Modeling and Forecasting Intraday Volatility with Unobserved

Introduction Model & Notation Estimation Forecasting Empirical Application

Modeling and ForecastingIntraday Volatility

with Unobserved Component Structures

Gabriele Fiorentini∗ and Christian Macaro∗∗

∗University of Florence∗∗University of Roma Tre and NYU

Preliminary and Incomplete

December 5, 2007


Motivation

The availability of UHF data for prices of financial assetshas fostered a bulk of research that makes use of thecontinuously recorded data to deal with different issues.

A non trivial portion of this research focused on themodeling and forecasting of intraday volatility

Forecasts of intraday volatility should assist trading deskoperators and market makers in placing limit orders andrebalancing portfolios


Motivation





Motivation





Motivation

It has long been recognized that conventional GARCHmodels are unsatisfactory for modeling returns sampled atintraday frequencies mainly because they fail to control forthe pronounced intraday (seasonal) patterns of volatility.

Alternative models have been suggested. In this project webuild on the previous literature and propose a new modelfor intraday volatility that largely borrows from existingproposals.


Motivation

It has long been recognized that conventional GARCHmodels are unsatisfactory for modeling returns sampled atintraday frequencies mainly because they fail to control forthe pronounced intraday (seasonal) patterns of volatility.

Alternative models have been suggested. In this project webuild on the previous literature and propose a new modelfor intraday volatility that largely borrows from existingproposals.


Proposal

The salient features of our model are

1 The volatility of high frequency returns is assumed todecompose into the product of stochastic components

2 The volatility of high frequency returns is estimated byusing higher frequency tick by tick data

3 The information brought by weakly exogenous variables iseasily incorporated

4 Focus is on stocks but extensions to other financial assetsis straightforward

5 Implementation is simple and the model is apt to be appliedto large data sets


Proposal








Proposal








Proposal








Proposal








Proposal








Related Literature

Andersen and Bollerslev 1997 JoEF, 1997 JoF, 1998 JoFGhose and Kroner 1996Taylor and Xu 1997 JoEFAndersen, Bollerslev and Das 2001 JoFGiot 2005 EJoFEngle, Sokalska and Chanda 2006Deo, Hurvich and Lu 2006 JoEDacorogna, Gencay, Muller, Olsen and Pictet 2001Beltratti and Morana 2001 EN

Mian and Adam 2001 AFEMartens, Chang and Taylor 2002 JoFRWorthington 2003Aradhyula and Ergun 2004 AFEMcMillan and Speight 2004 AFEWongswan 2006 RoFS


Related Literature

Andersen and Bollerslev 1997 JoEF, 1997 JoF, 1998 JoFGhose and Kroner 1996Taylor and Xu 1997 JoEFAndersen, Bollerslev and Das 2001 JoFGiot 2005 EJoFEngle, Sokalska and Chanda 2006Deo, Hurvich and Lu 2006 JoEDacorogna, Gencay, Muller, Olsen and Pictet 2001Beltratti and Morana 2001 EN

Mian and Adam 2001 AFEMartens, Chang and Taylor 2002 JoFRWorthington 2003Aradhyula and Ergun 2004 AFEMcMillan and Speight 2004 AFEWongswan 2006 RoFS


Notation

Let t = 1, . . . ,T denote days in the sampleEach day is divided into M intervals

Return in interval j of day t is defined as

Rt ,j = log Prct ,j − log Prct ,j−1 (t = 1, . . . ,T ) (j = 1, . . . ,M)

E.g. for NYSE stocks

if M = 12⇒ Rt ,j are 30-minute returnsif M = 36⇒ Rt ,j are 10-minute returns

Overnight returns are dropped and the total number ofobservations is T × N


Notation








Notation








Notation








Notation








Notation








Notation








Notation








Model

We assume that Rt ,j = σt ,j zt ,j zt ,j i.i.d. ∼ (0,1)

We further assume that σt ,j can be decomposed into

the product of components. Specifically

σt ,j = Pt ,j St ,j Ut ,j

where

P is the persistent component of volatility

S is the intraday (seasonal) periodic factor

U is the short-term irregular intraday factor


Model





where





Model





where





Model





where





Model





where





Model





where





Remarks σt ,j = Pt ,j St ,j Ut ,j

– Notice that each component is indexed by t and j

– Many popular models for high frequency volatility can bewritten in this unobserved component framework

– In A&B(97,98,...) and ESC(06) Pt ,j depends only on t .

