The Measurement of Market Risk: Modelling of Risk Factors, Asset Pricing, and Approximation of Portfolio Distributions

262 Pages·2001·8.23 MB·English

by  Dr. Pierre-Yves Moix (auth.)

Lecture Notes in Economics
and Mathematical Systems 504
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Springer-V erlag Berlin Heidelberg GmbH
Pierre-Yves Moix
The Measurement
of Market Risk
Modelling ofRisk Factors, Asset Pricing,
and Approximation ofPortfolio Distributions
Springer
Author
Dr. Pierre-Yves Moix, CFA
Untere Torfeldstrasse 52
5033 Buchs/AG
Switzerland
Cataloging-in-Publication data applied for
Die Deutsche Bibliothek -CIP-Einheitsaufnahme
Moix, Pierre-Yves:
The measurement of market risk : modelling of risk factors, asset
pricing, and approximation of portfolio distributions / Pierre-Yves
Moix. -Berlin; Heidelberg ; New York; Barcelona; Hong Kong ;
London ; Milan ; Paris; Singapore; Tokyo: Springer, 2001
(Lecture notes in economics and mathematical systems ; 504)
ISBN 978-3-540-42143-6 ISBN 978-3-642-56481-9 (eBook)
DOI 10.1007/978-3-642-56481-9
ISSN 0075-8450
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To
Sophie Rose, Susanne,
Marie-Rose and Gerard
To the memory ofmy onele
Ernest Moix
(1927-1999)
a
((L )absence est l)amour ce qu)est au feu le vent;
il eteint le petit et allume le grand))
Roger de Bussy-Rabuttin
"Histoire amoureuse de GauZes"
Preface
This book is a revised version of my doctoral dissertation submitted to the
University ofSt. Gallen in October 1999. I would like to thank Dr.oec. Marc
Wildi whose careful reading ofmuch ofthe text led to many improvements.
All errors remain mine.
Pfiiffikon SZ, Switzerland, March 2001 Pierre-Yves Moix
Preface to the dissertation
"Education is man's going forward from cocksure ignorance to
thoughtful uncertainty"
Don Clark's Scrapbook quoted in Wonnacott and Wonnacott (1990).
After several years of banking practice, I decided to give up some of my
certitudes and considered this thesis project a good opportunity to study
some of the quantitative tools necessary for the modelling of uncertainty.
lowe very much to Prof. Dr. Karl Frauendorfer, the referee of my thesis,
for the time he took to read the manuscript and for the numerous valuable
suggestions he made. I am also very grateful to Prof. Dr. Klaus Spremann
who kindly accepted to co-refer my thesis and who strengthened my inter
est in finance during my study period. During my time at the Institute for
Operations Research of the University of St. Gallen (lfU-HSG) I had the
opportunity to participate in the project "RiskLab" which provides a very
profitable link between finance practice and academics. I would especially
like to thank Dr. Christophe Rouvinez from Credit Suisse for his comments
and all the data he provided so generously. Likewise, I am very grateful to
my colleagues at the institute who have been a valuable source ofnew ideas.
I especially profited from Dr.oec. Christina Mahron's programming skills,
from the mathematical expertise of Dipl.-Math. Veronika Halder, and from
the numerous discussions with Dr.oec. Ralf Gaese, Dipl.-Math.oec. Guido
Haarbriicker, and Dipl.-Wirtsch.Ing. Stefan Scholz. Many thanks to Claudia
Rossi-Mayer, CELTA,whospent manydaysatcorrectingthemanuscriptand
improving my modest knowledge ofEnglish. Last but not least, my warmest
thanks go to my entire family for all the love and support they have given
me, and above all for their patience.
Contents
1. Introduction.............................................. 1
1.1 The Need for Risk Measurement. ......................... 1
1.2 The Nature of Financial Risk. ........................... 3
1.3 Formal Framework 5
1.3.1 Modelling the Uncertainty. ........................ 6
1.3.2 The Information Structure. ........................ 8
1.4 Problem Statement ..................................... 9
1.5 Structure ofthe Book. .................................. 11
1.6 Test Environment 16
1.6.1 Environment I ........................... 17
1.6.2 Environment II 18
2. Risk and Risk Measures. ................................. 21
2.1 The Investment Decision 22
2.1.1 Utility Theory and Expected Utility Hypothesis. ..... 23
2.1.2 Rules for the Ordering of Uncertain Prospects 33
2.2 The Capital Requirement Decision. ....................... 41
2.2.1 Value-at-Risk.................................... 42
2.2.2 Coherent Risk Measures. .......................... 44
2.3 Summary.............................................. 47
3. Modelling the Dynamics ofthe Risk Factors. ............. 49
3.1 Statistical Definitions. .................................. 50
3.1.1 Stochastic Processes: Basic Definitions. ............. 50
3.1.2 Properties of Stochastic Processes. ................. 52
3.1.3 Basic Stochastic Processes. ........................ 53
3.2 The Economic Assumption: the Efficient Market Hypothesis. 59
3.3 Empirical Evidence for the Returns. ...................... 62
3.3.1 Calendar Effects 62
3.3.2 Leptokurtosis and Weak Evidence of Skewness ....... 62
3.3.3 The Autocorrelation ofthe Squared Returns. ........ 64
3.4 Models for the Risk Factor Dynamics .. ................... 66
