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ARCH Models for
Financial Applications
Evdokia Xekalaki . Stavros DegiannakisDepartment of Statistics
Athens University of Economics and Business, Greece
ARCH Models for
Financial Applications
ARCH Models for
Financial Applications
Evdokia Xekalaki . Stavros DegiannakisDepartment of Statistics
Athens University of Economics and Business, Greece
This edition first published 2010
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Library of Congress Cataloguing-in-Publication Data
Xekalaki, Evdokia.
ARCH models for financial applications / Evdokia Xekalaki, Stavros Degiannakis.
p. cm.
Includes bibliographical references and index.
ISBN 978-0-470-06630-0 (cloth)
1. Finance–Mathematical models. 2. Autoregression (Statistics) I. Degiannakis, Stavros. II. Title.
HG106.X45 2010
332.01’519536–dc22
2009052104
A catalogue record for this book is available from the British Library.
ISBN 978-0-470-06630-0 (H/B)
Set in 10/12pt Times by Thomson Digital, Noida, India
Printed and bound in Great Britain by TJ International, Padstow, Cornwall
To my husband and my son, a wonderful family
Evdokia Xekalaki
To the memory of the most important person in my life, my father Antonis,
and to my mother and my brother
Stavros Degiannakis
Contents
Preface xi
Notation xv
1 What is an ARCH process? 1
1.1 Introduction 1
1.2 The autoregressive conditionally heteroscedastic process 8
1.3 The leverage effect 13
1.4 The non-trading period effect 15
1.5 The non-synchronous trading effect 15
1.6 The relationship between conditional variance and
conditional mean 16
1.6.1 The ARCH in mean model 16
1.6.2 Volatility and serial correlation 18
2 ARCH volatility specifications 19
2.1 Model specifications 19
2.2 Methods of estimation 23
2.2.1 Maximum likelihood estimation 23
2.2.2 Numerical estimation algorithms 25
2.2.3 Quasi-maximum likelihood estimation 28
2.2.4 Other estimation methods 29
2.3 Estimating the GARCH model with EViews 6:
an empirical example 31
2.4 Asymmetric conditional volatility specifications 42
2.5 Simulating ARCH models using EViews 49
2.6 Estimating asymmetric ARCH models with G@RCH 4.2
OxMetrics: an empirical example 55
2.7 Misspecification tests 66
2.7.1 The Box–Pierce and Ljung–Box Q statistics 66
2.7.2 Tse’s residual based diagnostic test for
conditional heteroscedasticity 67
2.7.3 Engle’s Lagrange multiplier test 67
2.7.4 Engle and Ng’s sign bias tests 68
2.7.5 The Breusch–Pagan, Godfrey, Glejser, Harvey and White tests 69
2.7.6 The Wald, likelihood ratio and Lagrange multiplier tests 69
2.8 Other ARCH volatility specifications 70
2.8.1 Regime-switching ARCH models 70
2.8.2 Extended ARCH models 72
2.9 Other methods of volatility modelling 76
2.10 Interpretation of the ARCH process 82
Appendix 86
3 Fractionally integrated ARCH models 107
3.1 Fractionally integrated ARCH model specifications 107
3.2 Estimating fractionally integrated ARCH models using
G@RCH 4.2 OxMetrics: an empirical example 111
3.3 A more detailed investigation of the normality of the
standardized residuals: goodness-of-fit tests 122
3.3.1 EDF tests 123
3.3.2 Chi-square tests 124
3.3.3 QQ plots 125
3.3.4 Goodness-of-fit tests using EViews and G@RCH 126
Appendix 129
4 Volatility forecasting: an empirical example using EViews 6 1434.1 One-step-ahead volatility forecasting 143
4.2 Ten-step-ahead volatility forecasting 150
Appendix 154
5 Other distributional assumptions 163
5.1 Non-normally distributed standardized innovations 163
5.2 Estimating ARCH models with non-normally distributed
standardized innovations using G@RCH 4.2 OxMetrics:
an empirical example 168
5.3 Estimating ARCH models with non-normally distributed
standardized innovations using EViews 6: an empirical example 174
5.4 Estimating ARCH models with non-normally distributed
standardized innovations using EViews 6: the logl object 176
Appendix 182
6 Volatility forecasting: an empirical example using G@RCH Ox 185
Appendix 195
7 Intraday realized volatility models 217
7.1 Realized volatility 217
7.2 Intraday volatility models 220
7.3 Intraday realized volatility and ARFIMAX models
in G@RCH 4.2 OxMetrics: an empirical example 223
7.3.1 Descriptive statistics 223
viii CONTENTS
7.3.2 In-sample analysis 228
7.3.3 Out-of-sample analysis 232
8 Applications in value-at-risk, expected shortfall and options pricing 2398.1 One-day-ahead value-at-risk forecasting 239
8.1.1 Value-at-risk 239
8.1.2 Parametric value-at-risk modelling 240
8.1.3 Intraday data and value-at-risk modelling 242
8.1.4 Non-parametric and semi-parametric value-at-risk modelling 244
8.1.5 Back-testing value-at-risk 245
8.1.6 Value-at-risk loss functions 248
