Statistical methods for financial engineering

While many financial engineering books are available, the statistical
aspects behind the implementation of stochastic models used in
the field are often overlooked or restricted to a few well-known
cases. Statistical Methods for Financial Engineering guides
current and future practitioners on implementing the most useful
stochastic models used in financial engineering.
After introducing properties of univariate and multivariate models
for asset dynamics as well as estimation techniques, the book
discusses limits of the Black-Scholes model, statistical tests to
verify some of its assumptions, and the challenges of dynamic
hedging in discrete time. It then covers the estimation of risk
and performance measures, the foundations of spot interest rate
modeling, Lévy processes and their financial applications, the
properties and parameter estimation of GARCH models, and the
importance of dependence models in hedge fund replication and
other applications. It concludes with the topic of filtering and its
financial applications.

K12677

Rémillard

This self-contained book offers a basic presentation of stochastic
models and addresses issues related to their implementation in the
financial industry. Each chapter introduces powerful and practical
statistical tools necessary to implement the models. The author
not only shows how to estimate parameters efficiently, but he also
demonstrates, whenever possible, how to test the validity of the
proposed models. Throughout the text, examples using MATLAB®
illustrate the application of the techniques to solve real-world
financial problems. MATLAB and R programs are available on the

author’s website.

Statistical Methods for
Financial Engineering

Finance


STATISTICAL
METHODS FOR
FINANCIAL
ENGINEERING


STATISTICAL
METHODS FOR
FINANCIAL
ENGINEERING

BRUNO RÉMILLARD


MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not
warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular
pedagogical approach or particular use of the MATLAB® software.

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Version Date: 20130214
International Standard Book Number-13: 978-1-4398-5695-6 (eBook - PDF)
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have attempted to trace the copyright holders of all material reproduced in this publication and apologize to
copyright holders if permission to publish in this form has not been obtained. If any copyright material has
not been acknowledged please write and let us know so we may rectify in any future reprint.
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Visit the Taylor & Francis Web site at

and the CRC Press Web site at



MATLAB® is a trademark of The MathWorks, Inc. and is used with permission. The MathWorks does not
warrant the accuracy of the text or exercises in this book. This book’s use or discussion of MATLAB® software or related products does not constitute endorsement or sponsorship by The MathWorks of a particular
pedagogical approach or particular use of the MATLAB® software.

CRC Press
Taylor & Francis Group
6000 Broken Sound Parkway NW, Suite 300
Boca Raton, FL 33487-2742
© 2013 by Taylor & Francis Group, LLC
CRC Press is an imprint of Taylor & Francis Group, an Informa business
No claim to original U.S. Government works
Printed on acid-free paper
Version Date: 20130214
International Standard Book Number-13: 978-1-4398-5694-9 (Hardback)
This book contains information obtained from authentic and highly regarded sources. Reasonable efforts
have been made to publish reliable data and information, but the author and publisher cannot assume
responsibility for the validity of all materials or the consequences of their use. The authors and publishers
have attempted to trace the copyright holders of all material reproduced in this publication and apologize to
copyright holders if permission to publish in this form has not been obtained. If any copyright material has
not been acknowledged please write and let us know so we may rectify in any future reprint.
Except as permitted under U.S. Copyright Law, no part of this book may be reprinted, reproduced, transmitted, or utilized in any form by any electronic, mechanical, or other means, now known or hereafter invented,
including photocopying, microfilming, and recording, or in any information storage or retrieval system,
without written permission from the publishers.
For permission to photocopy or use material electronically from this work, please access www.copyright.
com ( or contact the Copyright Clearance Center, Inc. (CCC), 222 Rosewood
Drive, Danvers, MA 01923, 978-750-8400. CCC is a not-for-profit organization that provides licenses and
registration for a variety of users. For organizations that have been granted a photocopy license by the CCC,
a separate system of payment has been arranged.
Trademark Notice: Product or corporate names may be trademarks or registered trademarks, and are used
only for identification and explanation without intent to infringe.
Library of Congress Cataloging-in-Publication Data
Remillard, Bruno.
Statistical methods for financial engineering / Bruno Remillard.

pages cm
Includes bibliographical references and index.
ISBN 978-1-4398-5694-9 (hardcover : alk. paper)
1. Financial engineering--Statistical methods. 2. Finance--Statistical methods. I.
Title.
HG176.7.R46 2013
332.01’5195--dc23
Visit the Taylor & Francis Web site at

and the CRC Press Web site at


2012050917


Contents

Preface

xxi

List of Figures

xxv

List of Tables

xxix

1 Black-Scholes Model

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.1 The Black-Scholes Model . . . . . . . . . . . . . . . . . . . .
1.2 Dynamic Model for an Asset . . . . . . . . . . . . . . . . . .
1.2.1 Stock Exchange Data . . . . . . . . . . . . . . . . . .
1.2.2 Continuous Time Models . . . . . . . . . . . . . . . .
1.2.3 Joint Distribution of Returns . . . . . . . . . . . . . .
1.2.4 Simulation of a Geometric Brownian Motion . . . . .
1.2.5 Joint Law of Prices . . . . . . . . . . . . . . . . . . . .
1.3 Estimation of Parameters . . . . . . . . . . . . . . . . . . . .
1.4 Estimation Errors . . . . . . . . . . . . . . . . . . . . . . . .
1.4.1 Estimation of Parameters for Apple . . . . . . . . . .
1.5 Black-Scholes Formula . . . . . . . . . . . . . . . . . . . . . .
1.5.1 European Call Option . . . . . . . . . . . . . . . . . .
1.5.1.1 Put-Call Parity . . . . . . . . . . . . . . . . .
1.5.1.2 Early Exercise of an American Call Option .
1.5.2 Partial Differential Equation for Option Values . . . .
1.5.3 Option Value as an Expectation . . . . . . . . . . . .
1.5.3.1 Equivalent Martingale Measures and Pricing
of Options . . . . . . . . . . . . . . . . . . .
1.5.4 Dividends . . . . . . . . . . . . . . . . . . . . . . . . .
1.5.4.1 Continuously Paid Dividends . . . . . . . . .
1.6 Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
1.6.1 Greeks for a European Call Option . . . . . . . . . . .
1.6.2 Implied Distribution . . . . . . . . . . . . . . . . . . .
1.6.3 Error on the Option Value . . . . . . . . . . . . . . . .
1.6.4 Implied Volatility . . . . . . . . . . . . . . . . . . . . .
1.6.4.1 Problems with Implied Volatility . . . . . . .
1.7 Estimation of Greeks using the Broadie-Glasserman Methodologies . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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Contents
1.7.1 Pathwise Method . . . . .
1.7.2 Likelihood Ratio Method
1.7.3 Discussion . . . . . . . . .
1.8 Suggested Reading . . . . . . . .
1.9 Exercises . . . . . . . . . . . . .
1.10 Assignment Questions . . . . . .
1.A Justification of the Black-Scholes
1.B Martingales . . . . . . . . . . . .
1.C Proof of the Results . . . . . . .
1.C.1 Proof of Proposition 1.3.1
1.C.2 Proof of Proposition 1.4.1
1.C.3 Proof of Proposition 1.6.1
Bibliography . . . . . . . . . . . . . .

