform