Modelling-Financial-Times-Series- Sthepen J. Taylor, Manuais, Projetos, Pesquisas de Estatística

Baixe Modelling-Financial-Times-Series- Sthepen J. Taylor e outras Manuais, Projetos, Pesquisas em PDF para Estatística, somente na Docsity!

n

M o d ellin g Fin an ci a

Time Series

Second Edition

h

STEPHEN J * TAYLOR

Imcaster University, UK

Modelling Financial Time Serices

world scientific

Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA oflice: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK ofice: 57 Shelton Street, Covent Garden, London WC2H 9HE

Library of Congress Cataloging-in-PublicationData Taylor, Stephen (Stephen J.) Modelling financial time series / b y Stephen J Taylor. -- 2nd ed.

Reprint of the edition originally published: Chichester [West Sussex] ; New York :

Includes bibliographical references and index. ISBN-I 3: 978-981-277-084- ISBN- 10: 98 1 -277-084-

p. cm.

Wiley, c1986.

1. Stocks--Prices--Mathematical models. 2. Commodity exchanges--Mathematical models. 3. Financial futures--Mathematical models. 4. Time-series analysis. I. Title. HG4636.T35 2007 332.63'22201 1 --dc 2007043574

British Library Cataloguing-in-PublicationData A catalogue record for this book is available from the British Library.

Copyright 0 2008 by World Scientific Publishing Co. Re. Ltd. All rights reserved. This book. or parts thereof; may not be reproduced in any form or by any means, electronic. or tnechanical, including photocopying, recording or any information storage and retrieval system now known or to be invented, without written permission,from the Publisher.

For photocopying of material in this volume, please pay a copying fee through the Copyright Clearance Center, Inc., 222 Rosewood Drive, Danvers, M A 01923, USA. In this case permission to photocopy is not required from the publisher.

Printed in Singapore by World Scientific Printers ( S ) Pte Ltd

This page intentionally left blankThis page intentionally left blank

Contents-

Preface to the 2nd edition

Preface to the 1st edition

1 INTRODUCTION

1 .I Financial time series

1.2 About this study

The world’s major financial markets

Examples of daily price series

A selective review of previous research

important questions

The random walk hypothesis

The efficient market hypothesis

1.6 Daily returns

1.7 Models

1.8 Models in this book

1.9 Stochastic processes

General remarks

Stationary processes

Autocorrelation

Spectral density

White noise

ARMA processes

Gaussian processes

1 .I0 Linear stochastic processes

Their definition

Autocorrelation tests

2 FEATURES OF FINANCIAL RETURNS

2.1 Constructing financial time series

Sources

Time scales

Additional information

Using futures contracts

vii

Page

xv

xxv

Contents i x

Step variances, continuous distributions Markov variances, continuous distributions

Notation

3.3 A general variance model

3.4 Modelling variance jumps

General models Stationary models The lognormal, autoregressive model

General concepts Caused b y past squared returns Caused b y past absolute returns ARMACH models

Variances not caused b y returns Variances caused b y returns

Modelling frequent variance changes not caused by prices

3.6 Modelling frequent variance changes caused b y past prices

3.7 Modelling autocorrelation and variance changes

Parameter estimation for variance models Parameter estimates for product processes Lognormal AR(1) Results 3.10 Parameter estimates for ARMACH processes Results 3.11 Summary

Appendix 3(A) Results for ARCH processes

4 FORECASTING STANDARD DEVIATIONS

4.1 Introduction 4.2 Key theoretical results Uncorrelated returns Correlated returns Relative mean square errors Stationary processes 4.3 Forecasts: methodology and methods Benchmark forecast Parametric forecasts Product process forecasts ARMACH forecasts EWMA forecasts Futures forecasts

Empirical RMSE

4.4 Forecasting results Absolute returns

a

a a

X Con tents

Conditional standard deviations Two leading forecasts M o r e distant forecasts Conclusions about stationarity Another approach

Examples

4.6 Summary

Recommended forecasts f o r the next day

5 THE ACCURACY^ OF^ AUTOCORRELATION^ ESTIMATES

5.1 Introduction

5.2 Extreme examples

5.3 A special null hypothesis

5.5 Some asymptotic results

Estimates of t h e variances o f sample autocorrelations

Linear processes Non-linear processes

5.6 Interpreting the estimates

5.7 The estimates for returns

5.8 Accurate autocorrelation estimates

Rescaled returns Variance estimates f o r recommended coefficients Exceptional series 5.9 Simulation results

5.10 Autocorrelations o f rescaled processes

5.11 Summary

6 TESTING THE R A N D O M WALK HYPOTHESIS

6.1 Introduction

6.2 Test methodology

6.3 Distributions o f sample autocorrelations

Asymptotic limits Finite samples

6.4 A selection o f test statistics

Autocorrelation tests Spectral tests The runs test

Price-trend autocorrelations A n example Price-trend spectral density

6.6 Tests f o r random walks versus price-trends

6.7 Consequences o f data errors

6.5 The price-trend hypothesis

xii Contents

7.7 Further forecasting theory Expected changes in prices Forecasting the direction of the trend Forecasting prices

7.8 Summary

8 EVIDENCE AGAINST THE EFFICIENCY^ OF^ FUTURES MARKETS

8.1 Introduction

8.2 The efficient market hypothesis

8.3 Problems raised by previous studies

Filter rules Benchmarks Significance Optimization

Returns Risk Necessary assumptions

8.4 Problems measuring risk and return

8.5 Trading conditions

8.6 Theoretical analysis

Trading strategies Assumptions Conditions for trading profits Inefficient regions Some implications