8.2 One-day-ahead expected shortfall forecasting 248
8.2.1 Historical simulation and filtered historical simulation
for expected shortfall 251
8.2.2 Loss functions for expected shortfall 251
8.3 FTSE100 index: one-step-ahead value-at-risk and expected
shortfall forecasting 252
8.4 Multi-period value-at-risk and expected shortfall forecasting 258
8.5 ARCH volatility forecasts in Black–Scholes option pricing 260
8.5.1 Options 261
8.5.2 Assessing the performance of volatility forecasting methods 269
8.5.3 Black–Scholes option pricing using a set of ARCH processes 270
8.5.4 Trading straddles based on a set of ARCH processes 271
8.5.5 Discussion 279
8.6 ARCH option pricing formulas 281
8.6.1 Computation of Duan’s ARCH option prices: an example 286
Appendix 288
9 Implied volatility indices and ARCH models 3419.1 Implied volatility 341
9.2 The VIX index 342
9.3 The implied volatility index as an explanatory variable 344
9.4 ARFIMAX model for implied volatility index 349
Appendix 352
10 ARCH model evaluation and selection 357
10.1 Evaluation of ARCH models 358
10.1.1 Model evaluation viewed in terms of information criteria 359
10.1.2 Model evaluation viewed in terms of statistical
loss functions 360
10.1.3 Consistent ranking 367
10.1.4 Simulation, estimation and evaluation 377
10.1.5 Point, interval and density forecasts 383
10.1.6 Model evaluation viewed in terms of loss functions based
on the use of volatility forecasts 384
CONTENTS ix
10.2 Selection of ARCH models 386
10.2.1 The Diebold–Mariano test 386
10.2.2 The Harvey–Leybourne–Newbold test 389
10.2.3 The Morgan–Granger–Newbold test 389
10.2.4 White’s reality check for data snooping 390
10.2.5 Hansen’s superior predictive ability test 390
10.2.6 The standardized prediction error criterion 393
10.2.7 Forecast encompassing tests 400
10.3 Application of loss functions as methods of model selection 401
10.3.1 Applying the SPEC model selection method 401
10.3.2 Applying loss functions as methods of model selection 402
10.3.3 Median values of loss functions as methods
of model selection 407
10.4 The SPA test for VaR and expected shortfall 408
Appendix 410
11 Multivariate ARCH models 445
11.1 Model Specifications 446
11.1.1 Symmetric model specifications 446
11.1.2 Asymmetric and long-memory model specifications 453
11.2 Maximum likelihood estimation 454
11.3 Estimating multivariate ARCH models using EViews 6 456
11.4 Estimating multivariate ARCH models using G@RCH 5.0 465
11.5 Evaluation of multivariate ARCH models 473
Appendix 475
References 479
Author Index 521
Subject Index 533
x CONTENTS
Preface
There has been wide interest throughout the financial literature on theoretical and
applied problems in the context of ARCH modelling. While a plethora of articles
exists in various international journals, the literature has been rather sparse when it
comes to books with an exclusive focus on ARCH models. As a result, students,
academics in the area of finance and economics, and professional economists with
only a superficial grounding in the theoretical aspects of econometric modelling,
while able to understand the basic theories about model construction, estimation and
forecasting, often fail to get a grasp of how these can be used in practice.
The present book addresses precisely these issues by interweaving practical
questions with approaches hinging on financial and statistical theory: we have
adopted an interactional exposition of the ARCH theory and its implementation
throughout. This is a book of practical orientation and applied nature intended for
readers with a basic knowledge of time series analysis wishing to gain an aptitude in
the applications of financial econometric modelling. Balancing statistical methodol-
ogy and structural descriptive modelling, it aims to introduce readers to the area of
discrete time applied stochastic volatility models and to help them acquire the ability
to deal with applied economic problems. It provides background on the theory of
ARCH models, but with a focus on practical implementation via applications to real
data (the accompanying CD-ROM provides programs and data) and via examples
worked with econometrics packages (EViews and the G@RCH module for the Ox
package) with step-by-step explanations of their use. Readers are familiarized with
theoretical issues of ARCH models from model construction, fitting and forecasting
through to model evaluation and selection, and will gain facility in employing these
models in the context of financial applications: volatility forecasting, value-at-risk
forecasting, expected shortfall estimation, and volatility forecasts for pricing options.
