dlfeb com modelling and forecasting high frequency financial data

Model
l
i
ngand
For
ecast
i
ng
Hi
ghFr
equency
Fi
nanci
alDat
a
St
avr
osDegi
annaki
sandChr
i
st
osFl
or
os


Modelling and Forecasting High Frequency Financial Data


This page intentionally left blank

Modelling and
Forecasting
High Frequency
Financial Data
Stavros Degiannakis and Christos Floros


© Stavros Degiannakis and Christos Floros 2015
All rights reserved. No reproduction, copy or transmission of this
publication may be made without written permission.
No portion of this publication may be reproduced, copied or transmitted
save with written permission or in accordance with the provisions of the
Copyright, Designs and Patents Act 1988, or under the terms of any licence
permitting limited copying issued by the Copyright Licensing Agency,
Saffron House, 6–10 Kirby Street, London EC1N 8TS.
Any person who does any unauthorized act in relation to this publication
may be liable to criminal prosecution and civil claims for damages.
The authors have asserted their rights to be identified as the authors of this work
in accordance with the Copyright, Designs and Patents Act 1988.
First published 2015 by
PALGRAVE MACMILLAN
Palgrave Macmillan in the UK is an imprint of Macmillan Publishers Limited,
registered in England, company number 785998, of Houndmills, Basingstoke,
Hampshire RG21 6XS.
Palgrave Macmillan in the US is a division of St Martin’s Press LLC,
175 Fifth Avenue, New York, NY 10010.
Palgrave Macmillan is the global academic imprint of the above companies
and has companies and representatives throughout the world.

Palgrave® and Macmillan® are registered trademarks in the United States,
the United Kingdom, Europe and other countries
ISBN 978-1-349-56690-7
ISBN 978-1-137-39649-5 (eBook)
DOI 10.1057/9781137396495

This book is printed on paper suitable for recycling and made from fully
managed and sustained forest sources. Logging, pulping and manufacturing
processes are expected to conform to the environmental regulations of the
country of origin.
A catalogue record for this book is available from the British Library.
Library of Congress Cataloging-in-Publication Data
Degiannakis, Stavros, author.
Modelling and forecasting high frequency financial data /
Stavros Degiannakis, Christos Floros.
pages cm
1. Finance–Mathematical models. 2. Speculation–Mathematical models.
3. Technical analysis (Investment analysis)–Mathematical models.
I. Floros, C. (Christos), author. II. Title.
HG106.D44 2015
2015013168
332.01 5195–dc23


To Aggelos, Andriana and Rebecca
Stavros Degiannakis
To Ioanna, Vasilis-Spyridon, Konstantina-Artemis and Christina-Ioanna
Christos Floros

This page intentionally left blank


Contents
List of Figures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xi
List of Tables . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xiv
Acknowledgments . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . xvii
List of Symbols and Operators . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .xviii
1

Introduction to High Frequency Financial Modelling .
1
The role of high frequency trading . . . . . . . . . .
2
Modelling volatility . . . . . . . . . . . . . . . . . . . .
3
Realized volatility . . . . . . . . . . . . . . . . . . . . . .
4
Volatility forecasting using high frequency data .
5
Volatility evidence . . . . . . . . . . . . . . . . . . . . .
6
Market microstructure . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.

.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.

.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.

.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.

1
2
10
11
14
14
15

2

Intraday Realized Volatility Measures . . . . . . . . . . . . . . . .
1
The theoretical framework behind the realized volatility
2

Theory of ultra-high frequency volatility modelling . . .
3
Equidistant price observations . . . . . . . . . . . . . . . . . .
3.1
Linear interpolation method . . . . . . . . . . . . . .
3.2
Previous tick method . . . . . . . . . . . . . . . . . . .
4
Methods of measuring realized volatility . . . . . . . . . . .
4.1
Conditional – inter-day – Variance . . . . . . . . . .
4.2
Realized variance . . . . . . . . . . . . . . . . . . . . . .
4.3
Price range . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4
Model-based duration . . . . . . . . . . . . . . . . . . .
4.5
Multiple grids . . . . . . . . . . . . . . . . . . . . . . . .
4.6
Scaled realized range . . . . . . . . . . . . . . . . . . . .
4.7
Price jumps . . . . . . . . . . . . . . . . . . . . . . . . . .
4.8
Microstructure frictions . . . . . . . . . . . . . . . . .
4.9
Autocorrelation of intraday returns . . . . . . . . . .
4.10 Interday adjustments . . . . . . . . . . . . . . . . . . .
5
Simulating the realized volatility . . . . . . . . . . . . . . . .

6
Optimal sampling frequency . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.

24
24
27
31
31
32
32
32
34
35
37
37
37
37
37
38
38
42
47

3

Methods of Volatility Estimation and Forecasting . . . . . . . . . . . . . . . . . 58
1
Daily volatility models – review . . . . . . . . . . . . . . . . . . . . . . . . . . 58

vii

viii

Contents

2

3

.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.

