An Introduction to Wavelet Theory in Finance: A Wavelet Multiscale Approach

213 Pages·2012·1.38 MB·English

by  Francis In, Sangbae Kim

An Introduction to
Wavelet Theory
in Finance
A Wavelet Multiscale Approach
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An Introduction to
Wavelet Theory
in Finance
A Wavelet Multiscale Approach
Francis In
Monash University, Australia
Sangbae Kim
Kyungpook National University, Korea
World Scientifc
NE W J E R S E Y • L O N D O N • S I N G A P O R E • BE IJ ING • S H A N G H A I • H O N G K O N G • TA I P E I • C H E N N A I
8431hc_9789814397834_tp.indd 1 16/8/12 8:34 AM
Published by
World Scientific Publishing Co. Pte. Ltd.
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USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601
UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE
Library of Congress Cataloging-in-Publication Data
In, Francis.
An introduction to wavelet theory in finance : a wavelet multiscale
approach / by Francis In & Sangbae Kim.
p. cm.
Includes bibliographical references and index.
ISBN 978-9814397834 -- ISBN 9814397830
1. Finance--Mathematical models. 2. Wavelets (Mathematics) I. Kim, Sangbae, 1965–
II. Title.
HG106.I5 2012
332.01'5152433--dc23
2012030894
British Library Cataloguing-in-Publication Data
A catalogue record for this book is available from the British Library.
Copyright © 2013 by World Scientific Publishing Co. Pte. Ltd.
All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means,
electronic or mechanical, including photocopying, recording or any information storage and retrieval
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Clearance Center, Inc., 222 Rosewood Drive, Danvers, MA 01923, USA. In this case permission to
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In-house Editor: Alisha Nguyen
Typeset by Stallion Press
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Contents
1. Methodology: Introduction to Wavelet Analysis 1
1.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 1
1.2 Fourier Analysis and Spectral Analysis . . . . . . . . . . 2
1.2.1 Fourier analysis . . . . . . . . . . . . . . . . . . . 2
1.2.2 Spectral analysis . . . . . . . . . . . . . . . . . . 5
1.2.3 Comparison between Fourier transform
and wavelet transform . . . . . . . . . . . . . . . 9
1.3 Wavelet Analysis . . . . . . . . . . . . . . . . . . . . . . 11
1.3.1 Continuous wavelet transform . . . . . . . . . . . 11
1.3.2 Discrete wavelet transform . . . . . . . . . . . . . 14
1.3.3 Maximal overlap discrete wavelet transform . . . 24
1.3.4 Boundary condition . . . . . . . . . . . . . . . . . 27
1.4 Wavelet Variance, Covariance and Correlation . . . . . . 29
1.4.1 Wavelet variance . . . . . . . . . . . . . . . . . . 29
1.4.2 Wavelet covariance and correlation . . . . . . . . 32
1.4.3 Cross wavelet covariance and correlation . . . . . 35
1.5 Long Memory Estimation Using Wavelet Analysis . . . . 36
1.5.1 Definitions of long memory . . . . . . . . . . . . . 36
1.5.2 Wavelet ordinary least square . . . . . . . . . . . 37
1.5.3 Approximate maximum-likelihood estimation
of the long memory parameter . . . . . . . . . . . 37
1.5.4 Another estimation method of the long
memory parameter . . . . . . . . . . . . . . . . . 38
v
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vi An Introduction to Wavelet Theory in Finance: A Wavelet Multiscale Approach
2. Multiscale Hedge Ratio Between the Stock and Futures
Markets: A New Approach Using Wavelet Analysis
and High Frequency Data 41
2.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 41
2.2 Minimum Variance Hedge . . . . . . . . . . . . . . . . . 44
2.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . 46
2.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . 53
3. Modeling the International Links Between the Dollar,
Euro and Yen Interest Rate Swap Markets Through
a Multiscaling Approach 57
3.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Data and Descriptive Statistics . . . . . . . . . . . . . . 60
3.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . 62
3.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . 73
4. Long Memory in Rates and Volatilities of LIBOR:
Wavelet Analysis 75
4.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 75
4.2 Data and Empirical Results . . . . . . . . . . . . . . . . 78
4.3 Summary and Concluding Remarks . . . . . . . . . . . . 84
5. Cross-Listing and Transmission of Pricing Information
of Dually-Listed Stocks: A New Approach Using Wavelet
Analysis 87
5.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 87
5.2 Data Description and Basic Statistics . . . . . . . . . . . 92
5.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . 98