Chapter 1 introduces the concept of an autoregressive conditionally heterosce-
dastic (ARCH) process and discusses the effects that various factors have on financial
time series such as the leverage effect, the non-trading period effect, and the non-
synchronous trading effect. Chapter 2 provides an anthology of representations of
ARCH models that have been considered in the literature. Estimation and simulation
of the models is discussed, and several misspecification tests are provided. Chapter 3
deals with fractionally integrated ARCH models and discusses a series of tests for
testing the hypothesis of normality of the standardized residuals. Chapter 4 famil-
iarizes readers with the use of EViews in obtaining volatility forecasts. Chapter 5
treats the case of ARCH models with non-normally distributed standardized
innovations – in particular, models with innovations with Student t, beta, Paretian or
Gram–Charlier type distributions, aswell as generalized error distributions. Chapter 6
acquaints readers with the use of G@RCH in volatility forecasting. Chapter 7
introduces realized volatility as an alternative volatility measure. The use of high-
frequency returns to compute volatility at a lower frequency and the prediction of
volatility with ARFIMAXmodels are presented. Chapter 8 illustrates applications of
volatility forecasting in risk management and options pricing. Step-by-step empirical
applications provide an insight into obtaining value-at-risk estimates and expected
shortfall forecasts. An options trading game driven by volatility forecasts produced by
various methods of ARCH model selection is illustrated, and option pricing models
for asset returns that conform to anARCHprocess are discussed. Chapter 9 introduces
the notion of implied volatility and discusses implied volatility indices and their use in
ARCH modelling. It also discusses techniques for forecasting implied volatility.
Chapter 10 deals with evaluation and selection of ARCH models for forecasting
applications. The topics of consistent ranking and of proxy measures for the actual
variance are extensively discussed and illustrated via simulated examples. Statistical
tests for testing whether a model yields statistically significantly more accurate
volatility forecasts than its competitors are presented, and several examples illustrat-
ing methods of model selection are given. Finally, Chapter 11 introduces multivariate
extensions of ARCH models and illustrates their estimation using EViews and
G@RCH.
The contents of the book have evolved from lectures given to postgraduate and
final-year undergraduate students at the Athens University of Economics and
Business and at the University of Central Greece. Readers are not expected to have
prior knowledge of ARCH models and financial markets; they only need a basic
knowledge of time series analysis or econometrics, alongwith some exposure to basic
statistical topics such as inference and regression, at undergraduate level.
The book is primarily intended as a text for postgraduate and final-year under-
graduate students of economic, financial, business and statistics courses. It is also
intended as a reference book for academics and researchers in applied statistics and
econometrics, and doctoral students dealing with volatility forecasting, risk evalua-
tion, option pricing, model selection methods and predictability. It can also serve as a
handbook for consultants as well as traders, financial market practitioners and
professional economists wishing to gain up-to-date expertise in practical issues of
financial econometric modelling. Finally, graduate students on master’s courses
holding degrees from different disciplines may also benefit from the practical
orientation and applied nature of the book.
Writing this book has been both exciting and perplexing.We have tried to compile
notions, theories and practical issues that by nature lie in areas that are intrinsically
complex and ambiguous such as those of financial applications. From numerous
possible topics, we chose to include those that we judged most essential. For each of
these,we have provided an extensive bibliography for the readerwishing to go beyond
the material covered in the book.
Wewould like to extend our thanks to the several classes of studentswhose queries
and comments in the course of the preparation of the book helped in planning our
xii PREFACE
approach to different issues, in deciding the depth to which chosen topics should be
covered and in clearing up ambiguities. We are also grateful to the many people
(university colleagues, researchers, traders and financial market practitioners), who
by occasional informal exchange of views have had an influence on these aspects as
well. Much of the material in this book was developed while the first author was on
sabbatical leave at the Department of Statistics, University of California, Berkeley,
which she gratefully acknowledges for providing her with a wonderful work
environment. Moreover, it has been a pleasure to work on this project with Kathryn
Sharples, Simon Lightfoot, Susan Barclay, Richard Davies, Heather Kay and Ilaria
Meliconi at John Wiley & Sons, Ltd, copy editor Richard Leigh who so attentively
read the manuscript and made useful editing suggestions, and Poirei Sanasam at
Thomson Digital. Most importantly, we wish to express our gratitude to our families
for their support.
May 2009 Evdokia Xekalaki
Stavros Degiannakis
PREFACE xiii
Notation
� Hadamard (elementwise) product
A Lð Þ Polynomial of ARCH
AD Anderson–Darling statistic
AIC Akaike information criterion
B Matrix of unknown parameters in a multivariate
regression model.
B :; :ð Þ Cumulative distribution function of the binomial distribution
B Lð Þ Polynomial of GARCH
B tð Þ Standard Brownian motion
c :ð Þ Smooth function on [0,1].
ci Autoregressive coefficients
C Lð Þ Polynomial of AR
Ct Matrix of conditional correlations
Ctð Þt Call option at time t, with t days to maturity
Ctð Þtþ1jt Call option at time tþ 1 given the information available at
time t, with t days to maturity
CGR Correlated gamma ratio distribution
CM Cram�er–von Mises statistic
d Exponent of the fractional differencing operator 1�Lð Þd in
FIGARCHmodels~d Exponent of the fractional differencing operator 1�Lð Þ~d in
ARFIMAX models~~d Integer differencing operator
di Moving average coefficients
dti Duration or interval between two transactions, dti � ti�ti�1
D Lð Þ Polynomial of MA
DM A;Bð Þ Diebold–Mariano statistic
ESð1�pÞtþtjt Expected shortfall t days ahead.