.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.

59
59
60
60
60

61
61
61
62
62
63
63
64
64

. . . 64
. . . 67
.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.

.
.

67
70
70
70
72
73

Multiple Model Comparison and Hypothesis Framework Construction
1
Statistical methods of comparing the forecasting ability of models
1.1
Diebold and Mariano test of equal forecast accuracy . . . . .
1.2
Reality check for data snooping . . . . . . . . . . . . . . . . . . .
1.3
Superior Predictive Ability test . . . . . . . . . . . . . . . . . . . .
1.4
SPEC model selection method . . . . . . . . . . . . . . . . . . . .
2
Theoretical framework: distribution functions . . . . . . . . . . . . . .
3
A framework to compare the predictive ability of two competing
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
A framework to compare the predictive ability of
n competing models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1

Generic model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
Regression model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
Regression model with time varying conditional variance .
4.4
Fractionally integrated ARMA model with time varying
conditional variance . . . . . . . . . . . . . . . . . . . . . . . . . . .
5
Intraday realized volatility application . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.

.
.
.
.
.
.
.

110
110
111

111
112
112
113

4

4

1.1
ARCH(q) model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.2
GARCH(p, q) model . . . . . . . . . . . . . . . . . . . . . . . . . .
1.3
APARCH(p, q) model . . . . . . . . . . . . . . . . . . . . . . . . .
1.4
FIGARCH(p, d, q) model . . . . . . . . . . . . . . . . . . . . . . .
1.5
FIAPARCH(p, d, q) model . . . . . . . . . . . . . . . . . . . . . .
1.6
Other methods of interday volatility modelling . . . . . . . .
Intraday volatility models: review . . . . . . . . . . . . . . . . . . . . . .
2.1
ARFIMA k, d , l model . . . . . . . . . . . . . . . . . . . . . . . .
2.2
ARFIMA k, d , l - GARCH p, q model . . . . . . . . . . . . .
2.3
HAR-RV model . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.4
HAR-sqRV model . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2.5
HAR-GARCH(p, q) model . . . . . . . . . . . . . . . . . . . . . .
2.6
Other methods of intraday volatility modelling . . . . . . .
Volatility forecasting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1
One-step-ahead volatility forecasting: Interday volatility
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.2
Daily volatility models: program construction . . . . . . . .
3.3
One-step-ahead volatility forecasting: intraday volatility
models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.4
Intraday volatility models: program construction . . . . . .
The construction of loss functions . . . . . . . . . . . . . . . . . . . . .
4.1
Evaluation or loss functions . . . . . . . . . . . . . . . . . . . . .
4.2
Information criteria . . . . . . . . . . . . . . . . . . . . . . . . . .
4.3
Loss functions depend on the aim of a specific. application
........

. . 115
.
.
.
.

.
.
.
.

119
119
121
121

. . 122
. . 123


ix

Contents

6

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

128
128
130
133

Realized Volatility Forecasting: Applications . . . . . . . . . . . . . . . .
1
Measuring realized volatility . . . . . . . . . . . . . . . . . . . . . . .
1.1
Volatility signature plot . . . . . . . . . . . . . . . . . . . . . .

1.2
Interday adjustment of the realized volatility . . . . . . .
1.3
Distributional properties of realized volatility . . . . . . .
2
Forecasting realized volatility . . . . . . . . . . . . . . . . . . . . . . .
3
Programs construction . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Realized volatility forecasts comparison: SPEC criterion . . . .
5
Logarithmic realized volatility forecasts comparison: SPA and
DM Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.1
SPA test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.2
DM test . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.

.
.
.

.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.

.
.

161
161
162
165
174
176
178
190

6

Recent Methods: A Review . . . . . . . . . . . . . . . . . . . . . . . . .
1
Modelling jumps . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1
Jump volatility measure and jump tests . . . . . . . .
1.2
Daily jump tests . . . . . . . . . . . . . . . . . . . . . . . .
1.3
Intraday jump tests . . . . . . . . . . . . . . . . . . . . . .
1.4
Using OxMetrics (Re@lized under G@RCH 6.1) . .
2
The RealGARCH model . . . . . . . . . . . . . . . . . . . . . . .
2.1
Realized GARCH forecasting . . . . . . . . . . . . . . .
2.2

Leverage effect . . . . . . . . . . . . . . . . . . . . . . . . .
2.3
Realized EGARCH . . . . . . . . . . . . . . . . . . . . . .
3
Volatility forecasting with HAR-RV-J and HEAVY models
3.1
The HAR-RV-J model . . . . . . . . . . . . . . . . . . .
3.2
The HEAVY model . . . . . . . . . . . . . . . . . . . . .
4
Financial risk measurements . . . . . . . . . . . . . . . . . . . .
4.1
The method . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.

.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.

.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.

.
.
.
.
.
.
.
.
.
.
.
.
.
.
.