P2t ,j =

σ2t

M , where σ2t is daily volatility (e.g. from a GARCH model)

– In G&K(96) and ESC(06) S2t ,j = S2

j = 1T∑T

t=1 R?2t ,j .

where R?t,j = Rt,j/σt

– In the FFF approach of A&B St ,j may either depend only onj or also on t via the interaction with the daily component.

– In T&X(97) St ,j depends on j and also on t but only for theday of the week effect






P2t ,j =

σ2t


– In G&K(96) and ESC(06) S2t ,j = S2

j = 1T∑T

t=1 R?2t ,j .









P2t ,j =

σ2t


– In G&K(96) and ESC(06) S2t ,j = S2

j = 1T∑T

t=1 R?2t ,j .









P2t ,j =

σ2t


– In G&K(96) and ESC(06) S2t ,j = S2

j = 1T∑T

t=1 R?2t ,j .









P2t ,j =

σ2t


– In G&K(96) and ESC(06) S2t ,j = S2

j = 1T∑T

t=1 R?2t ,j .









P2t ,j =

σ2t


– In G&K(96) and ESC(06) S2t ,j = S2

j = 1T∑T

t=1 R?2t ,j .





Specification

We assume that logσt ,j and its components follow somestochastic linear processes (augmented with regressors)

The model can then be written in two related ways.

1 Arima Model Based (AMB) decomposition: Specify ageneral REG-AR(F)IMA model for logσt ,j and derivecompatible models for the components.

2 Structural Time Series (STS) model à la Harvey: Use aState-Space formulation in which the models for thecomponents are specified directly.

The two approaches are equivalent and differ only for theestimation (filtering-smoothing) algorithm (W-K vs. Kalman)and, sometimes, for the identification assumptions.


Specification







Specification







Specification







Specification







Specification







AMB Specification

logσt ,j = pt ,j + st ,j + ut ,j = logσ?t ,j + f (Xt ,j , β)

logσ?t ,j = p?t ,j + s?t ,j + u?t ,j

Φ(L)[logσt ,j − f (Xt ,j , β)

]= Φ(L) logσ?t ,j = Θ(L)at ,j

Φp(L)p?t,j = Θp(L)aP t,j ; Φs(L)s?

t,j = Θs(L)aS t,j ; Φu(L)u?t,j = Θu(L)aU t,j ;

where f (Xt,j , β) is a known regression function of weakly exogenousvariables Xt,j and unknown coefficients β.

Φ•(L) and Θ•(L) are polynomials in the lag operator L with

Φ(L) = Φp(L)Φs(L)Φu(L)

Θ(L)at = Θp(L)Φs(L)Φu(L)aP t,j +Θs(L)Φp(L)Φu(L)aS t,j +Θu(L)Φp(L)Φs(L)aU t,j


AMB Specification












AMB Specification












AMB Specification












AMB Specification












AMB Specification












AMB Identification

Consider the following set of sufficient identificationassumptions

1 Factorization of Φ(L) into Φp(L)Φs(L)Φu(L) isstraightforward and is obtained through a simple rootallocation.

2 Orthogonality of components p? ⊥ s? ⊥ u?

3 Canonical decomposition: The model for the componentsare balanced and Var(aP t ,j) and Var(aS t ,j) are chosen assmall as possible. In this way p? and s? are termedcanonical components.

Remark: For any other admissible choice of, say, Var(aS t,j ) thecorresponding periodic component can be written as the sum ofthe canonical one plus an uncorrelated white noise. (Hillmer andTiao 1982)


AMB Identification







AMB Identification







AMB Identification







AMB Example: Quarterly Airline Model

Example: Consider the following typical model for quarterlydata

(1− L)(1− L4)Yt = (1 + θ1L)(1 + θ4L4)at

Yt = Trend + Seasonal + Noise

TREND: (1− 2L + L2)pt = (1 + θp1L + θp2L2)ap t

SEASONAL: (1 + L + L2 + L3)st = (1 + θs1L + θs2L2 + θs3L3)as t




(1− L)(1− L4)Yt = (1 + θ1L)(1 + θ4L4)at







(1− L)(1− L4)Yt = (1 + θ1L)(1 + θ4L4)at







(1− L)(1− L4)Yt = (1 + θ1L)(1 + θ4L4)at






0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

θ1 = -0.1 θ

4 = -0.3

Pseudo-Spectra of Series and Components

SERIESTRENDSEASONALNOISE



0 0.5 1 1.5 2 2.5 30

0.2

0.4

0.6

0.8

1

1.2

θ1 = -0.1 θ

4 = -0.3

Pseudo-Spectra of Series and (Canonical) Components

SERIESTRENDSEASONALNOISECAN. TRENDCAN. SEASONALCAN. NOISE


Estimation of σ2t ,j

Suppose that each one of the M intervals is furtherpartitioned into N sub-intervals.Return in sub-interval i of bin j of day t is defined as