ESð1�pÞtþ1jt Expected shortfall forecast at time tþ 1 based on information
available at time t, at 1�pð Þ probability level.
f :ð Þ Probability density function
fað:Þ a-quantile of the distribution with density function f :ð Þf BGð Þ x; y;_T ; r
� �Probability density function of the bivariate gamma
distribution
f CGRð Þ x; 2�1 _T ; r
� �Probability density function of the correlated gamma ratio
distribution
fd 0ð Þ Spectral density at frequency zero
f GCð Þ zt; n; gð Þ Probability density function of Gram–Charlier distribution
f GEDð Þ zt; nð Þ Probability density function of generalized error distribution
f GTð Þ zt; n; gð Þ Probability density function of generalized t distribution
f SGEDð Þ zt; n; yð Þ Probability density function of skewed generalized error
distribution
f skTð Þ zt; n; gð Þ Probability density function of skewed Student t distribution
f tð Þ zt; nð Þ Probability density function of standardized Student
t distribution.
Fð:Þ Cumulative distribution function
2F2 :; :; :; :; :ð Þ Generalized hypergeometric function
Fil Filter
FoEn 1;2ð Þ Forecast encompassing statistic of Harvey et al. (1998)
g :ð Þ Functional form of conditional variance
gst Constant multiplier in SWARCH model
GED Generalized error distribution
GT Generalized t distribution
Ht Conditional covariance matrix of multivariate stochastic
process, Ht � Vt�1 ytð ÞHitt Index of VaR violation minus expected ratio of violations,
Hitt ¼ ~It�p
HQ Hannan and Quinn information criterion
i Vector of ones.
It Information set~It Index of VaR violations
JB Jarque–Bera test
k Dimension of vector of unknown parameters b~k Number of slices
K Exercise (or strike) price~K Number of regimes in SWARCH model� � �K Number of factors in the
� � �K Factor ARCH p; qð Þ model
KS Kolmogorov–Smirnov statistic
KS* Kuiper statistic
Ku Kurtosis
l Order of the moving average model
li Eigenvalue
lt :ð Þ Log-likelihood function for the tth observation
log Natural logarithm
L Lag operator
LT :ð Þ Full-sample log-likelihood function based on a sample
of size T
LRcc Christoffersen’s likelihood ratio statistic for conditional
coverage.
xvi NOTATION
LRin Christoffersen’s likelihood ratio statistic for independence of
violations
LRun Kupiec’s likelihood ratio statistic for unconditional coverage.
m Number of intraday observations per day
mDM A;Bð Þ Modified Diebold–Mariano statistic
MGN A;Bð Þ Morgan–Granger–Newbold statistic
n Dimension of the multivariate stochastic process, ytf gnij Number of points in time with value i followed by j
N Total number of VaR violations, N ¼ S~Tt¼1~It
N :ð Þ Cumulative distribution function of the standard normal
distribution
NRTt Net rate of return at time t from trading an option.
p Order of GARCH form
pi;t Filtered probability in regime switch models (the probability
of the market being in regime i at time t)
pt Switching probability in regime switch models
P :ð Þ Probability
Ptð Þt Put option at time t, with t days to maturity
Ptð Þtþ1jt Put option at time tþ 1 given the information available at time
t, with t days to maturity
~q Order of BEKK p; q; ~qð Þ model.
qt Switching probability in regime switch models
rft Rate of return on a riskless asset
R2 Coefficient of multiple determination
RTt Rate of return at time t from trading an option
st Regime in SWARCH model.
y �it Risk-neutral log-returns at time t.
q Order of ARCH form
Q LBð Þ Ljung–Box statistic
rj Autocorrelation of squared standardized residuals at j lags
skT Skewed Student t distribution
St Market closing price of asset at time t~S Tð Þ Terminal stock price adjusted for risk neutrality
SBC Schwarz information criterion
SGED Skewed generalized error distribution
SH Shibata information criterion
Sk Skewness
SPA i*ð Þ Superior predictive ability statistic for the benchmarkmodel i�
T Number of total observations, T ¼ ~T þ T^
~T Number of observations for out-of-sample forecasting
T^
Number of observations for rolling sample_T Number of observations for model selection methods in out-
of-sample evaluation vt ¼ e2t�s2tVaR
ð1�pÞt VaR at 1�p probability level at time t
NOTATION xvii
VaRð1�pÞtþ1jt VaR forecast at time tþ 1 based on information available at
time t, at 1�pð Þ probability level
VaRð1�pÞtþtjt VaR t days ahead
dVaRð1�pÞt VaR in-sample estimate at time t, at 1�p probability level
vech :ð Þ Operator stacks the columns of square matrix.
w Vector of estimated parameters for the density function f
w^
Number of parameters of vector w
wi Weight
X Transaction cost
yt Log-returns
yt Multivariate stochastic process
ytþ1jt ið Þ One-step-ahead conditional mean at time tþ 1 based on
information available at time t, from model i
yBCð Þt Box–Cox transformed variable of ytztþ1jt One-step-ahead standardized prediction errors at time tþ 1
based on information available at time t.