217
217
218
219
220
221
230
232

234
234
235
235
236
238
238

7

Intraday Hedge Ratios and Option Pricing . . . . . . . . . . . . .
1
Introduction to intraday hedge ratios . . . . . . . . . . . . .
2
Definition of hedge ratios . . . . . . . . . . . . . . . . . . . . .
2.1
BEKK model . . . . . . . . . . . . . . . . . . . . . . . . .
2.2
Asymmetric BEKK model . . . . . . . . . . . . . . . .
2.3
Constant Conditional Correlation (CCC) model
2.4
Dynamic Conditional Correlation (DCC) model
2.5
Estimation of the models . . . . . . . . . . . . . . . . .
3
Data . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4
Estimated hedge ratios . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.

.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.
.
.

243
243

246
248
248
249
250
251
251
253

5

Simulate the SPEC criterion . . . .
6.1
ARMA(1,0) simulation . . .
6.2
Repeat the simulation . . . .
6.3
Intraday simulated process

.
.
.
.

.
.
.
.

.

.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.

.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.
.
.
.

.

.
.
.

.
.
.
.

.
.
.
.
.
.
.
.
.
.

.
.
.
.

.
.
.
.

. . . . . 200
. . . . . 200
. . . . . 202


x

Contents

5
6
7
8

9

Hedging effectiveness . . . . . . . . . . . . . . . . . . . . . . . .
Other models for intraday hedge ratios . . . . . . . . . . . .
Introduction to intraday option pricings . . . . . . . . . . .
Price movement models . . . . . . . . . . . . . . . . . . . . . .
8.1
The approach of Merton . . . . . . . . . . . . . . . . .
8.2
The approach of Scalas and Politi . . . . . . . . . . .
8.3
Relation between the distributions of the epochs
and durations . . . . . . . . . . . . . . . . . . . . . . . .
8.4
Price movement . . . . . . . . . . . . . . . . . . . . . . .
Option pricing . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

9.1
The approach of Merton . . . . . . . . . . . . . . . . .
9.2
The approach of Scalas and Politi . . . . . . . . . . .
9.3
Time t is an epoch . . . . . . . . . . . . . . . . . . . . .
9.4
Time t is not an epoch . . . . . . . . . . . . . . . . . .
9.5
Other models for intraday option pricing . . . . .

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.

.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

.
.
.
.
.
.

256
259
259
260
261
261

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.

.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.
.
.

.
.
.
.
.
.

.
.

.
.
.
.
.
.
.
.

262
263
265
265
265
266
267
269

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 274


List of Figures
1.1
1.2
2.1

Flash crash of May 6, 2010 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Knight Capital collapse (August 2012) . . . . . . . . . . . . . . . . . . . . . . .
Determination of realized variance for day t,
(τ )

RVt
4.1

4.2

=

τ

j=1

8
9

2

log Ptj − log Ptj−1 , when 1000 intraday observations

are available and τ = 100 equidistant points in time are considered . . . . 45
The cumulative density function of the tri-variate minimum
multivariate gamma distribution, FX(1) (x; a, C123 ) =
3FX1 (x) − 2i1 =1 3i2 =2 FXi1 ,Xi2 (x, x) + FX1 ,X2 ,X3 (x, x, x) . . . . . . . . . . . 118
The logarithmic of EURONEXT 100 one-trading-day realized
(τ )

volatility, log 252RVt(HL∗ ) , from 12 February 2001 to 23 March 2006 . . 124

4.3

4.4

(τ )

The log 252RVt(HL∗ ) of EURONEXT 100 against the
one-trading-day-ahead realized volatility forecasts, for the period
from 25 January 2005 to 23 March 2006 . . . . . . . . . . . . . . . . . . . . . . 127
ARMA(1,0) data generated process, with the number of points in
time that the ARMA(1,0) and ARMA(0,1) models are selected by
the SPEC algorithm, for various values of T, and in particular for

5.1

5.2

T = 1,...,70 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 130
Average daily squared log-returns, T −1 Tt=1 yt2 , average daily
(τ )
realized volatility, T −1 Tt=1 RVt , and average intraday
T
τ −1
τ
−1
autocovariance, 2T
t=1
i=j+1 yti yti−j , for sampling
j=1
frequency of m = 1, 2, . . . , 40 minutes . . . . . . . . . . . . . . . . . . . . . . . . 163

The annualized one-trading-day realized standard deviation,
(τ )

5.3

5.4

252RVt . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 165
The ω1 and ω2 estimates (for interday adjustment of realized
volatility), for 200 iterations, excluding at each iteration either the
highest value of the closed-to-open interday volatility or the highest
value of the intraday volatility . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 167
The annualized one-trading-day interday adjusted realized standard
(τ )