Strategies Assumptions Notes on objectives

Com mod ities Currencies

Calibration contracts Test contracts Portfolio results

8.7 Realistic strategies and assumptions

8.8 Trading simulated contracts

8.9 Trading results for futures

8.10 Towards conclusions

8.11 Summary

9 VALUING OPTIONS

9.1 Introduction

9.2 Black-Scholes option pricing formulae

9.3 Evaluating standard formulae

... Contents X l l l

9.4 Call values when conditional variances change

Formulae for a stationary process Examples Non-stationary processes Conclusions Price trends and call values A formula for trend models Examples

9.6 Summary

10 CONCLUDING REMARKS

10.1 Price behaviour

10.2 Advice t o traders

10.3 Further research

10.4 Stationary models

Random walks Price trends

APPENDIX: A COMPUTER PROGRAM F O R MODELLING FINANCIAL

TIME SERIES 243

O u t p u t produced 243

User-defined parameters 244

Optional parameters 245

I n p u t req u i re ments 245

About the subroutines 247

FORTRAN program 248

Computer time required 244

References 256

Author index Subject index

Preface to the 2nd Edition

Modelling Financial Time Series was first published by John Wiley and Sons in 1986. The first edition appeared when relatively few people were engaged in research into financial market prices. In the 198Os, it required a considerable effort to obtain long time series of prices, while further effort was required to write computer programs that could test hypotheses about price behaviour. Furthermore, the potential for modelling and forecasting

volatility was only appreciated by a handful of researchers. All this has

changed during the last 20 years - now there are a considerable number of

empirical finance researchers, price data are freely available on the internet,

financial software i s^ widely^ available^ and^ the^ implications^ of^ stochastic

volatility for risk managers and traders are generally understood. The first edition made a particular contribution to the development of financial econometrics, by describing the empirical characteristics of market prices and several models that could explain the early empirical evidence. Some of the people cited in the first edition have continued to expand our understanding of the dynamic properties of asset prices, most notably Robert Engle, Clive Granger and George Tauchen. Important new ideas subsequently flowed from a talented generation of researchers, many of whom are named in the additional references at the end of this preface. I am pleased to acknowledge these contributions, of which several have been made by Tim Bollerslev and Neil Shephard. The remainder of this preface summarises both the innovative material included in the first edition and the related key innovations in more recent years. A detailed, contemporary presentation of models for financial time series can be found in Taylor (2005).

Chapter 2 documents the statistical features of daily asset returns. The

three most important features are summarised on page 58: daily returns are there described as having firstly low autocorrelations, secondly non-normal distributions and thirdly a non-linear, generating process. The third feature was deduced by observing that there i s “substantially more correlation

xv

xvi Preface to the 2 n d edition

between absolute or squared returns than there i s between the returns

themselves” (p. 55). This third ‘stylized fact’ seems to have been generally

unknown when my book was published, in contrast with the first and second facts which were well-known from Fama’s (1965) paper and earlier studies. Some people refer to the third fact as the ‘Taylor effect’and, additionally, observe that absolute returns seem to have higher correlations than squared

returns (from Tables 2.1 1 and 2.12). The third fact i s important because it

shows that we need price models which do not assume that returns have independent and identical distributions. The non-normality of returns, their almost-zero autocorrelations and the positive dependence among absolute and squared returns remain the

three most important ‘stylized facts’ about daily returns. A few studies have

discussed the power A that maximises the autocorrelations of the time series { l r t l ) , with r, denoting the return for day t in this preface. The results in Ding,

Granger and Engle (1 9931, Granger and Ding (1995) and Taylor (2005) show

that the maximum autocorrelation most often occurs when A i s approximately one, i.e., for absolute returns. Chapter 3 provides reasons for changes in volatility, and it states and compares several stochastic volatility models. The prior work of Clark and

Tauchen & Pitts had already related volatility to the number of intra-day

price movements (or news events), as shown and discussed on page 71. By

supposing these measure of market activity are correlated across trading days,

I obtained explanations for both the stochastic behaviour of volatility and

the correlation among absolute and squared returns. The simplest interesting model for volatility supposes that it follows a two-state Markov chain, as considered on page 67, but more volatility states are required to provide a

realistic description of market prices (Taylor, 1999).