5.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . 103
6. On the Relationship Between Stock Returns and Risk
Factors: New Evidence From Wavelet Analysis 105
6.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 105
6.2 Data and Basic Statistics . . . . . . . . . . . . . . . . . . 108
6.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . 110
6.3.1 Results from the traditional CAPM . . . . . . . . 111
6.3.2 Results using two risk factors: Excess market
returns and SMB . . . . . . . . . . . . . . . . . . 115
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Contents vii
6.3.3 Results using three factors: Excess market returns,
SMB and HML . . . . . . . . . . . . . . . . . . . 118
6.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . 122
7. Can the Risk Factors Explain the Cross-Section
of Average Stock Returns in the Long Run? 125
7.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 125
7.2 Data and Basic Statistics . . . . . . . . . . . . . . . . . . 128
7.3 Empirical Results . . . . . . . . . . . . . . . . . . . . . . 132
7.3.1 Traditional CAPM context . . . . . . . . . . . . . 132
7.3.2 Fama–French three factor model . . . . . . . . . . 135
7.3.3 Fama–French three-factor model augmented
by the momentum factor . . . . . . . . . . . . . . 141
7.4 Conclusion . . . . . . . . . . . . . . . . . . . . . . . . . . 146
8. Multiscale Relationships Between Stock Returns
and Inflations: International Evidence 147
8.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 147
8.2 Research Methodologies . . . . . . . . . . . . . . . . . . 149
8.2.1 The multi-scale hedge ratio . . . . . . . . . . . . 149
8.2.2 The bootstrap approach . . . . . . . . . . . . . . 150
8.3 Data and Empirical Results . . . . . . . . . . . . . . . . 151
8.4 Summary and Concluding Remarks . . . . . . . . . . . . 157
9. Mutual Fund Performance and Investment Horizon 161
9.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 161
9.2 Sharpe Ratio at Different Investment Horizons . . . . . . 164
9.3 Data and Empirical Results . . . . . . . . . . . . . . . . 164
9.3.1 Data . . . . . . . . . . . . . . . . . . . . . . . . . 164
9.3.2 Rank correlation between investment horizons . . 165
9.3.3 Robustness of the findings . . . . . . . . . . . . . 167
9.4 Concluding Remarks . . . . . . . . . . . . . . . . . . . . 174
10. A New Assessment of US Mutual Fund Returns Through
a Multiscaling Approach 177
10.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . 177
10.2 Empirical Method . . . . . . . . . . . . . . . . . . . . . . 179
10.2.1 Multiscaling approach . . . . . . . . . . . . . . . 179
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viii An Introduction to Wavelet Theory in Finance: A Wavelet Multiscale Approach
10.2.2 The bootstrap approach . . . . . . . . . . . . . . 180
10.3 Data and Empirical Results . . . . . . . . . . . . . . . . 181
10.3.1 Estimation results for the three aggregate groups 183
10.3.2 Estimation results for individual mutual funds . . 186
10.4 Summary and Concluding Remarks . . . . . . . . . . . . 190
References 191
Index 203
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Chapter 1
Methodology: Introduction to Wavelet
Analysis
1.1. Introduction
The multiscale relationship is important in economics and finance because
each investor has a different investment horizon. Consider the large number
of investors who participate in the stock market and make decisions over
different time scales. Stock market participants are a diverse group that
include intraday traders, hedging strategists, international portfolio man-
agers, commercial banks, large multinational corporations, and national
central banks. It is notable that these market participants operate on
very different time scales. In fact, due to the different decision-making
time scales among traders, the true dynamic structure of the relationship
between variables will vary over different time scales associated with those
different horizons. However, most previous studies focus on a two-scale
analysis — short-run and long-run. The reason being for this is mainly
a lack of empirical tools. Recently, wavelet analysis has attracted attention
in the fields of economic and finance as a means of filling this gap.
Wavelet analysis is relatively new in economics and finance, although
the literature on wavelets is growing rapidly. The studies, related to
economics and finance, can be divided into four categories: general
wavelet transform, stationary process (long memory), denoising, and
variance/covariance analysis. The first category includes Davidson et al.
(1998), Pan and Wang (1998), Ramsey and Lampart (1998a, 1998b), and
Chew (2001). Another stream of research is related to the long memory
1