3.4.1 The Generic Model for the Log-returns. ............. 67
3.4.2 ARCH Models ................................... 68
X Contents
3.4.3 Stochastic Variance Models. ....................... 71
3.5 Empirical Analysis of the Returns on Swiss Stocks. ......... 80
3.5.1 The Data 80
3.5.2 Descriptive Statistics and Correlation. .............. 80
3.5.3 Implementation of an Alternative Model 82
3.5.4 Impact ofthe Alternative Modelling. ............... 90
3.6 Continuous-Time Models. ............................... 92
3.7 Summary.............................................. 97
4. Valuation ofFinancial Instruments 99
4.1 Principles ofValuation 100
4.1.1 Valuation by Arbitrage 102
4.2 Cash Instruments 116
4.2.1 Equities 116
4.2.2 Fixed-Income Instruments 117
4.3 Futures and Forwards 119
4.4 Options 121
4.4.1 The Black-Scholes Analysis 122
4.4.2 Risk-Neutral Valuation 124
4.4.3 Numerical Approaches 126
4.5 Approximation ofthe Value Function 133
4.5.1 Global Taylor Approximation for Option Pricing 135
4.5.2 Piecewise Taylor Approximations 136
5. Approximation ofthe Portfolio Distribution 141
5.1 Analytical Methods 142
5.1.1 Delta Approximation 144
5.1.2 Delta-Gamma Approximation 146
5.2 Generation ofScenarios 151
5.2.1 The Pseudo-Random Method 152
5.2.2 The Quasi-Random Method 153
5.2.3 Generation ofDistributions for the Risk Factors 159
5.3 Monte Carlo Simulation 165
5.3.1 Error Analysis 166
5.3.2 Variance Reduction Techniques 173
5.4 The BDPQA 177
5.4.1 Simplices 178
5.4.2 Simplicial Coverage ofthe Risk Factor Distribution 180
5.4.3 Barycentric Discretisation 183
5.4.4 Approximation ofthe Portfolio Distribution 185
5.4.5 Refinement Strategies 188
5.4.6 Numerical Example 191
5.5 Benchmarking the BDPQA 195
5.5.1 The Choice ofthe Holding Period 201
5.6 Summary 202
Contents XI
6. Sample Estimation of Risk Measures 205
6.1 Introduction 205
6.2 Order Statistics 206
6.2.1 Distribution of Order Statistics 206
6.2.2 Moments of Order Statistics 209
6.2.3 Confidence Interval for Population Quantiles 210
6.3 Quantile Estimators Based on Order Statistics 213
6.3.1 Linear Combination ofSeveral Order Statistics 214
6.4 Kernel-Based Estimators 215
6.4.1 Accuracy ofthe Estimate Density 217
6.4.2 Bandwidth Selection 221
6.4.3 Quantile Estimation Based on the Kernel Density
Method 223
6.5 Comparison ofthe Quantile Estimators 224
6.6 Summary 227
7. Conclusion and Outlook 229
7.1 Summary 229
7.2 The Issue of Credit Risk 232
7.3 Outlook 233
A. Probability and Statistics 235
A.l Probabilistic Modelling 235
A.2 Random Variable 237
A.2.1 Distribution Function 238
A.2.2 Moments 239
A.2.3 Independence and Correlation 241
A.2.4 Conditional Probability and Expectation 242
A.2.5 Stochastic Processes and Information Structure 243
A.2.6 Martingales 244
A.3 Selected Distributions 245
A.3.1 Basic Distributions 245
A.3.2 Elliptically Contoured Distributions 246
A.3.3 Stable Distribution 248
A.4 Types of Convergence 249
A.5 Sampling Theory 251
Bibliography 253
List ofFigures ................................................ 265
List of Tables 267
Index 269
1. Introduction
1.1 The Need for Risk Measurement
Financial risk, the financial exposure to uncertainty, has become a crucial
issuefor both financial and non-financial corporations. Recent scandals (Bar
ings, Orange County, Metallgesellschaft, LTCM) show the need for powerful
riskmanagementtools. Inadditiontotherecentdebacles,severalfactorshave
contributed to the development of a consistent and integrated treatment of
financial risk:
• the increasing volatility ofthe markets
• the development ofinformation technology
• the deregulation offinancial markets
• the processes ofsecurisation and disintermediation
• the shift in the legal and regulatory environment.
It is generally accepted that the financial world has become riskier over
the pastdecades. Whilethevolatilityoftheequitymarketshasbeenobserved
at least since the first crash of 1929, other markets have experienced an in
crease of uncertainty more recently. The breakdown of the Bretton Woods
agreement has led to an evident increase in exchange rate volatility. As the
central banks began to target money supply growth to restrain inflation, in
terest rates in turn have become more volatile. Finally, the price of several
basic commodities, especially petroleum, began to fluctuate heavily in the
70's.
The general increase in volatility of the financial markets can be partly
explained by the growing uncertainty ofthe economic environment. It is, on
the other hand, a reflection of the growing efficiency and integration of the
financial markets. The development of information technology and the inte
gration of financial markets allow the almost instantaneous move of capital
from one market place to another and therefore the rapid assimilation of
any new, available information. In efficient markets, all current information,
which issupposedto arriveat random, isfully reflected in the prices. As a re
sult, future prices are hard to predict and uncertainty increaseswith the time
P-Y Moix, The Measurement of Market Risk
© Springer-Verlag Berlin Heidelberg 2001