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2 Multivariate Black-Scholes Model
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.1 Black-Scholes Model for Several Assets . . . . . . . . . . . .
2.1.1 Representation of a Multivariate Brownian Motion . .
2.1.2 Simulation of Correlated Geometric Brownian Motions
2.1.3 Volatility Vector . . . . . . . . . . . . . . . . . . . . .
2.1.4 Joint Distribution of the Returns . . . . . . . . . . . .
2.2 Estimation of Parameters . . . . . . . . . . . . . . . . . . . .
2.2.1 Explicit Method . . . . . . . . . . . . . . . . . . . . .
2.2.2 Numerical Method . . . . . . . . . . . . . . . . . . . .
2.3 Estimation Errors . . . . . . . . . . . . . . . . . . . . . . . .
2.3.1 Parametrization with the Correlation Matrix . . . . .
2.3.2 Parametrization with the Volatility Vector . . . . . . .
2.3.3 Estimation of Parameters for Apple and Microsoft . .
2.4 Evaluation of Options on Several Assets . . . . . . . . . . . .
2.4.1 Partial Differential Equation for Option Values . . . .
2.4.2 Option Value as an Expectation . . . . . . . . . . . .
2.4.2.1 Vanilla Options . . . . . . . . . . . . . . . .
2.4.3 Exchange Option . . . . . . . . . . . . . . . . . . . . .
2.4.4 Quanto Options . . . . . . . . . . . . . . . . . . . . .
2.5 Greeks . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.5.1 Error on the Option Value . . . . . . . . . . . . . . . .
2.5.2 Extension of the Broadie-Glasserman Methodologies for
Options on Several Assets . . . . . . . . . . . . . . . .
2.6 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . .

2.7 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
2.8 Assignment Questions . . . . . . . . . . . . . . . . . . . . . .
2.A Auxiliary Result . . . . . . . . . . . . . . . . . . . . . . . . .
2.A.1 Evaluation of E eaZ N (b + cZ) . . . . . . . . . . . .
2.B Proofs of the Results . . . . . . . . . . . . . . . . . . . . . .
2.B.1 Proof of Proposition 2.1.1 . . . . . . . . . . . . . . . .

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Contents
2.B.2 Proof of Proposition 2.2.1
2.B.3 Proof of Proposition 2.3.1
2.B.4 Proof of Proposition 2.3.2
2.B.5 Proof of Proposition 2.4.1
2.B.6 Proof of Proposition 2.4.2
2.B.7 Proof of Proposition 2.5.1
2.B.8 Proof of Proposition 2.5.3
Bibliography . . . . . . . . . . . . . .

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3 Discussion of the Black-Scholes Model
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.1 Critiques of the Model . . . . . . . . . . . . . . . . . . . . .
3.1.1 Independence . . . . . . . . . . . . . . . . . . . . . . .
3.1.2 Distribution of Returns and Goodness-of-Fit Tests of
Normality . . . . . . . . . . . . . . . . . . . . . . . . .
3.1.3 Volatility Smile . . . . . . . . . . . . . . . . . . . . . .

3.1.4 Transaction Costs . . . . . . . . . . . . . . . . . . . .
3.2 Some Extensions of the Black-Scholes Model . . . . . . . . .
3.2.1 Time-Dependent Coefficients . . . . . . . . . . . . . .
3.2.1.1 Extended Black-Scholes Formula . . . . . . .
3.2.2 Diffusion Processes . . . . . . . . . . . . . . . . . . . .
3.3 Discrete Time Hedging . . . . . . . . . . . . . . . . . . . . .
3.3.1 Discrete Delta Hedging . . . . . . . . . . . . . . . . .
3.4 Optimal Quadratic Mean Hedging . . . . . . . . . . . . . . .
3.4.1 Offline Computations . . . . . . . . . . . . . . . . . .
3.4.2 Optimal Solution of the Hedging Problem . . . . . . .
3.4.3 Relationship with Martingales . . . . . . . . . . . . .
3.4.3.1 Market Price vs Theoretical Price . . . . . .
3.4.4 Markovian Models . . . . . . . . . . . . . . . . . . . .
3.4.5 Application to Geometric Random Walks . . . . . . .
3.4.5.1 Illustrations . . . . . . . . . . . . . . . . . .
3.4.6 Incomplete Markovian Models . . . . . . . . . . . . .
3.4.7 Limiting Behavior . . . . . . . . . . . . . . . . . . . .
3.5 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . .
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
3.7 Assignment Questions . . . . . . . . . . . . . . . . . . . . . .
3.A Tests of Serial Independence . . . . . . . . . . . . . . . . . .
3.B Goodness-of-Fit Tests . . . . . . . . . . . . . . . . . . . . . .
3.B.1 Cram´er-von Mises Test . . . . . . . . . . . . . . . . .
3.B.1.1 Algorithms for Approximating the P -Value .
3.B.2 Lilliefors Test . . . . . . . . . . . . . . . . . . . . . . .
3.C Density Estimation . . . . . . . . . . . . . . . . . . . . . . .
3.C.1 Examples of Kernels . . . . . . . . . . . . . . . . . . .
3.D Limiting Behavior of the Delta Hedging Strategy . . . . . . .
3.E Optimal Hedging for the Binomial Tree . . . . . . . . . . . .