ac Measure of cost of capital opportunity
b Vector of unknown parameters in a regression model
gd ið Þ Sample autocovariance of ith order
gt Dividend yield
et Innovation process for the conditional mean of multivariate
stochastic process, et � yt�ltetþ1jt One-step-ahead standardized prediction errors at time tþ 1
based on information available at time t
~et Innovation process in SWARCH model
y Vector of estimated parameters for the conditional mean and
variance
yðtÞið Þ Vector of estimated parameters for the conditional mean and
variance at time t from model i
y^
Number of parameters of vector yk Order of the autoregressive model
m Instantaneous expected rate of return
m :ð Þ Functional form of conditional mean
mt Predictable component of conditional mean
lt Conditional mean of multivariate stochastic process,
lt � Et�1 ytð Þn Tail-thickness parameter
pij Percentage of points in time with value i followed by j,
pij ¼ nij=Sjnijr Correlation coefficient
s Instantaneous variance of the rate of return
si;j;t Conditional covariance between asset returns i and j at time t.
s2tþ1 True, but unobservable, value of the variance at time tþ 1
xviii NOTATION
s2tþ1jt One-step-ahead conditional variance at time tþ 1 based on
information available at time t.
s2tþ1jt ið Þ One-step-ahead conditional variance at time tþ 1 based on
information available at time t, from model i.
s2tþsjt s-days-ahead conditional variance at time tþ s based on
information available at time t
s2 tð Þtþ1 True, but unobservable, value of variance for a period of t
days, from tþ 1 until tþ ts2 tð Þtþ1jt Variance forecast for a period of t days, from tþ 1 until tþ t,
given the information available at time t
~s2 tð Þtþ1 Proxy for true, but unobservable, value of variance for a period
of t days, from tþ 1 until tþ t�s tð Þtþ1jt Average standard deviation forecasts from tþ 1 up to tþ t,
given the information available at time t,
�s tð Þtþ1jt ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit�1
Pti¼1
ps2tþijt ¼
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffit�1s2 tð Þ
tþ1jtq
s2tþ1 In-sample conditional variance at time tþ 1 based on the
entire available data set T
s2 tð Þtþ1 In-sample conditional variance for a period of t days, from
tþ 1 until tþ t, based on the entire available data set T
s2 RVð Þunð Þ;tþ1
In-sample realized volatility at time tþ 1 based on the entire
available data set T
s2 RVð Þtþ1 Observable value of the realized variance at time tþ 1
s2 RVð Þ tð Þtþ1 Observable value of the realized variance for a period of t
days, from tþ 1 until tþ ts2 RVð Þ
unð Þ;tþ1jt One-day-ahead conditional realized variance at time tþ 1
based on information available at time t
t Point in time (i.e. days) for out-of-sample forecasting. Also
days to maturity for options
ut Vector of predetermined variables included in Itf :ð Þ Functional form of conditional variance in conditional mean
in GARCH-M model
F :; :; :ð Þ Confluent hypergeometric function
F Lð Þ Polynomial of FIGARCH
j t; a; b; s; mð Þ Characteristic function of stable Paretian distribution
w2gð Þ Pearson’s chi-square statistic
c Vector of estimated parameters for the conditional mean,
variance and density function, c0 ¼ y0;w0ð Þc :ð Þ Euler psi function
c^
Number of parameters of vector c, c^¼ y
^þw^
cTð Þ
Maximum likelihood estimator of c based on a sample
of size T
NOTATION xix
�C tð ÞSEð Þ ið Þ Mean squared error loss function for model i volatility
forecasts t days ahead,i.e. �C tð Þ
SEð Þ ið Þ ¼ ~T�1 P
t¼1~T s2 tð Þtþ1jt ið Þ�s2 tð Þ
tþ1 ið Þ� �2
C tð ÞSEð Þt ið Þ Squared error loss function for models i volatility forecasts t
days ahead�C tð Þ
Average of a loss function for t-days-ahead volatility
forecasts, i.e. �C tð Þ ¼ ~T�1 P~T
t¼1 Ctð Þt
C tð Þt ið Þ Loss function that measures the distance between actual
volatility over a t-day period and model i volatility forecast
over the same period
Ct ið Þ Loss function that measures the distance between one-day
actual volatility and its forecast by model i, i.e. Ct ið Þ � C 1ð Þt ið Þ
C tð Þt i�;ið Þ Difference of loss functions (loss differential) ofmodels i� and
i, C tð Þt i�;ið Þ ¼ C tð Þ
t i�ð Þ�C tð Þt ið Þ
�C tð Þi�;ið Þ Sample mean loss differential of models i� and i, �C tð Þ
i�;ið Þ ¼ ~T�1P~T
t¼1 Ctð Þt i�;ið Þ
xx NOTATION
1
What is an ARCH process?