5.5

deviation, 252RVt(HL∗ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 170
The estimated density of annualized one-trading-day interday
(τ )
adjusted realized daily variances, 252RVt(HL∗ ) . . . . . . . . . . . . . . . . . . . 171

xi


xii

List of Figures

5.6

The estimated density of annualized one-trading-day interday

5.7

(τ )
adjusted realized daily standard deviations, 252RVt(HL
∗ ) . . . . . . . . . . 172
The estimated density of annualized interday adjusted realized daily
(τ )

5.8
5.9

logarithmic standard deviations, log 252RVt(HL∗ ) . . . . . . . . . . . . . . . 172
The estimated density of log-return series, yt . . . . . . . . . . . . . . . . . . . 173
The estimated density of standardized log-return series,
standardized with the annualized one-trading-day interday
(τ )

adjusted realized standard deviation, yt / 252RVt(HL∗ ) . . . . . . . . . . . . 173
(τ )

5.10 The log 252RVt+1(HL∗ ) against the annualized interday adjusted
realized daily logarithmic standard deviation forecasts,
(τ )

log 252RVt+1|t(HL∗ ) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 179
5.11 The parameter estimates of the three models across time . . . . . . . . . . . 182
5.12 The annualized interday adjusted realized volatility and its forecast,

(τ )

252RV(un),t+1|t(HL∗ ) from the model with the lowest half-sum of
the squared standardized prediction errors . . . . . . . . . . . . . . . . .
5.13 The estimated density of the standardized one-step-ahead
prediction errors, zt+1|t , from the ARFIMA(1, d , 1)-TARCH(1,1)
model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5.14 The estimated density of the standardized one-step-ahead
prediction errors, zt+1|t , from the HAR-TARCH(1,1) model . . . .
5.15 The estimated density of the standardized one-step-ahead
prediction errors, zt+1|t , from the AR(2) model . . . . . . . . . . . . .
6.1 Simulated DAX30 instantaneous log-prices, 1-min log-prices and
daily prices of a continuous-time GARCH diffusion model . . . . .
6.2 Simulated DAX30 instantaneous returns, 1-min returns and daily
returns of a continuous-time GARCH diffusion model . . . . . . . .
2 , integrated
6.3 DAX30 volatility measures: simulated volatility σt+
volatility IVt , conditional volatility from GARCH(1,1) on daily
returns, and daily squared log-returns . . . . . . . . . . . . . . . . . . . .
6.4 DAX30 continuous-time GARCH diffusion process with jumps . .
6.5 DAX30 simulated and detected jumps . . . . . . . . . . . . . . . . . . . .
6.6 DAX30 integrated volatility and realized volatility for a
continuous-time GARCH model with jumps . . . . . . . . . . . . . . .
6.7 DAX30 integrated volatility and Bi-power variation for a
continuous-GARCH jump process . . . . . . . . . . . . . . . . . . . . . .
6.8 DAX30 integrated and realized jumps (using bipower variation) . .
6.9 DAX30 integrated volatility and realized outlyingness
weighted variance . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

. . . . 194

. . . . 197
. . . . 198
. . . . 199
. . . . 222
. . . . 223

. . . . 223
. . . . 224
. . . . 225
. . . . 225
. . . . 227
. . . . 227
. . . . 228


List of Figures

xiii

6.10 DAX30 integrated and realized jumps (using realized outlyingness
weighted variance) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 229
6.11 DAX30 simulated and detected jumps (Intraday jumps) . . . . . . . . . . . 229
7.1 Evolution of the hourly hedge ratios of the DAX 30 index, from 3
May 2000 to 29 November 2013, (9:00 a.m. to 5:00 p.m.) . . . . . . . . . . . 255


List of Tables
2.1

2.2

Values of the MSE loss functions. The data generating process is
the continuous time diffusion
√
log p (t + dt) = log p (t) + σ (t) dtW1 (t) ,
√
σ 2 (t + dt) = 0.00108dt + σ 2 (t) 1 − 0.083dt + 0.084dtW2 (t) . . . 46
Averages of the values of the MSE loss functions of the 200
simulations. The data generating process is the
√ continuous time
diffusion log p (t + dt) = log p (t) + σ (t) dtW1 (t) ,
√
σ 2 (t + dt) = 0.00108dt + σ 2 (t) 1 − 0.083dt + 0.084dtW2 (t) . . . 46

4.1(A) The probability 1 − p that the minimum X(1) of a trivariate
gamma vector is less than or equal to ω1−p for 2 ≥ ω1−p ≥ 50,
5 ≥ a ≥ 50, and ρ1,2 = 30%, ρ1,3 = 60% and ρ2,3 = 95%, the
non-diagonal elements of C123 . . . . . . . . . . . . . . . . . . . . . . . . .
4.1(B) The probability 1 − p that the minimum X(1) of a trivariate
gamma vector is less than or equal to ω1−p for 2 ≥ ω1−p ≥ 50,
5 ≥ a ≥ 50, and ρ1,2 = 95%, ρ1,3 = 95% andρ2,3 = 95%, the
non-diagonal elements of C123 . . . . . . . . . . . . . . . . . . . . . . . . .
4.2
The half-sum of squared standardized one-day-ahead prediction
errors of the three estimated realized volatility models,
2(mi )
Xmi ≡ 2−1 60
t=1 zt+1|t , for i = 1, 2, 3 . . . . . . . . . . . . . . . . . . . . .
4.3