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viii

Contents
3.F A Useful Result . . . . . . . . . . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

4 Measures of Risk and Performance
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1 Measures of Risk . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.1 Portfolio Model . . . . . . . . . . . . . . . . . . . . . .
4.1.2 VaR . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.3 Expected Shortfall . . . . . . . . . . . . . . . . . . . .
4.1.4 Coherent Measures of Risk . . . . . . . . . . . . . . .
4.1.4.1 Comments . . . . . . . . . . . . . . . . . . .
4.1.5 Coherent Measures of Risk with Respect to a Stochastic
Order . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.1.5.1 Simple Order . . . . . . . . . . . . . . . . . .
4.1.5.2 Hazard Rate Order . . . . . . . . . . . . . .
4.2 Estimation of Measures of Risk by Monte Carlo Methods . .
4.2.1 Methodology . . . . . . . . . . . . . . . . . . . . . . .
4.2.2 Nonparametric Estimation of the Distribution Function
4.2.2.1 Precision of the Estimation of the Distribution
Function . . . . . . . . . . . . . . . . . . . .

4.2.3 Nonparametric Estimation of the VaR . . . . . . . . .
4.2.3.1 Uniform Estimation of Quantiles . . . . . . .
4.2.4 Estimation of Expected Shortfall . . . . . . . . . . . .
4.2.5 Advantages and Disadvantages of the Monte Carlo
Methodology . . . . . . . . . . . . . . . . . . . . . . .
4.3 Measures of Risk and the Delta-Gamma Approximation . . .
4.3.1 Delta-Gamma Approximation . . . . . . . . . . . . . .
4.3.2 Delta-Gamma-Normal Approximation . . . . . . . . .
4.3.3 Moment Generating Function and Characteristic Function of Q . . . . . . . . . . . . . . . . . . . . . . . . .
4.3.4 Partial Monte Carlo Method . . . . . . . . . . . . . .
4.3.4.1 Advantages and Disadvantages of the Methodology . . . . . . . . . . . . . . . . . . . . . .
4.3.5 Edgeworth and Cornish-Fisher Expansions . . . . . .
4.3.5.1 Edgeworth Expansion for the Distribution
Function . . . . . . . . . . . . . . . . . . . .
4.3.5.2 Advantages and Disadvantages of the Edgeworth Expansion . . . . . . . . . . . . . . . .
4.3.5.3 Cornish-Fisher Expansion . . . . . . . . . . .
4.3.5.4 Advantages and Disadvantages of the CornishFisher Expansion . . . . . . . . . . . . . . .
4.3.6 Saddlepoint Approximation . . . . . . . . . . . . . . .
4.3.6.1 Approximation of the Density . . . . . . . .
4.3.6.2 Approximation of the Distribution Function

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Contents

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4.3.6.3

Advantages and Disadvantages of the Methodology . . . . . . . . . . . . . . . . . . . . . .
4.3.7 Inversion of the Characteristic Function . . . . . . . .
4.3.7.1 Davies Approximation . . . . . . . . . . . . .
4.3.7.2 Implementation . . . . . . . . . . . . . . . .
4.4 Performance Measures . . . . . . . . . . . . . . . . . . . . . .
4.4.1 Axiomatic Approach of Cherny-Madan . . . . . . . . .
4.4.2 The Sharpe Ratio . . . . . . . . . . . . . . . . . . . .
4.4.3 The Sortino Ratio . . . . . . . . . . . . . . . . . . . .
4.4.4 The Omega Ratio . . . . . . . . . . . . . . . . . . . .
4.4.4.1 Relationship with Expectiles . . . . . . . . .
4.4.4.2 Gaussian Case and the Sharpe Ratio . . . .
4.4.4.3 Relationship with Stochastic Dominance . .
¯ . . . . . . . . .
4.4.4.4 Estimation of Omega and G
4.5 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . .
4.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.7 Assignment Questions . . . . . . . . . . . . . . . . . . . . . .
4.A Brownian Bridge . . . . . . . . . . . . . . . . . . . . . . . . .
4.B Quantiles . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.C Mean Excess Function . . . . . . . . . . . . . . . . . . . . . .
4.C.1 Estimation of the Mean Excess Function . . . . . . . .
4.D Bootstrap Methodology . . . . . . . . . . . . . . . . . . . . .
4.D.1 Bootstrap Algorithm . . . . . . . . . . . . . . . . . . .
4.E Simulation of QF,a,b . . . . . . . . . . . . . . . . . . . . . . .
4.F Saddlepoint Approximation of a Continuous Distribution Function . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
4.G Complex Numbers in MATLAB . . . . . . . . . . . . . . . .

4.H Gil-Pelaez Formula . . . . . . . . . . . . . . . . . . . . . . .
4.I Proofs of the Results . . . . . . . . . . . . . . . . . . . . . .
4.I.1 Proof of Proposition 4.1.1 . . . . . . . . . . . . . . . .
4.I.2 Proof of Proposition 4.1.3 . . . . . . . . . . . . . . . .
4.I.3 Proof of Proposition 4.2.1 . . . . . . . . . . . . . . . .
4.I.4 Proof of Proposition 4.2.2 . . . . . . . . . . . . . . . .
4.I.5 Proof of Proposition 4.3.1 . . . . . . . . . . . . . . . .
4.I.6 Proof of Proposition 4.4.1 . . . . . . . . . . . . . . . .
4.I.7 Proof of Proposition 4.4.2 . . . . . . . . . . . . . . . .
4.I.8 Proof of Proposition 4.4.4 . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
5 Modeling Interest Rates
Summary . . . . . . . . . . . . . . . . . .
5.1 Introduction . . . . . . . . . . . . .
5.1.1 Vasicek Result . . . . . . . .
5.2 Vasicek Model . . . . . . . . . . . .
5.2.1 Ornstein-Uhlenbeck Processes