1.1 Introduction
Since the first decades of the twentieth century, asset returns have been assumed to
form an independently and identically distributed (i.i.d.) random process with zero
mean and constant variance. Bachellier (1900) was the first to contribute to the theory
of random walk models for the analysis of speculative prices. If Ptf g denotes the
discrete time asset price process and ytf g the process of continuously compounded
returns, defined by yt ¼ 100 log Pt=Pt�1ð Þ, the early literature viewed the system that
generates the asset price process as a fully unpredictable random walk process:
Pt ¼ Pt�1 þ etet �i:i:d: N 0; s2ð Þ; ð1:1Þ
where et is a zero-mean i.i.d. normal process. Figures 1.1 and 1.2 show simulated
Ptf gTt¼1 and ytf gTt¼1 processes for T ¼ 5000, P1 ¼ 1000 and s2 ¼ 1.
However, the assumptions of normality, independence and homoscedasticity do
not always hold with real data.
Figures 1.3 and 1.4 show the daily closing prices of the London Financial Times
Stock Exchange 100 (FTSE100) index and the Chicago Standard and Poor’s 500
Composite (S&P500) index. The data cover the period from 4April 1988 until 5 April
2005. At first glance, one might say that equation (1.1) could be regarded as the data-
generating process of a stock index. The simulated process Ptf gTt¼1 shares common
characteristics with the FTSE100 and the S&P500 indices.1 As they are clearly
ARCH Models for Financial Applications Evdokia Xekalaki and Stavros Degiannakis
� 2010 John Wiley & Sons, Ltd
1 The aim of the visual comparison here is not to ascertain a model that is closest to the realization of the
stochastic process (in fact another simulated realization of the process may result in a path quite different
from that depicted in Figure 1.1). It ismerely intended as a first step towards enhancing the reader’s thinking
about or conceiving of these notions by translating them into visual images. Higher-order quantities, such as
the correlation, absolute correlation and so forth, aremuchmore important tools in the analysis of stochastic
process than their paths.
non-stationary, the autocorrelations presented in Figure 1.5 are marginally less than
unity in any lag order. Figure 1.6 plots the distributions of the daily FTSE100 and
S&P500 indices as well as the distribution of the simulated process Ptf gTt¼1. The
density estimates are based on the normal kernel with bandwidths method calculated
according to equation (3.31) of Silverman (1986). S&P500 closing prices and the
simulated process Ptf gTt¼1 have similar density functions.
However, this is not the case for the daily returns. Figures 1.7 and 1.8 depict the
FTSE100 and S&P500 continuously compounded daily returns, yFTSE100;t� �T
t¼1and
−.4
−.3
−.2
−.1
.0
.1
.2
.3
.4
1000 2000 3000 4000 5000
Figure 1.2 Simulated ytf g process, where yt¼ 100 log(Pt=Pt�1), Pt ¼ Pt�1 þ et,P1 ¼ 1000 and et �i:i:d: N 0; 1ð Þ.
980
1000
1020
1040
1060
1080
1100
1120
1140
1000 2000 3000 4000 5000
Figure 1.1 Simulated Ptf g process, where Pt ¼ Pt�1 þ et, P1 ¼ 1000 and et �i:i:d:N 0; 1ð Þ.
2 ARCH MODELS FOR FINANCIAL APPLICATIONS
ySP500;t� �T
t¼1, while Figure 1.9 presents the autocorrelations of ytf gTt¼1, yFTSE100;t
� �Tt¼1
and ySP500;t� �T
t¼1for lags of order 1, . . ., 35. The 95% confidence interval for the
estimated sample autocorrelation is given by�1:96=ffiffiffiffiT
p, in the case of a process with
independently and identically normally distributed components. The autocorrelations
of the FTSE100 and the S&P500 daily returns differ from those of the simulated
process. In both cases, more than 5% of the estimated autocorrelations are outside the
above 95% confidence interval. Visual inspection of Figures 1.7 and 1.8 shows clearly
that themean is constant, but thevariance keeps changing over time, so the return series
does not appear to be a sequence of i.i.d. random variables. A characteristic of asset
returns, which is noticeable from the figures, is the volatility clustering first noted by
1000
2000
3000
4000
5000
6000
7000
1000 2000 3000 4000
Figure 1.4 FTSE100 equity index closing prices from 4 April 1988 to 5 April 2005.
200
400
600
800
1000
1200
1400
1600
1000 2000 3000 4000
Figure 1.3 S&P500 equity index closing prices from 4 April 1988 to 5 April 2005.
WHAT IS AN ARCH PROCESS? 3
Mandelbrot (1963): ‘Large changes tend to be followed by large changes, of either sign,
and small changes tend to be followed by small changes’. Fama (1970) also observed
the alternation between periods of high and low volatility: ‘Large price changes are
followed by large price changes, but of unpredictable sign’.
Figure 1.10 presents the histograms of the stock market series. Asset returns are
highly peaked (leptokurtic) and slightly asymmetric, a phenomenon observed by
Mandelbrot (1963):
The empirical distributions of price changes are usually too peaked to be
relative to samples fromGaussian populations . . . the histograms of price
changes are indeed unimodal and their central bells [recall] the Gaussian
ogive. But, there are typically so many outliers that ogives fitted to the
mean square of price changes are much lower and flatter than the
distribution of the data themselves.