Selected values of the cumulative density function,
FX(1) ω1−p ; a = 30, C123 . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.4
ARMA(1,0) data generated process. The half sum of the squared
2
standardized one-step-ahead prediction errors: 12 1000
t=71 zt+1|t . . .
4.5
The ARMA(1,0) data generated process. The average values (100
iterations) of the loss functions and the percentages of times a
model achieves the lowest value of the loss function . . . . . . . . . .
4.6
The HAR-RV data generated process, with the half-sum of the
squared standardized one-step-ahead prediction errors:
1000 2
1
t=1 zt+1|t . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2
4.7
The HAR-RV data generated process, with the average values (100
iterations) of the loss functions and the percentages of times a
model achieves the lowest value of the loss function. . . . . . . . . . .
5.1
Information for the intraday data . . . . . . . . . . . . . . . . . . . . . . .
(τ )
5.2
Estimation of the interday adjusted realized volatility, RVt(HL∗ ) . .
xiv

. . . 116

. . . 117

. . . 128
. . . 128
. . . 129

. . . 132

. . . 133

. . . 134
. . . 162
. . . 169


List of Tables

5.3

xv

5.4

Descriptive statistics of annualized one-trading-day interday
(τ )
adjusted realized daily variances, 252RVt(HL∗ ) . . . . . . . . . . . . . . . . . 175
Descriptive statistics of annualized one-trading-day interday

5.5

adjusted realized daily standard deviations, 252RVt(HL∗ ) . . . . . . . . . 175
Descriptive statistics of annualized interday adjusted realized daily

(τ )

(τ )

5.6
5.7

logarithmic standard deviations, log 252RVt(HL∗ ) . . . . . . . . . . . . . . 175
Descriptive statistics of daily log-returns, yt . . . . . . . . . . . . . . . . . . . 175
Descriptive statistics of standardized log-returns, standardized
with the annualized one-trading-day interday adjusted realized
(τ )

5.8
5.9
5.10
5.11
5.12
5.13

standard deviation, yt / 252RVt(HL∗ ) . . . . . . . . . . . . . . . . . . . . . . . 176
The half-sums of squared standardized one-trading-day-ahead
˜

2(m )

prediction errors of the three models, Xmi ≡ 2−1 Tt=1 zt+1|ti . . .
Descriptive statistics of the residuals, εt|t , from the three models
Descriptive statistics of the one-step-ahead prediction errors,
εt+1|t , from the three models . . . . . . . . . . . . . . . . . . . . . . . . .
Descriptive statistics of the standardized one-step-ahead
prediction errors zt+1|t , from the three models . . . . . . . . . . . . .
The percentage of times a model achieves the lowest value of the
2
, for T = 60 . . . . . . . . . . . . . . . . .
loss function, 12 Tt=1 zt+1|t

. . . . 194
. . . . 195
. . . . 196
. . . . 196
. . . . 197

At each trading day T + 1, select the model with
min

1
2

Tˆ
2
t=1 zt+1|t

. Then compute the half-sum of z 2ˆ

, for

T+1|Tˆ
the total of T˜ − T˘ − T trading days. Each z 2ˆ ˆ is computed
T+1|T
ˆ
2
from the model with min 12 Tt=1 zt+1|t
at each trading day T.
˜ T−T
˘
2
The first column presents the loss function 12 T−
zt+1|t
t=1

5.14

from the strategy of selecting at each trading day the model
proposed by the SPEC criterion. The last three columns present
˜ T−T
˘
2
zt+1|t
of each model . . . . . . . . . . . . . . . 199
the loss function 12 T−
t=1
The average of the squared predictive errors of the three models,
˜
(mi )
¯ (mi ) = T˜ −1 Tt=1

, for
t
(MSE)
(mi )
t

5.15
5.16

(τ )(m )

(τ )

i
= log 252RVt+1|t(HL
252RVt(HL∗ )
∗ ) − log

2

. . . . . . . . . . . 201

The output produced by the program SPA Test Euronext100.ox . . . . . 201
The SPA test consistent p-values of the null hypothesis that the
(mi )
is statistically superior to its competitors . . 201
model with min ¯ (MSE)


xvi

5.17

List of Tables

The DM test statistic for testing the null hypothesis that the m1
model has equal predictive ability with m2 model, or
(m1 )
(m2 )
= 0, against the alternative hypothesis
− (MSE)t
E (MSE)t
(m )

6.1
6.2
6.3
7.1
7.2

7.3
7.4

(m )