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Contents
5.2.2

5.2.3

5.3

5.4
5.5
5.6
5.7
5.A
5.B
5.C
5.D

Change of Measurement and Time Scales . . . . . . . 149
Properties of Ornstein-Uhlenbeck Processes . . . . . . 150
5.2.3.1 Moments of the Ornstein-Uhlenbeck Process
150
5.2.3.2 Stationary Distribution of the OrnsteinUhlenbeck Process . . . . . . . . . . . . . . . 151
5.2.4 Value of Zero-Coupon Bonds under a Vasicek Model . 151
5.2.4.1 Vasicek Formula for the Value of a Bond . . 152
5.2.4.2 Annualized Bond Yields . . . . . . . . . . . . 152
5.2.5 Estimation of the Parameters of the Vasicek Model Using Zero-Coupon Bonds . . . . . . . . . . . . . . . . . 153
5.2.5.1 Measurement and Time Scales . . . . . . . . 154
5.2.5.2 Duan Approach for the Estimation of Non Observable Data . . . . . . . . . . . . . . . . . . 154
5.2.5.3 Joint Conditional Density of the Implied Rates 155
5.2.5.4 Change of Variables Formula . . . . . . . . . 156
5.2.5.5 Application of the Change of Variable Formula
to the Vasicek Model . . . . . . . . . . . . . 156
5.2.5.6 Precision of the Estimation . . . . . . . . . . 158
Cox-Ingersoll-Ross (CIR) Model . . . . . . . . . . . . . . . . 160

5.3.1 Representation of the Feller Process . . . . . . . . . . 160
5.3.1.1 Properties of the Feller Process . . . . . . . . 162
5.3.1.2 Measurement and Time Scales . . . . . . . . 163
5.3.2 Value of Zero-Coupon Bonds under a CIR Model . . . 163
5.3.2.1 Formula for the Value of a Zero-Coupon Bond
under the CIR Model . . . . . . . . . . . . . 164
5.3.2.2 Annualized Bond Yields . . . . . . . . . . . . 165
5.3.2.3 Value of a Call Option on a Zero-Coupon Bond 165
5.3.2.4 Put-Call Parity . . . . . . . . . . . . . . . . . 166
5.3.3 Parameters Estimation of the CIR Model Using ZeroCoupon Bonds . . . . . . . . . . . . . . . . . . . . . . 166
5.3.3.1 Measurement and Time Scales . . . . . . . . 167
5.3.3.2 Joint Conditional Density of the Implied Rates 167
5.3.3.3 Application of the Change of Variable Formula
for the CIR Model . . . . . . . . . . . . . . . 168
5.3.3.4 Precision of the Estimation . . . . . . . . . . 169
Other Models for the Spot Rates . . . . . . . . . . . . . . . . 170
5.4.1 Affine Models . . . . . . . . . . . . . . . . . . . . . . . 171
Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . 171
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 172
Assignment Questions . . . . . . . . . . . . . . . . . . . . . . 175
Interpretation of the Stochastic Integral . . . . . . . . . . . . 175
Integral of a Gaussian Process . . . . . . . . . . . . . . . . . 176
Estimation Error for a Ornstein-Uhlenbeck Process . . . . . 176
Proofs of the Results . . . . . . . . . . . . . . . . . . . . . . 178
5.D.1 Proof of Proposition 5.2.1 . . . . . . . . . . . . . . . . 178


Contents
5.D.2 Proof of Proposition 5.2.2
5.D.3 Proof of Proposition 5.3.1

5.D.4 Proof of Proposition 5.3.2
5.D.5 Proof of Proposition 5.3.3
Bibliography . . . . . . . . . . . . . .

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6 L´

evy Models
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.1 Complete Models . . . . . . . . . . . . . . . . . . . . . .
6.2 Stochastic Processes with Jumps . . . . . . . . . . . . . .
6.2.1 Simulation of a Poisson Process over a Fixed Time
terval . . . . . . . . . . . . . . . . . . . . . . . . .
6.2.2 Jump-Diffusion Models . . . . . . . . . . . . . . .
6.2.3 Merton Model . . . . . . . . . . . . . . . . . . . .
6.2.4 Kou Jump-Diffusion Model . . . . . . . . . . . . .
6.2.5 Weighted-Symmetric Models for the Jumps . . . .
6.3 L´evy Processes . . . . . . . . . . . . . . . . . . . . . . . .
6.3.1 Random Walk Representation . . . . . . . . . . . .
6.3.2 Characteristics . . . . . . . . . . . . . . . . . . . .
6.3.3 Infinitely Divisible Distributions . . . . . . . . . .
6.3.4 Sample Path Properties . . . . . . . . . . . . . . .
6.3.4.1 Number of Jumps of a L´evy Process . . .
6.3.4.2 Finite Variation . . . . . . . . . . . . . .
6.4 Examples of L´evy Processes . . . . . . . . . . . . . . . .
6.4.1 Gamma Process . . . . . . . . . . . . . . . . . . .
6.4.2 Inverse Gaussian Process . . . . . . . . . . . . . .
6.4.2.1 Simulation of Tα,β . . . . . . . . . . . . .
6.4.3 Generalized Inverse Gaussian Distribution . . . . .
6.4.4 Variance Gamma Process . . . . . . . . . . . . . .
6.4.5 L´evy Subordinators . . . . . . . . . . . . . . . . .
6.5 Change of Distribution . . . . . . . . . . . . . . . . . . .
6.5.1 Esscher Transforms . . . . . . . . . . . . . . . . . .
6.5.2 Examples of Application . . . . . . . . . . . . . . .
6.5.2.1 Merton Model . . . . . . . . . . . . . . .
6.5.2.2 Kou Model . . . . . . . . . . . . . . . . .
6.5.2.3 Variance Gamma Process . . . . . . . . .

6.5.2.4 Normal Inverse Gaussian Process . . . . .
6.5.3 Application to Option Pricing . . . . . . . . . . . .
6.5.4 General Change of Measure . . . . . . . . . . . . .
6.5.5 Incompleteness . . . . . . . . . . . . . . . . . . . .
6.6 Model Implementation and Estimation of Parameters . .
6.6.1 Distributional Properties . . . . . . . . . . . . . .
6.6.1.1 Serial Independence . . . . . . . . . . . .
6.6.1.2 L´evy Process vs Brownian Motion . . . .
6.6.2 Estimation Based on the Cumulants . . . . . . . .