FTSE100 S&P500 }{ Tt=1tP
.0000
.0004
.0008
.0012
.0016
.0020
400 800 1200 1600.00000
.00004
.00008
.00012
.00016
.00020
.00024
.00028
.00032
2000 3000 4000 5000 6000 7000.000
.004
.008
.012
.016
.020
1000 1040 1080 1120
Figure 1.6 Density estimate of the S&P500 and FTSE100 closing prices, and of the
simulated process Ptf gTt¼1.
Cor(FTSE100t, FTSE100t−τ )Cor(SP500t, SP500t−τ ) Cor(Pt, Pt−τ)
−10%
10%
30%
50%
70%
90%
332925211713951Order of lag
Aut
ocor
rela
tion
−10%
10%
30%
50%
70%
90%
332925211713951
Order of lag
Aut
ocor
rela
tion
−10%
10%
30%
50%
70%
90%
332925211713951Order of lag
Aut
ocor
rela
tion
Figure 1.5 Autocorrelation of the S&P500 and the FTSE100 closing prices and of
the simulated process Ptf gTt¼1, for t ¼ 1 1ð Þ35 lags. Dashed lines present the 95%
confidence interval for the estimated sample autocorrelations given by �1:96=ffiffiffiffiT
p.
4 ARCH MODELS FOR FINANCIAL APPLICATIONS
According to Table 1.1, for estimated kurtosis2 equal to 7.221 (or 6.241) and an
estimated skewness3 equal to �0.162 (or �0.117), the distribution of returns is flat
(platykurtic) and has a long left tail relative to the normal distribution. The Jarque and
Bera (1980, 1987) test is usually used to test the null hypothesis that the series is
normally distributed. The test statistic measures the size of the difference between the
skewness, Sk, and kurtosis, Ku, of the series and those of the normal distribution. It is
computed as JB ¼ T�Sk2 þ � Ku�3ð Þ2=4��=6,whereT is the number of observations.
Under the null hypothesis of a normal distribution, the JB statistic is w2 distributed
−8
−6
−4
−2
0
2
4
6
1000 2000 3000 4000
Figure 1.7 S&P500 equity index continuously compounded daily returns from
5 April 1988 to 5 April 2005.
2 Kurtosis is a measure of the degree of peakedness of a distribution of values, defined in terms of a
normalized form of its fourth central moment by m4=m22 (it is in fact the expected value of quartic
standardized scores) and estimated by
Ku ¼ TXTt¼1
yt��yð Þ4, XT
t¼1
yt��yð Þ2 !2
;
where T is the number of observations and�y is the samplemean,�y ¼PTt¼1 yt. The normal distribution has a
kurtosis equal to 3 and is calledmesokurtic. A distribution with a kurtosis greater than 3 has a higher peak
and is called leptokurtic, while a distribution with a kurtosis less than 3 has a flatter peak and is called
platykurtic. Some writers talk about excess kurtosis, whereby 3 is deducted from the kurtosis so that the
normal distribution has an excess kurtosis of 0 (see Alexander, 2008, p. 82).3 Skewness is a measure of the degree of asymmetry of a distribution, defined in terms of a normalized
form of its third central moment of a distribution by m3=m3=22 (it is in fact the expected value of cubed
standardized scores) and estimated by
Sk ¼ffiffiffiffiT
p XTt¼1
yt��yð Þ3, XT
t¼1
yt��yð Þ2 !3=2
:
The normal distribution has a skewness equal to 0.A distributionwith a skewness greater than 0 has a longer
right tail is described as skewed to the right, while a distributionwith a skewness less than 0 has a longer left
tail and is described as skewed to the left.
WHAT IS AN ARCH PROCESS? 5
with 2 degrees of freedom. The two return series were tested for normality using JB
resulting in a p-value that was practically zero, thus signaling non-validity of the
hypothesis. Due to the fact that the JB statistic frequently rejects the hypothesis
of normality, especially in the presence of serially correlated observations, a series of
more powerful test statistics (e.g. the Anderson–Darling and the Cram�er–von Mises
statistics) were also computed with similar results. A detailed discussion of the
computation of the aforementioned test statistics is given in Section 3.3.
Cor(ySP500,t, ySP500,t−τ) Cor(yFTSE100,t, yFTSE100,t−τ) Cor(yt, yt−τ)
−8%
−6%
−4%
−2%
0%
2%
4%
6%
−8%
−6%
−4%
−2%
0%
2%
4%
6%
−8%
−6%
−4%
−2%
0%
2%
4%
6%
332925211713951Order of lag
Aut
ocor
rela
tion
332925211713951Order of lag
Aut
ocor
rela
tion
332925211713951
Order of lag
Aut
ocor
rela
tion
Figure 1.9 Autocorrelation of the S&P500 and FTSE100 continuously com-
pounded daily returns and of the simulated process ytf gTt¼1, for t ¼ 1 1ð Þ35 lags.