Rt ,j,i (t = 1, . . . ,T ) (j = 1, . . . ,M), (i = 1, . . . ,N)

E.g. for NYSEif M = 12 and N = 30⇒ Rt ,j,i are 1-minute returnsif M = 36 and N = 20⇒ Rt ,j,i are half-minute returns

We can, thus, define an estimator of σ2t ,j based on some

realized measure that uses N observations.




Rt ,j,i (t = 1, . . . ,T ) (j = 1, . . . ,M), (i = 1, . . . ,N)







Rt ,j,i (t = 1, . . . ,T ) (j = 1, . . . ,M), (i = 1, . . . ,N)







Rt ,j,i (t = 1, . . . ,T ) (j = 1, . . . ,M), (i = 1, . . . ,N)







Rt ,j,i (t = 1, . . . ,T ) (j = 1, . . . ,M), (i = 1, . . . ,N)







Rt ,j,i (t = 1, . . . ,T ) (j = 1, . . . ,M), (i = 1, . . . ,N)







Rt ,j,i (t = 1, . . . ,T ) (j = 1, . . . ,M), (i = 1, . . . ,N)






For example to keep matters simple consider realizedvolatility

σ2t ,j =

N∑i=1

R2t ,j,i

In principle we could take M as large as desired and letN →∞

In practice there are limits to the benefits attainable fromUHF data and M × N cannot be too large




σ2t ,j =

N∑i=1

R2t ,j,i






σ2t ,j =

N∑i=1

R2t ,j,i




REGARIMA specification and estimation

Let lrvt ,j = 12 log σ2

t ,j . We specify and estimate the model

Φ(L)[lrvt ,j − f (Xt ,j , β)

]= Φ(L)lrv?t ,j = Θ(L)at ,j

Regression variables Xt,j may include:

1) Macro and sector/firm specific announcement variables

2) Day of the week dummies3) (A non-negative function of) Overnight returns

M regressors or 1 regressor with distributed effects

4) Detected outliers AO and TC

5) . . .

Model is identified with BIC and estimated by PML












5) . . .













5) . . .













5) . . .













5) . . .













5) . . .













5) . . .













5) . . .



Estimation of Components

Next, we derive the (canonical) models for the components thatcan be estimated by linear projections with the Wiener -Kolmogorov symmetric filter

p?t ,j = WKp(L) lrv?(E)

t ,j

where lrv?(E)

t ,j is the series extended with forecasts andbackcasts

WKp(L) =Θp(L)Θp(L−1)

Φp(L)Φp(L−1)

Φ(L)Φ(L−1)

Θ(L)Θ(L−1)

The frequency domain representation of the filter is given bythe Ratio of Spectra. Thus the squared gain will always be lessor equal than one and proportional to the relative power of thecomponent spectrum.



Next, we derive the (canonical) models for the components thatcan be estimated by linear projections with the Wiener -Kolmogorov symmetric filter

p?t ,j = WKp(L) lrv?(E)

t ,j

where lrv?(E)

t ,j is the series extended with forecasts andbackcasts

WKp(L) =Θp(L)Θp(L−1)

Φp(L)Φp(L−1)

Φ(L)Φ(L−1)

Θ(L)Θ(L−1)

The frequency domain representation of the filter is given bythe Ratio of Spectra. Thus the squared gain will always be lessor equal than one and proportional to the relative power of thecomponent spectrum.



The estimated regression effects are assigned to thecomponents according to their properties.

Forecasts of the components are also readily obtained

Finally the multiplicative factors Pt ,j , St ,j and Ut ,j are computed

Remark: When the intraday periodic component of volatility ispronounced and the “seasonal” factors are large, a biascorrection is needed to correct the underestimation of the“seasonally adjusted” volatility caused by the fact thatgeometric means underestimate arithmetic means.




