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xii

Contents
6.6.2.1 Estimation of the Cumulants . . . . . . . . .
6.6.2.2 Application . . . . . . . . . . . . . . . . . . .
6.6.2.3 Discussion . . . . . . . . . . . . . . . . . . .
6.6.3 Estimation Based on the Maximum Likelihood Method
6.7 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . .
6.8 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
6.9 Assignment Questions . . . . . . . . . . . . . . . . . . . . . .
6.A Modified Bessel Functions of the Second Kind . . . . . . . .
6.B Asymptotic Behavior of the Cumulants . . . . . . . . . . . .
6.C Proofs of the Results . . . . . . . . . . . . . . . . . . . . . .
6.C.1 Proof of Lemma 6.5.1 . . . . . . . . . . . . . . . . . .
6.C.2 Proof of Corollary 6.5.2 . . . . . . . . . . . . . . . . .

6.C.3 Proof of Proposition 6.6.1 . . . . . . . . . . . . . . . .
6.C.4 Proof of Proposition 6.4.1 . . . . . . . . . . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

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7 Stochastic Volatility Models
223
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.1 GARCH Models . . . . . . . . . . . . . . . . . . . . . . . . . 223
7.1.1 GARCH(1,1) . . . . . . . . . . . . . . . . . . . . . . . 224
7.1.2 GARCH(p,q) . . . . . . . . . . . . . . . . . . . . . . . 226
7.1.3 EGARCH . . . . . . . . . . . . . . . . . . . . . . . . . 226
7.1.4 NGARCH . . . . . . . . . . . . . . . . . . . . . . . . . 227
7.1.5 GJR-GARCH . . . . . . . . . . . . . . . . . . . . . . . 227
7.1.6 Augmented GARCH . . . . . . . . . . . . . . . . . . . 227

7.2 Estimation of Parameters . . . . . . . . . . . . . . . . . . . . 228
7.2.1 Application for GARCH(p,q) Models . . . . . . . . . . 229
7.2.2 Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . . 230
7.2.3 Goodness-of-Fit and Pseudo-Observations . . . . . . . 230
7.2.4 Estimation and Goodness-of-Fit When the Innovations
Are Not Gaussian . . . . . . . . . . . . . . . . . . . . 232
7.3 Duan Methodology of Option Pricing . . . . . . . . . . . . . 235
7.3.1 LRNVR Criterion . . . . . . . . . . . . . . . . . . . . 235
7.3.2 Continuous Time Limit . . . . . . . . . . . . . . . . . 237
7.3.2.1 A New Parametrization . . . . . . . . . . . . 238
7.4 Stochastic Volatility Model of Hull-White . . . . . . . . . . . 239
7.4.1 Market Price of Volatility Risk . . . . . . . . . . . . . 239
7.4.2 Expectations vs Partial Differential Equations . . . . . 240
7.4.3 Option Price as an Expectation . . . . . . . . . . . . . 240
7.4.4 Approximation of Expectations . . . . . . . . . . . . . 242
7.4.4.1 Monte Carlo Methods . . . . . . . . . . . . . 242
7.4.4.2 Taylor Series Expansion . . . . . . . . . . . . 242
7.4.4.3 Edgeworth and Gram-Charlier Expansions . 243
7.4.4.4 Approximate Distribution . . . . . . . . . . . 245
7.5 Stochastic Volatility Model of Heston . . . . . . . . . . . . . 246


Contents
7.6
7.7
7.8
7.A

Suggested Reading . . . . . . . .
Exercises . . . . . . . . . . . . .

Assignment Questions . . . . . .
Khmaladze Transform . . . . . .
7.A.1 Implementation Issues . .
7.B Proofs of the Results . . . . . .
7.B.1 Proof of Proposition 7.1.1
7.B.2 Proof of Proposition 7.4.1
7.B.3 Proof of Proposition 7.4.2
Bibliography . . . . . . . . . . . . . .

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8 Copulas and Applications
257
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 257
8.1 Weak Replication of Hedge Funds . . . . . . . . . . . . . . . 257
8.1.1 Computation of g . . . . . . . . . . . . . . . . . . . . . 258
8.2 Default Risk . . . . . . . . . . . . . . . . . . . . . . . . . . . 259
8.2.1 n-th to Default Swap . . . . . . . . . . . . . . . . . . . 259
8.2.2 Simple Model for Default Time . . . . . . . . . . . . . 260
8.2.3 Joint Dynamics of Xi and Yi . . . . . . . . . . . . . . 261
8.2.4 Simultaneous Evolution of Several Markov Chains . . 262
8.2.4.1 CreditMetrics . . . . . . . . . . . . . . . . . 262
8.2.5 Continuous Time Model . . . . . . . . . . . . . . . . . 264
8.2.5.1 Modeling the Default Time of a Firm . . . . 266

8.2.6 Modeling Dependence Between Several Default Times
266
8.3 Modeling Dependence . . . . . . . . . . . . . . . . . . . . . . 266
8.3.1 An Image is Worth a Thousand Words . . . . . . . . . 267
8.3.2 Joint Distribution, Margins and Copulas . . . . . . . . 269
8.3.3 Visualizing Dependence . . . . . . . . . . . . . . . . . 269
8.4 Bivariate Copulas . . . . . . . . . . . . . . . . . . . . . . . . 271
8.4.1 Examples of Copulas . . . . . . . . . . . . . . . . . . . 271
8.4.2 Sklar Theorem in the Bivariate Case . . . . . . . . . . 272
8.4.3 Applications for Simulation . . . . . . . . . . . . . . . 274
8.4.4 Simulation of (U1 , U2 ) ∼ C . . . . . . . . . . . . . . . 274
8.4.5 Modeling Dependence with Copulas . . . . . . . . . . 275
8.4.6 Positive Quadrant Dependence (PQD) Order . . . . . 276
8.5 Measures of Dependence . . . . . . . . . . . . . . . . . . . . 276
8.5.1 Estimation of a Bivariate Copula . . . . . . . . . . . . 278
8.5.1.1 Precision of the Estimation of the Empirical
Copula . . . . . . . . . . . . . . . . . . . . . 278
8.5.1.2 Tests of Independence Based on the Empirical
Copula . . . . . . . . . . . . . . . . . . . . . 278
8.5.2 Kendall Function . . . . . . . . . . . . . . . . . . . . . 280
8.5.2.1 Estimation of Kendall Function . . . . . . . 281
8.5.2.2 Precision of the Estimation of the Kendall
Function . . . . . . . . . . . . . . . . . . . . 282