Two of the volatility models described in Chapter 3 have been used in numerous empirical studies. The lognormal, autoregressive model for a latent

volatility variable was first published in Taylor (1982) and it i s included in

Section 3.5 and introduced on page 73. It i s often called the SV (stochastic

volatility) model. In modern notation, it defines the return r, as its expectation

,u plus the product of the volatility o,and an i.i.d. standard normal variable u,,

with

Because this model has two random components (u, and 7,) per unit time, it i s impossible to exactly observe the realisations of the volatility process and relatively difficult to estimate the parameters of the AR(1) process for log(o,).

xviii Preface to the 2nd edition

conditional normal distribution, and it i s also contradicted by the empirical

forecasting evidence presented in Chapter 4 (see p. 107). I soon appreciated,

however, that my arguments against ARCH do not apply to the more

general specifications of Engle and Bollerslev (1 986) and Bollerslev (1987)

that permitted the Z, to have a fat-tailed distribution; their Student-t choice i s a mixture of normal distributions that can be motivated by uncertainty at time t - 1 about the amount of relevant news on day t. There are many similarities between SV models and general ARCH models, which are surveyed in Taylor

(1994a). It i s very difficult, and possibly of minimal value, to say which of

these two types of models provides the best description of market returns. A general ARCH model has a general specification for the conditional variance h, and a very considerable number of specifications have been

suggested. In particular, the general GARCH(p,q) model of Bollerslev (1 986)

makes h, a linear function of the p previous squared excess returns and the

q terms

We should note that the signs of the previous residuals, r,-;-p, are

irrelevant in the GARCH model. Nelson (1991) made a very important

contribution to volatility modelling by demonstrating that the symmetric

treatment of positive and negative residuals i s not appropriate for U.S. index

returns. The estimates for his exponential GARCH model show that h,

i s higher when the most recent residual i s negative than when it has the

same magnitude and i s positive. Many people have subsequently evaluated

asymmetric ARCH models, with one of the most popular being the following simple extension of GARCH(1, I ) defined by Glosten, Jagannathan and

Runkle (1993):

1 I i 5 q.

with S,, = 1 if rt-l < p and otherwise S,, = 0. Other notable ARCH variations include working with linear functions

of absolute values rather than squares (Chapter 3 and Schwert, 1989),

making the conditional mean a function of h, (Engle, Lilien and Robins,

1987), and specifying a long-memory process for h, (Baillie, Bollerslev

and Mikkelsen, 1996). The survey by Bollerslev, Chou and Kroner (1992) covers a considerable number of applications, while the subsequent survey

by Bollerslev, Engle and Nelson (1994) i s more theoretical but also contains

detailed examples of the specification of conditional densities for daily

returns from U.S. indices, going back as far as 1885.

Chapter 4 compares some methods for estimating and forecasting volatility. inferences about volatility forecasting methods are obtained indirectly by

comparing methods for forecasting absolute returns. It i s recommended that

exponentially weighted moving averages of absolute returns are used to measure volatility, as they are robust against changes in the long-run level

Preface to the 2nd edition xix

of volatility and are also empirically as accurate as alternative methods. Volatility forecasts are of particular interest to risk managers and option traders, and a vast literature now compares forecasting methods; for a recent

survey see Poon and Granger (2003). The best forecasting methods rely upon

volatility levels implied by option prices and upon high-frequency returns, typically by using measures such as the daily sum of squared five-minute returns (e.g., Blair, Poon and Taylor, 2001 1.

Chapter 5 evaluates the accuracy of autocorrelations estimated from

returns. The classical standard error formula i s 1/& for a sample of n

observations from independent and identical distributions. This formula fails

for the uncorrelated processes introduced in Chapter 3; a suitable formula i s

b/& , with b > 1 estimated from the observed data. The formula evaluated comes from my 1984 paper and it i s also derived (using different

assumptions) in Lo and MacKinlay (1 988). Rescaled returns, defined as

returns minus their average divided by an estimated conditional standard deviation, are shown to have autocorrelations whose sample variances are much closer to the classical result.

Chapter 6 presents methods and results for tests of the random walk

hypothesis. One of the test statistics i s derived in one of my 1982 papers and

it i s powerful when the alternative to randomness i s trends in the price

process. It i s argued that any trends will create small positive autocorrelation

among returns, that declines as the lag increases. An ARMA(1,l) model is

the simplest that has these autocorrelation properties. The appropriate test statistic i s then proportional to &$rfir, with 9 a suitable AR parameter value, f i r the sample autocorrelation for lag r a n d the sum taken over the first k lags.

The trend test i s shown to reject the random walk hypothesis for commodity

and currency returns, for choices of 4 and k that have high test power for plausible trend parameters. This conclusion does not hold, however, for the tested equity series, that appear instead to be best described by MA(1) processes.

Results from the trend test have only been reported in a few papers, such

as Kariya et a/. (1995). Many papers have instead employed the variance- ratio test statistic of Lo and MacKinlay (1 988). This test compares the variance of k-day returns with k multiplied by the variance of one-day returns. It i s equivalent to basing the test upon the linear combination X ( k - r ) f i r , with

the sum over the first k - 1 lags. As both my trend test and the variance-ratio

test are based upon sums cwrfir, with w, > w, > w, > > 0, they are both expected to be powerful for the same alternatives to the random walk hypothesis. These alternatives include the ARMA(1,l) specifications provided

by my price-trend models, and then all the dependence i s positive. They also

include the models of Fama and French (1988) and Poterba and Summers (1988) that assert market prices revert towards rational levels that follow a