1
2
that E (MSE)t
<0 . . . . . . . . . . . . . . . . . . . . . . . .
− (MSE)t

Z jump statistic on log(RV) . . . . . . . . . . . . . . . . . . . . . . . . . . .
Z jump statistic on RV-ROWVar . . . . . . . . . . . . . . . . . . . . . . .
Lee & Mykland test for jumps . . . . . . . . . . . . . . . . . . . . . . . . .
Summary statistics for the hourly returns, from 3 May 2000 to
29 November 2013 (9:00 a.m. to 5:00 p.m.) . . . . . . . . . . . . . . . .
Estimates of the different models for the sample of hourly
observations from 9:00 a.m. to 5:00 p.m. of the DAX 30 index and
its corresponding future contracts (3 May 2000 to 29
November 2013) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Summary statistics for the hourly hedge ratios . . . . . . . . . . . . . .
Effectiveness analysis of the strategies. The variance reduction, the
(a)
(VaR)
loss
percentage of VaRt violations and the 10 Tt=1 t
function, for 1 − a at 5%, 1% and 0.1%, are computed for the
hourly DAX 30 index for the period 21 June 2001 to 29 November
2013 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

.
.
.
.

.
.
.
.

.

.
.
.

203
226
226
228

. . . 252

. . . 254
. . . 255

. . . 258


Acknowledgments
Dr. Stavros Degiannakis and Dr. Christos Floros acknowledge the support
from the European Community’s Seventh Framework Programme (Marie Curie
FP7-PEOPLE-IEF & FP7-PEOPLE-RG) funded under grant agreements no.
PIEF-GA-2009-237022 & PERG08-GA-2010-276904. We would like to thank
Dr. Enrique Salvador and Dr. Thomas Poufinas for their constructive contribution
to the 7th chapter. We are also grateful to the many people (colleagues at
the university, researchers, traders and financial market practitioners), who by
occasional informal exchange of views have had an influence on these aspects as
well. Most importantly, we wish to express our gratitude to our families for their
support.

xvii

List of Symbols and Operators
d(.)
rf
AIC
rannual
Pask,ti
γi
δ
ci
¯ (.)

indicator function; i.e. d(yt > 0) = 1 if yt > 0, and
d(yt > 0) = 0 otherwise.
risk-free rate.
Akaike’s information criterion.
annual risk-free interest rate.
ask price.
asymmetric parameter in ARCH model.
asymmetric power parameter in ARCH model.
autoregressive coefficients.
average of predictive loss/evaluation function, i.e.
¯ (MSE) = T˜ −1

μ2
μ1
Pbid,ti
BVt (m)
CTS

wi
i)
μ(m
t
μt
μ(.)
σt
σˆ t2
g(.)
CGR
ρ
dqt
σm,i
η12

T˜
t=1

(τ )
(τ ) 2
(RVt+1|t
− RVt+1
) .

average of the daily realized variances.
average of the squared closed-to-open log-returns.
bid price.
Bi-Power variation at m sampling frequency.
calendar time sampling.
coefficients of HAR model.

conditional mean estimation of model mi .
conditional mean.
conditional mean’s functional form.
conditional standard deviation.
conditional variance estimate.
conditional variance’s functional form.
Correlated Gamma Ratio distribution.
correlation coefficient.
counting process.
covariance of market returns and asset.
covariance of realized variance and squared closed-to-open
log-returns.
FX(1) (x; a, C12...n ) cumulative distribution function of X(1) .
cumulative distribution function of durations J.
FJ∗ n (t)

xviii


List of Symbols and Operators

FTn (t)
F(.)
DM(.,.)
(.,.)
t

k (mi )
Ji = Ti − Ti−1
{Ti }M

i=1
τ
ω1 , ω2
θˆ (T)
d
d
(m)

yˆt
LT (.)
(.)
Xmi
HQ
c inf
It
εt−i|t
yt−i|t
p (t)
log p (t)
σ (t)
2(IQ)

cumulative distribution function of epoch Tn .
cumulative distribution function.
Diebold Mariano Statistic.
difference of loss/evaluation functions (evaluation
differential).
dimension of vector of unknown parameters β (mi ) .
durations.
epochs.

equidistance points (sub-intervals) in time
(τ )
estimated parameters of RVt(HL∗ ) .
estimator of θ based on a sample of size T.
exponent of the fractional differencing operator(1 − L)d in
ARFIMA models.
exponent of the fractional differencing operator(1 − L)d in
FIGARCH & FIAPARCH models.
forecasts of yt from model m.
full sample log-likelihood function based on a sample of
size T.
Gamma function.
half sum of squared standardized one-step-ahead prediction
errors of model mi .
Hannan and Quinn information criterion.
infimum.
information set.
in-sample fitted error at time t − i based on information
available at time t.
in-sample fitted value of conditional mean at time t − i
based on information available at time t.
instantaneous (unobserved) asset price.
instantaneous logarithmic asset price.
instantaneous variance of the rate of return.

σ[a,b]

integrated quarticity.