xiv

Contents
8.5.2.3

8.6

8.7

Tests of Independence Based on the Empirical
Kendall Function . . . . . . . . . . . . . . . 282
8.5.3 Kendall Tau . . . . . . . . . . . . . . . . . . . . . . . . 286
8.5.3.1 Estimation of Kendall Tau . . . . . . . . . . 286
8.5.3.2 Precision of the Estimation of Kendall Tau . 287
8.5.4 Spearman Rho . . . . . . . . . . . . . . . . . . . . . . 287
8.5.4.1 Estimation of Spearman Rho . . . . . . . . . 288
8.5.4.2 Precision of the Estimation of Spearman Rho 288
8.5.5 van der Waerden Rho . . . . . . . . . . . . . . . . . . 289
8.5.5.1 Estimation of van der Waerden Rho . . . . . 290
8.5.5.2 Precision of the Estimation of van der Waerden Rho . . . . . . . . . . . . . . . . . . . . 290
8.5.6 Other Measures of Dependence . . . . . . . . . . . . . 291
8.5.6.1 Estimation of ρ(J) . . . . . . . . . . . . . . . 291
8.5.6.2 Precision of the Estimation of ρ(J) . . . . . . 292
8.5.7 Serial Dependence . . . . . . . . . . . . . . . . . . . . 292
Multivariate Copulas . . . . . . . . . . . . . . . . . . . . . . 293
8.6.1 Kendall Function . . . . . . . . . . . . . . . . . . . . . 294
8.6.2 Conditional Distributions . . . . . . . . . . . . . . . . 294
8.6.2.1 Applications of Theorem 8.6.2 . . . . . . . . 294
8.6.3 Stochastic Orders for Dependence . . . . . . . . . . . 295
8.6.3.1 Fr´echet-Hoeffding Bounds . . . . . . . . . . . 295
8.6.3.2 Application . . . . . . . . . . . . . . . . . . . 296
8.6.3.3 Supermodular Order . . . . . . . . . . . . . . 296
Families of Copulas . . . . . . . . . . . . . . . . . . . . . . . 297
8.7.1 Independence Copula . . . . . . . . . . . . . . . . . . 297
8.7.2 Elliptical Copulas . . . . . . . . . . . . . . . . . . . . 297

8.7.2.1 Estimation of ρ . . . . . . . . . . . . . . . . . 298
8.7.3 Gaussian Copula . . . . . . . . . . . . . . . . . . . . . 298
8.7.3.1 Simulation of Observations from a Gaussian
Copula . . . . . . . . . . . . . . . . . . . . . 299
8.7.4 Student Copula . . . . . . . . . . . . . . . . . . . . . . 299
8.7.4.1 Simulation of Observations from a Student
Copula . . . . . . . . . . . . . . . . . . . . . 300
8.7.5 Other Elliptical Copulas . . . . . . . . . . . . . . . . . 300
8.7.6 Archimedean Copulas . . . . . . . . . . . . . . . . . . 301
8.7.6.1 Financial Modeling . . . . . . . . . . . . . . 301
8.7.6.2 Recursive Formulas . . . . . . . . . . . . . . 301
8.7.6.3 Conjecture . . . . . . . . . . . . . . . . . . . 303
8.7.6.4 Kendall Tau for Archimedean Copulas . . . . 303
8.7.6.5 Simulation of Observations from an Archimedean
Copula . . . . . . . . . . . . . . . . . . . . . 304
8.7.7 Clayton Family . . . . . . . . . . . . . . . . . . . . . . 304
8.7.7.1 Simulation of Observations from a Clayton
Copula . . . . . . . . . . . . . . . . . . . . . 305


Contents
8.7.8

8.8

8.9
8.10

8.11

8.12
8.13

Gumbel Family . . . . . . . . . . . . . . . . . . . . . .
8.7.8.1 Simulation of Observations from a Gumbel
Copula . . . . . . . . . . . . . . . . . . . . .
8.7.9 Frank Family . . . . . . . . . . . . . . . . . . . . . . .
8.7.9.1 Simulation of Observations from a Frank Copula . . . . . . . . . . . . . . . . . . . . . . . .
8.7.10 Ali-Mikhail-Haq Family . . . . . . . . . . . . . . . . .
8.7.10.1 Simulation of Observations from an AliMikhail-Haq Copula . . . . . . . . . . . . . .
8.7.11 PQD Order for Archimedean Copula Families . . . . .
8.7.12 Farlie-Gumbel-Morgenstern Family . . . . . . . . . . .
8.7.13 Plackett Family . . . . . . . . . . . . . . . . . . . . . .
8.7.14 Other Copula Families . . . . . . . . . . . . . . . . . .
Estimation of the Parameters of Copula Models . . . . . . .
8.8.1 Considering Serial Dependence . . . . . . . . . . . . .
8.8.2 Estimation of Parameters: The Parametric Approach .
8.8.2.1 Advantages and Disadvantages . . . . . . . .
8.8.3 Estimation of Parameters: The Semiparametric Approach . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.8.3.1 Advantages and Disadvantages . . . . . . . .
8.8.4 Estimation of ρ for the Gaussian Copula . . . . . . . .
8.8.5 Estimation of ρ and ν for the Student Copula . . . . .
8.8.6 Estimation for an Archimedean Copula Family . . . .
8.8.7 Nonparametric Estimation of a Copula . . . . . . . . .
8.8.8 Nonparametric Estimation of Kendall Function . . . .
Tests of Independence . . . . . . . . . . . . . . . . . . . . . .
8.9.1 Test of Independence Based on the Copula . . . . . .
Tests of Goodness-of-Fit . . . . . . . . . . . . . . . . . . . .
8.10.1 Computation of P -Values . . . . . . . . . . . . . . . .
8.10.2 Using the Rosenblatt Transform for Goodness-of-Fit