Forecasting Methods

We consider alternative forecasting methods

1) Forecast directly lrvt ,j with the REG-ARIMA model

2) Use the AMB decomposition to compute forecasts of theintraday periodic factor St ,j and of the persistentcomponent Pt ,j . Next, apply a GARCH type model to thestandardized returns Rt ,j/(Pt ,jSt ,j).

3) Same as method 2) but the forecast of the persistentcomponent is computed from:

A) GARCH type model on daily dataB) Forecasts of daily realized volatility


Forecasting Methods







Forecasting Methods







Forecasting Methods







Forecasting Methods







Forecasting Comparisons

Method 3) is analogous to Engle Sokalska and Chanda (2006)(ESC). Differences are confined to the estimation andforecasting of the intraday periodic (diurnal) factors.

An FFF based alternative to Method 3) can also be formulated

ESC and the FFF based variation are used as benchmarksagainst which we test the forecasting performance of ourmodel.












Data

NYSE quotes for BOEING, EXXON, GENERAL ELECTRIC andIBM are obtained from the TAQ database

Data spans a 16-month period in January 2006 - April 2007

Monday July 3rd and Friday November 24th are deleted fromthe sample

Each day we remove the first and last 15-minute period andconsider M = 12 and N = 30.

Rt ,j,i are then 1-minute returns on day t between 9:45 and15:45. We aim at modeling and forecasting 30-minute volatility

Prices are defined as size weighted bid/ask averages (otherdefinitions have been considered).

We use four months for the “historical” analysis and retaintwelve months for the rolling forecasting exercise.


Data









Data









Data









Data









Data









Data









Data

The total sample length is 3948 observations on 30-minutereturns (329 days).

In the period January 2006 - April 2006 there are 972observations (81 days)

In the out-of-sample forecasting exercise we maintain thesample length equal to 972 and we evaluate 2976 k -periodsahead forecasts (k = 1,2,3).

Daily data on prices (since January 2003) are downloaded fromhttp://finance.yahoo.com/

Overnight returns are computed as log(Opent )− log(Closet−1).Open prices are adjusted for dividends using an adjustmentfactor calculated from closing prices.


Data







Data







Data







Data






0 10 20 30 40 50 60 70 80-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

BOEING

Days0 10 20 30 40 50 60 70 80

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

EXXON

Days

0 10 20 30 40 50 60 70 80-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

GENERAL ELECTRIC

Days0 10 20 30 40 50 60 70 80

-2

-1.5

-1

-0.5

0

0.5

1

1.5

2

IBM

Days

30-minute returns (January-April 2006)

0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1BOEING

Days0 10 20 30 40 50 60 70 80

0

0.2

0.4

0.6

0.8

1EXXON

Days

0 10 20 30 40 50 60 70 800

0.2

0.4

0.6

0.8

1GENERAL ELECTRIC

Days0 10 20 30 40 50 60 70 80

0

0.2

0.4

0.6

0.8

1IBM

Days

30-minute realized volatilities (January-April 2006)

0 12 24 36 48 60 72 84 96 108 120-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Squared Returns

Lags0 12 24 36 48 60 72 84 96 108 120

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Realized Volatilities

Lags

0 12 24 36 48 60 72 84 96 108 120-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Log of Squared Returns

Lags0 12 24 36 48 60 72 84 96 108 120

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Log of Realized Volatilities

Lags

ACF plots BOEING (January-April 2006)

0 12 24 36 48 60 72 84 96 108 120-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Squared Returns

Lags0 12 24 36 48 60 72 84 96 108 120

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Lags

0 12 24 36 48 60 72 84 96 108 120-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Lags0 12 24 36 48 60 72 84 96 108 120

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Lags

ACF plots EXXON (January-April 2006)

0 12 24 36 48 60 72 84 96 108 120-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Squared Returns

Lags0 12 24 36 48 60 72 84 96 108 120

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Lags

0 12 24 36 48 60 72 84 96 108 120-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Lags0 12 24 36 48 60 72 84 96 108 120

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Lags

ACF plots GENERAL ELECTRIC (January-April 2006)

0 12 24 36 48 60 72 84 96 108 120-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Squared Returns

Lags0 12 24 36 48 60 72 84 96 108 120

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Lags

0 12 24 36 48 60 72 84 96 108 120-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Lags0 12 24 36 48 60 72 84 96 108 120

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Lags

ACF plots IBM (January-April 2006)

0 12 24 36 48 60 72 84 96 108 120-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5