Plip,tj

Ku
(LM)
Jt,α
L
ytj
yt

integrated variance over the interval [a, b].
interpolated price.
kurtosis.
L&M statistic.
lag operator.
log-return over the sub-interval tj − tj−1 .
log-returnsover the sub-interval [t, t − 1].

2(IV )
σ[a,b]

xix


xx

List of Symbols and Operators
(.)
t

εtj = log Ptj − log ptj
(m )
Xt i

(MAD)
σi,t
MMG
X(1)
m(1)
di
MSE (τ )

loss/evaluation function that measures the distance between
volatility and its forecast.
market microstructure noise.
(m )
matrix of xt i explanatory variables.
Median Absolute Deviation.
Minimum Multivariate Gamma distribution.
minimum value of Xmi .
model with the lowest value of Xmi .
moving average coefficients.
MSE loss (or evaluation) function.

T
n
T˜
T˘
θ˘
T
P tj

number of forecasts for out-of-sample evaluation.
number of models or variables.

number of observations for out-of-sample forecasting.
number of observations for rolling sample.
number of parameters of vector θ .
number of total observations.
observed asset price.

τ
j=1
(τ )
RVt+1|t
(τ )

RV(un),t+1|t
yt+1|t
(m )

i
yt+1|t

2
σt+1|t
2
ht+1|t

εt+1|t
zt+1|t
q
p
k
l

di,t
C (L)
B (L)
(L)
D (L)

one-day-ahead realized variance at time t + 1 based on
information available at time t.
one-day-ahead realized variance at time t + 1 based on
information available at time t (unbiased estimator).
one-step-ahead conditional mean at time t + 1 based on
information available at time t.
one-step-ahead conditional mean at time t + 1 based on
information available at time t of model mi .
one-step-ahead conditional variance at time t + 1 based on
information available at time t.
one-step-ahead estimate of integrated quarticity given the
information available at time t.
one-step-ahead prediction error at time t + 1 based on
information available at time t.
one-step-ahead standardized prediction error at time
t + 1based on information available at time t.
order of ARCH form.
order of GARCH form.
order of the autoregressive model.
order of the moving average model.
outlyingness measure.
polynomial of autoregressive model - AR.
polynomial of FIGARCH & FIAPARCH models.
polynomial of FIGARCH & FIAPARCH models.

polynomial of moving average model - MA.


List of Symbols and Operators

xxi

Ppre,tj
previous tick price.
price log-range.
lRange[a,b]
price of the European call option at time t.
c E (t)
Range[4],[a,b]
price range, four-data-points.
price range.
Range[a,b]
f (.)
probability density function.
Quad-power quarticity at m sampling frequency.
QQt (m)
K
realized jumps at m sampling frequency.
RJt (m)
Realized Outlyingness Weighted Covariation.
ROWQCovt
ROWQuarticityt (m) Realized Outlyingness Weighted Quarticity at m sampling
frequency.
Realized Outlyingness Weighted Variance.
ROWVart

[2q]
realized power variation of order 2q.
RV[a,b]
RCovt
realized quadratic covariation.
(τ )
RVt
realized volatility at time t, divided in τ points in time.
realized volatility for the time interval [a, b].
RV[a,b]
(τ )
RVt(n)
realized volatility of n-trading-days.
(τ )
realized volatility with Fleming’s et al dynamic scaling.
RVt(FKO)
(τ )
realized volatility with Hansen and Lunde’s interday
RVt(HL∗ )
adjustment.
(τ )
realized volatility with Marten’s interday adjustment.
RVt(Martens)
γd (i)
sample autocovariance of ith order.
sampling frequency.
m = τb−a
−1
SBC
Schwarz information criterion.

SH
Shibata information criterion.
spectral density at frequency zero.
fd (0)
N (.)
standard normal density function.
standard normal distribution.
zt ∼ N (0, 1)
W (t)
standard Wiener process.
SPA
Superior Predictive Ability statistic.
time index, tj [a, b].
tj
time interval.
[a, b]
Tri-power quarticity at m sampling frequency.
TQt (m)
(m )
zt i
unpredictable component of model mi .
unpredictable component.
εt
variance of the daily realized variances.
η2
variance of the market.
σm2
variance of the squared closed-to-open log-returns.
η1
vector of dependent variable yt .

Yt
θ
vector of estimated parameters for the conditional mean
and variance.


xxii

List of Symbols and Operators

θ (t)

w
(m )
xt−1i
β
β (mi )
υt

(ES)
t
(VaR)
t
(a)
ESt
(a)
VaRt

rh,t
Rt

rt
zt
ρij
σf2,t
σs,t σf ,t
2
σs,t
t (.)
εt
μt (.)
R2
Cov (., .)
MVHR
diag (.)

vector of estimated parameters for the conditional mean
and variance
at time t.
vector of estimated parameters for the density function f .
vector of explanatory variables of mi regression model.
vector of parameters for estimation in regression model.
vector of parameters for estimation of mi regression model.
vector of predetermined variables included in It .
loss/evaluation function for Expected Shortfall.
loss/evaluation function for VaR.
Expected Shortfall of a portfolio at confidence level a.
Value-at-Risk of a portfolio at confidence level a.
log-return of the hedged portfolio at time t.
conditional correlation matrix.
vector of returns.

vector of standardized error term (residuals).
constant correlation of spot and future price returns.
variance of future price returns at time t.
covariance of spot and future price returns at time t.
variance of spot price returns at time t.
conditional variance-covariance matrix.
vector of error term (residuals).
vector of conditional mean.
first order log-difference.
log difference.
coefficient of determination.
covariance.
minimum-variance hedge ratio.
diagonal matrix.