Tests . . . . . . . . . . . . . . . . . . . . . . . . . . . .
8.10.2.1 Computation of P -Values . . . . . . . . . . .
Example of Implementation of a Copula Model . . . . . . . .
8.11.1 Change Point Tests . . . . . . . . . . . . . . . . . . . .
8.11.2 Serial Independence . . . . . . . . . . . . . . . . . . .
8.11.3 Modeling Serial Dependence . . . . . . . . . . . . . . .
8.11.3.1 Change Point Tests for the Residuals . . . .
8.11.3.2 Goodness-of-Fit for the Distribution of Innovations . . . . . . . . . . . . . . . . . . . . .
8.11.4 Modeling Dependence Between Innovations . . . . . .
8.11.4.1 Test of Independence for the Innovations . .
8.11.4.2 Goodness-of-Fit for the Copula of the Innovations . . . . . . . . . . . . . . . . . . . . . . .
Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . .
Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

xv
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306
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308
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310
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312

313
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318
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320
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321
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325
326


xvi

Contents
8.14
8.A

8.B
8.C
8.D
8.E

Assignment Questions . . . . . . . . . . . .
Continuous Time Markov Chains . . . . .
Tests of Independence . . . . . . . . . . . .
Polynomials Related to the Gumbel Copula
Polynomials Related to the Frank Copula .
Change Point Tests . . . . . . . . . . . . .
8.E.1 Change Point Test for the Copula .
8.F Auxiliary Results . . . . . . . . . . . . . .
8.G Proofs of the Results . . . . . . . . . . . .
8.G.1 Proof of Proposition 8.4.1 . . . . . .
8.G.2 Proof of Proposition 8.4.2 . . . . . .
8.G.3 Proof of Proposition 8.5.1 . . . . . .
8.G.4 Proof of Theorem 8.7.1 . . . . . . .
Bibliography . . . . . . . . . . . . . . . . . . . .

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331
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337
338
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339

9 Filtering
345
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 345
9.1 Description of the Filtering Problem . . . . . . . . . . . . . . 345
9.2 Kalman Filter . . . . . . . . . . . . . . . . . . . . . . . . . . 346
9.2.1 Model . . . . . . . . . . . . . . . . . . . . . . . . . . . 346
9.2.2 Filter Initialization . . . . . . . . . . . . . . . . . . . . 347
9.2.3 Estimation of Parameters . . . . . . . . . . . . . . . . 348
9.2.4 Implementation of the Kalman Filter . . . . . . . . . . 348
9.2.4.1 Solution . . . . . . . . . . . . . . . . . . . . . 348
9.2.5 The Kalman Filter for General Linear Models . . . . . 353
9.3 IMM Filter . . . . . . . . . . . . . . . . . . . . . . . . . . . . 354
9.3.1 IMM Algorithm . . . . . . . . . . . . . . . . . . . . . 354
9.3.2 Implementation of the IMM Filter . . . . . . . . . . . 356
9.4 General Filtering Problem . . . . . . . . . . . . . . . . . . . 356
9.4.1 Kallianpur-Striebel Formula . . . . . . . . . . . . . . . 356
9.4.2 Recursivity . . . . . . . . . . . . . . . . . . . . . . . . 357
9.4.3 Implementing the Recursive Zakai Equation . . . . . . 358
9.4.4 Solving the Filtering Problem . . . . . . . . . . . . . . 358
9.5 Computation of the Conditional Densities . . . . . . . . . . . 358
9.5.1 Convolution Method . . . . . . . . . . . . . . . . . . . 359
9.5.2 Kolmogorov Equation . . . . . . . . . . . . . . . . . . 360
9.6 Particle Filters . . . . . . . . . . . . . . . . . . . . . . . . . . 360
9.6.1 Implementation of a Particle Filter . . . . . . . . . . . 360
9.6.2 Implementation of an Auxiliary Sampling/Importance

Resampling (ASIR) Particle Filter . . . . . . . . . . . 361
9.6.2.1 ASIR0 . . . . . . . . . . . . . . . . . . . . . . 363
9.6.2.2 ASIR1 . . . . . . . . . . . . . . . . . . . . . . 363
9.6.2.3 ASIR2 . . . . . . . . . . . . . . . . . . . . . . 364
9.6.3 Estimation of Parameters . . . . . . . . . . . . . . . . 365
9.6.3.1 Smoothed Likelihood . . . . . . . . . . . . . 365


Contents
9.7
9.8
9.9
9.A
9.B
9.C
9.D

Suggested Reading . . . . . . . .
Exercises . . . . . . . . . . . . .
Assignment Questions . . . . . .
Schwartz Model . . . . . . . . .
Auxiliary Results . . . . . . . .
Fourier Transform . . . . . . . .
Proofs of the Results . . . . . .
9.D.1 Proof of Proposition 9.2.1
Bibliography . . . . . . . . . . . . . .

xvii
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10 Applications of Filtering
375
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 375
10.1 Estimation of ARMA Models . . . . . . . . . . . . . . . . . . 375
10.1.1 AR(p) Processes . . . . . . . . . . . . . . . . . . . . . 375
10.1.1.1 MA(q) Processes . . . . . . . . . . . . . . . . 376
10.1.2 MA Representation . . . . . . . . . . . . . . . . . . . . 376
10.1.3 ARMA Processes and Filtering . . . . . . . . . . . . . 377
10.1.3.1 Implementation of the Kalman Filter in the
Gaussian Case . . . . . . . . . . . . . . . . . 378
10.1.4 Estimation of Parameters of ARMA Models . . . . . . 379
10.2 Regime-Switching Markov Models . . . . . . . . . . . . . . . 380