Dashed lines present the 95% confidence interval for the estimated sample auto-
correlations given by �1:96=ffiffiffiffiT
p.
−6
−4
−2
0
2
4
6
1000 2000 3000 4000
Figure 1.8 FTSE100 equity index continuously compounded daily returns from
5 April 1988 to 5 April 2005.
6 ARCH MODELS FOR FINANCIAL APPLICATIONS
In the 1960s and 1970s, the regularity of leptokurtosis led to a literature on
modelling asset returns as i.i.d. random variables having some thick-tailed distribu-
tion (Blattberg and Gonedes, 1974; Clark, 1973; Hagerman, 1978;Mandelbrot, 1963,
1964; Officer, 1972; Praetz, 1972). These models, although able to capture the
leptokurtosis, could not account for the existence of non-linear temporal dependence
such as volatility clustering observed from the data. For example, applying an
autoregressive model to remove the linear dependence from an asset returns series
and testing the residuals for a higher-order dependence using the Brock–Dechert–
Scheinkman (BDS) test (Brock et al., 1987, 1991, 1996), the null hypothesis of i.i.d.
residuals was rejected.
Table 1.1 Descriptive statistics of the S&P500 and the FTSE100 equity index returns
S&P500 FTSE100
Mean 0.034% 0.024%
Standard deviation 15.81% 15.94%
Skewness �0.162 �0.117
Kurtosis 7.221 6.241
Jarque–Bera 3312.9 1945.6
[p-value] [0.00] [0.00]
Anderson–Darling 44.3 28.7
[p-value] [0.00] [0.00]
Cram�er–von Mises 8.1 4.6
[p-value] [0.00] [0.00]
The annualized standard deviation is computed by multiplying the standard deviation of daily
returns by 2521/2, the square root of the number of trading days per year. The Jarque–Bera,
Anderson–Darling and Cram�er–von Mises statistics test the null hypothesis that the daily
returns are normally distributed.
FTSE100 S&P500
0
200
400
600
800
1000
1200
1400
−6 −4 −2 0 2 4 60
200
400
600
800
1000
1200
−6 −4 −2 0 2 4 6
Figure 1.10 Histogram of the S&P500 and FTSE100 index log-returns.
WHAT IS AN ARCH PROCESS? 7
1.2 The autoregressive conditionally
heteroscedastic process
Autoregressive conditional heteroscedasticity (ARCH) models have been widely
used in financial time series analysis and particularly in analysing the risk of
holding an asset, evaluating the price of an option, forecasting time-varying
confidence intervals and obtaining more efficient estimators under the existence
of heteroscedasticity.
Before we proceed to the definition of the ARCHmodel, let us simulate a process
able to capture the volatility clustering of asset returns. Assume that the true data-
generating process of continuously compounded returns, yt, has a fully unpredictable
conditional mean and a time-varying conditional variance:
yt ¼ et;
et ¼ ztffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia0 þ a1e2t�1
p;
ð1:2Þ
where zt �i:i:d: N 0; 1ð Þ, zt is independent of et, a0 > 0 and 0 < a1 < 1. The uncondi-
tional mean of yt is E ytð Þ ¼ E ztð ÞE� ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia0 þ a1e2t�1
p � ¼ 0, as E ztð Þ ¼ 0 and zt and et�1
are independent of each other. The conditional mean of yt given the lag values of etis E ytjet�1; . . . ; e1ð Þ ¼ 0. The unconditional variance is V ytð Þ ¼ Eðe2t Þ�E etð Þ2 ¼E z2t a0 þ a1e2t�1
� �� � ¼ a0 þ a1Eðe2t�1Þ. AsEðe2t Þ ¼ Eðe2t�1Þ,V ytð Þ ¼ a0 1�a1ð Þ�1. The
conditional variance is V ytjet�1; . . . ; e1ð Þ ¼ E e2t jet�1; . . . ; e1� � ¼ a0 þ a1e2t�1. The
kurtosis of the unconditional distribution equals Eðe4t Þ=Eðe2t Þ2¼ 3ð1�a21Þ=ð1�3a21Þ.Note that the kurtosis exceeds 3 for a1 > 0 and diverges if a1 approaches
ffiffiffiffiffiffiffiffi1=3
p.
Figure 1.11 plots the unconditional kurtosis for 0 � a1 < 1=ffiffiffi3
p.
Both the unconditional and conditional means and the unconditional variance of
asset returns remain constant, but the conditional variance has a time-varying
character as it depends on the previous values of et. Let us consider equation (1.2)
as the true data-generating function and produce 5000 values of yt. Figure 1.12
0
50
100
150
200
250
300
350
0.540.480.420.360.300.240.180.120.060.00
Parameter a1
Unc
ondi
tiona
l Kur
tosi
s
Figure 1.11 Theunconditional kurtosis of et ¼ ztffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffia0 þ a1e2t�1
pfor 0 � a1 < 1=
ffiffiffi3
pand a0 > 0.
8 ARCH MODELS FOR FINANCIAL APPLICATIONS