1

Introduction to High
Frequency Financial
Modelling

The chapter presents an introduction to High Frequency Trading (HFT) and
focuses on the role of volatility using case studies. Further, we discuss recent
empirical researches on volatility forecasting and market microstructure.
Figlewski (2004) argues that a financial market is an institution set up by
human beings, and the behaviour of security prices established in it depend on
human activity. Since financial markets change continuously due to the uncertain
behaviour of investors, the accuracy of forecasting market behaviour is possible
only to the extent that the change of a financial instrument is relatively gradual

most of the time; forecasting the financial market (and its products) is a challenge
for financial modellers and economists. A basic financial instrument is referred to
as equity, stock or shares. Further, security is an instrument representing ownership
(stocks), a debt agreement (bonds), or the rights to ownership (derivatives). Stock,
for example, is the ownership of a small price of a company or firm; i.e. it gives
stockholders a share of ownership in a company. Its price is determined by the value
of the company and by the expectations of the performance of the company. These
expectations (i.e. behaviour of bid and ask prices) give an uncertainty to the future
price development of the stock. The stock value is either higher or lower than the
expected value. Therefore, the amount in which the stock value can differ from the
expected value is determined by the so-called volatility.
Volatility of returns is a key variable for researchers and financial analysts. Most
financial institutions make volatility assessments by monitoring their risk exposure.
Volatility defines the variability of an asset’s price over time (measured in percentage
terms). According to Figlewski (2004), volatility “is simply synonymous with risk:
high volatility is thought of as a symptom of market disruption ... [it] means that
securities are not being priced fairly and the capital market is not functioning as
well as it should”.
Volatility is a statistical measure of the tendency of a market or security price
to rise or fall sharply within a period of time. It can be measured by using the
variance of the price returns. Return is the gain or loss of a security in a particular
period, quoted as a percentage. The return consists of the income and the capital

1


2

Modelling and Forecasting High Frequency Financial Data

gains relative to an investment. A highly volatile market means that prices have
huge swings (moves) in very short periods of time (Tsay, 2005). Volatility dynamics
have been modelled to account for several features (stylized facts): clustering, slowly
decaying autocorrelation, and nonlinear responses to previous market information
of a different type (Corsi et al., 2012). Moreover, according to Foresight (2011),
price volatility is an indicator of financial instability in the market.
Financial volatility is time-varying, and therefore is a key term in asset pricing,
portfolio allocation and market risk management. Financial analysts are concerned
with modelling volatility, i.e. the covariance structure of asset returns. Further,
the subject of financial econometrics pays high attention to the modelling and
forecasting of time-varying volatility, i.e. the measurement and management of
risk. According to Tsay (2005, pp. 97–98), modelling the volatility of a time
series can improve the efficiency in parameter estimation and the accuracy in
interval forecast. The finance literature examined the so-called Autoregressive
Conditional Heteroscedasticity1 (ARCH) class of models of volatility (see Engle,
1982; Bollerslev, 1986), while in recent years this literature has benefited from the
availability of high-frequency data (1-second, 1-minute, 1-hour data, etc.). Since
the seminal paper of Andersen and Bollerslev (1998), much of the literature deals
with the development of the realized volatility as well as bi-power variation and
jumps tests. These techniques improved the measures of volatility, but also the
efficiency of the financial markets (i.e. market prices reflect the true underlying
value of the asset). In particular, the huge amount of intraday data provides
important information regarding fluctuations of assets and their co-movements;
this helps in understanding dynamics of financial markets and volatility behaviour,
while it may yield to more accurate measurements of volatility. However, the use of
high-frequency data (and its trading algorithms) may give several problems such as
the observation asynchronicity and/or market microstructure noise; i.e. academics
now are interested in estimating consistently the variance of noise, which can be
regarded as a measure of the liquidity2 of the market, or the quality of the trade
execution in a given exchange or market structure (see Ait-Sahalia and Yu, 2009).

In other words, new models (techniques) for describing high frequency strategies
under several trading conditions are necessary.

1 The role of high frequency trading
High frequency trading (henceforth HFT) strategies update orders very fast and
have no over night positions. HFT realizes profits per trade, and hence focuses
mainly on highly liquid instruments. Therefore, HFT relies on high speed access
to markets and their data using advanced computing technology. In other words,