0.6Squared Returns

Lags0 12 24 36 48 60 72 84 96 108 120

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Lags

0 12 24 36 48 60 72 84 96 108 120-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Lags0 12 24 36 48 60 72 84 96 108 120

-0.2

-0.1

0

0.1

0.2

0.3

0.4

0.5


Lags

ACF plots BOEING (January 2006 -April 2007) Full Sample


Summary: ARIMA (p,d ,q)× (bp,bd ,bq)

ORDERS OF ESTIMATED MODELS

NO REG. DoW DoW + OR DoW + 12OR

BOEING (1, 0, 1)× (0, 1, 1) (1, 0, 1)× (0, 1, 1) (1, 0, 1)× (0, 1, 1) (1, 0, 1)× (0, 1, 1)

EXXON (1, 0, 1)× (0, 1, 1) (2, 0, 1)× (0, 1, 1) (2, 0, 1)× (0, 1, 1) (3, 0, 1)× (0, 1, 1)

G.E. (2, 0, 1)× (0, 1, 1) (2, 0, 1)× (0, 1, 1) (2, 0, 1)× (0, 1, 1) (2, 0, 1)× (0, 1, 1)

IBM (1, 0, 1)× (0, 1, 1) (1, 0, 1)× (0, 1, 1) (1, 0, 1)× (0, 1, 1) (0, 1, 3)× (0, 1, 1)

DETECTED OUTLIERS

NO REG. DoW DoW + OR DoW + 12OR

BOEING — — — —

EXXON 3 AO + 2 TC 2 AO + 2 TC 2 AO + 2 TC 2 AO + 2 TC

G.E. 5 AO + 2 TC 6 AO + 2 TC 9 AO + 3 TC 6 AO + 1 TC

IBM 1 TC — — 1 AO + 4 TC

0 10 20 30 40 50 60 70 800.6

0.8

1

1.2

1.4

1.6

1.8

WITHOUT REGRESSORS

Days0 10 20 30 40 50 60 70 80

0.6

0.8

1

1.2

1.4

1.6

1.8

DAY OF THE WEEK

Days

0 10 20 30 40 50 60 70 800.6

0.8

1

1.2

1.4

1.6

1.8

DAY OF THE WEEK and SINGLE OVERNIGHT RETURNS

Days0 10 20 30 40 50 60 70 80

0.6

0.8

1

1.2

1.4

1.6

1.8

DAY OF THE WEEK and MULTIPLE OVERNIGHT RETURNS

Days

INTRADAY PERIODIC FACTORS BOEING (January-April 2006)

0 10 20 30 40 50 60 70 800.6

0.8

1

1.2

1.4

1.6

1.8

WITHOUT REGRESSORS

Days0 10 20 30 40 50 60 70 80

0.6

0.8

1

1.2

1.4

1.6

1.8

DAY OF THE WEEK

Days

0 10 20 30 40 50 60 70 800.6

0.8

1

1.2

1.4

1.6

1.8


Days0 10 20 30 40 50 60 70 80

0.6

0.8

1

1.2

1.4

1.6

1.8


Days

INTRADAY PERIODIC FACTORS EXXON (January-April 2006)

0 10 20 30 40 50 60 70 800.6

0.8

1

1.2

1.4

1.6

1.8

WITHOUT REGRESSORS

Days0 10 20 30 40 50 60 70 80

0.6

0.8

1

1.2

1.4

1.6

1.8

DAY OF THE WEEK

Days

0 10 20 30 40 50 60 70 800.6

0.8

1

1.2

1.4

1.6

1.8


Days0 10 20 30 40 50 60 70 80

0.6

0.8

1

1.2

1.4

1.6

1.8


Days

INTRADAY PERIODIC FACTORS GENERAL ELECTRIC (January-April 2006)

0 10 20 30 40 50 60 70 800.6

0.8

1

1.2

1.4

1.6

1.8

WITHOUT REGRESSORS

Days0 10 20 30 40 50 60 70 80

0.6

0.8

1

1.2

1.4

1.6

1.8

DAY OF THE WEEK

Days

0 10 20 30 40 50 60 70 800.6

0.8

1

1.2

1.4

1.6

1.8


Days0 10 20 30 40 50 60 70 80

0.6

0.8

1

1.2

1.4

1.6

1.8


Days

INTRADAY PERIODIC FACTORS IBM (January-April 2006)


Forecast Evaluation

FORECASTING

Documents

Modeling and Forecasting Intraday Volatility with Unobserved