10.2.1 Serial Dependence . . . . . . . . . . . . . . . . . . . . 380
10.2.2 Prediction of the Regimes . . . . . . . . . . . . . . . . 381
10.2.3 Conditional Densities and Predictions . . . . . . . . . 382
10.2.4 Estimation of the Parameters . . . . . . . . . . . . . . 383
10.2.4.1 Implementation of the E-step . . . . . . . . . 383
10.2.5 M-step in the Gaussian Case . . . . . . . . . . . . . . 384
10.2.6 Tests of Goodness-of-Fit . . . . . . . . . . . . . . . . . 385
10.2.7 Continuous Time Regime-Switching Markov Processes 388
10.3 Replication of Hedge Funds . . . . . . . . . . . . . . . . . . . 389
10.3.0.1 Measurement of Errors . . . . . . . . . . . . 390
10.3.1 Replication by Regression . . . . . . . . . . . . . . . . 391
10.3.2 Replication by Kalman Filter . . . . . . . . . . . . . . 391
10.3.3 Example of Application . . . . . . . . . . . . . . . . . 391
10.4 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . 395
10.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 396
10.6 Assignment Questions . . . . . . . . . . . . . . . . . . . . . . 397
10.A EM Algorithm . . . . . . . . . . . . . . . . . . . . . . . . . . 398
10.B Sampling Moments vs Theoretical Moments . . . . . . . . . 401
10.C Rosenblatt Transform for the Regime-Switching Model . . . 401
10.D Proofs of the Results . . . . . . . . . . . . . . . . . . . . . . 403
10.D.1 Proof of Proposition 10.1.1 . . . . . . . . . . . . . . . 403
10.D.2 Proof of Proposition 10.1.2 . . . . . . . . . . . . . . . 404
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 404


xviii

Contents

A Probability Distributions

Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.1 Introduction . . . . . . . . . . . . . . . . . . . . . . . . . . .
A.2 Discrete Distributions and Densities . . . . . . . . . . . . . .
A.2.1 Expected Value and Moments of Discrete Distributions
A.3 Absolutely Continuous Distributions and Densities . . . . . .
A.3.1 Expected Value and Moments of Absolutely Continuous
Distributions . . . . . . . . . . . . . . . . . . . . . . .
A.4 Characteristic Functions . . . . . . . . . . . . . . . . . . . .
A.4.1 Inversion Formula . . . . . . . . . . . . . . . . . . . .
A.5 Moments Generating Functions and Laplace Transform . . .
A.5.1 Cumulants . . . . . . . . . . . . . . . . . . . . . . . .
A.5.1.1 Extension . . . . . . . . . . . . . . . . . . . .
A.6 Families of Distributions . . . . . . . . . . . . . . . . . . . .
A.6.1 Bernoulli Distribution . . . . . . . . . . . . . . . . . .
A.6.2 Binomial Distribution . . . . . . . . . . . . . . . . . .
A.6.3 Poisson Distribution . . . . . . . . . . . . . . . . . . .
A.6.4 Geometric Distribution . . . . . . . . . . . . . . . . .
A.6.5 Negative Binomial Distribution . . . . . . . . . . . . .
A.6.6 Uniform Distribution . . . . . . . . . . . . . . . . . . .
A.6.7 Gaussian Distribution . . . . . . . . . . . . . . . . . .
A.6.8 Log-Normal Distribution . . . . . . . . . . . . . . . . .
A.6.9 Exponential Distribution . . . . . . . . . . . . . . . .
A.6.10 Gamma Distribution . . . . . . . . . . . . . . . . . . .
A.6.10.1 Properties of the Gamma Function . . . . . .
A.6.11 Chi-Square Distribution . . . . . . . . . . . . . . . . .
A.6.12 Non-Central Chi-Square Distribution . . . . . . . . . .
A.6.12.1 Simulation of Non-Central Chi-Square Variables . . . . . . . . . . . . . . . . . . . . . .
A.6.13 Student Distribution . . . . . . . . . . . . . . . . . . .
A.6.14 Johnson SU Type Distributions . . . . . . . . . . . . .
A.6.15 Beta Distribution . . . . . . . . . . . . . . . . . . . . .

A.6.16 Cauchy Distribution . . . . . . . . . . . . . . . . . . .
A.6.17 Generalized Error Distribution . . . . . . . . . . . . .
A.6.18 Multivariate Gaussian Distribution . . . . . . . . . . .
A.6.18.1 Representation of a Random Gaussian Vector
A.6.19 Multivariate Student Distribution . . . . . . . . . . .
A.6.20 Elliptical Distributions . . . . . . . . . . . . . . . . . .
A.6.21 Simulation of an Elliptic Distribution . . . . . . . . .
A.7 Conditional Densities and Joint Distributions . . . . . . . . .
A.7.1 Multiplication Formula . . . . . . . . . . . . . . . . .
A.7.2 Conditional Distribution in the Markovian Case . . .
A.7.3 Rosenblatt Transform . . . . . . . . . . . . . . . . . .
A.8 Functions of Random Vectors . . . . . . . . . . . . . . . . . .
A.9 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

407
407
407
408
408
410
410
412
413
413
414
415
415
415
416
416

417
417
417
418
418
419
420
420
421
421
421
422
423
423
424
424
425
425
426
426
429
429
429
430
430
430
433


Contents

xix

Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

434

B Estimation of Parameters
435
Summary . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 435
B.1 Maximum Likelihood Principle . . . . . . . . . . . . . . . . . 435
B.2 Precision of Estimators . . . . . . . . . . . . . . . . . . . . . 437
B.2.1 Confidence Intervals and Confidence Regions . . . . . 437
B.2.2 Nonparametric Prediction Interval . . . . . . . . . . . 437
B.3 Properties of Estimators . . . . . . . . . . . . . . . . . . . . 438
B.3.1 Almost Sure Convergence . . . . . . . . . . . . . . . . 438
B.3.2 Convergence in Probability . . . . . . . . . . . . . . . 438
B.3.3 Convergence in Mean Square . . . . . . . . . . . . . . 438
B.3.4 Convergence in Law . . . . . . . . . . . . . . . . . . . 439
B.3.4.1 Delta Method . . . . . . . . . . . . . . . . . 440
B.3.5 Bias and Consistency . . . . . . . . . . . . . . . . . . 441
B.4 Central Limit Theorem for Independent Observations . . . . 441
B.4.1 Consistency of the Empirical Mean . . . . . . . . . . . 442
B.4.2 Consistency of the Empirical Coefficients of Skewness
and Kurtosis . . . . . . . . . . . . . . . . . . . . . . . 442
B.4.3 Confidence Intervals I . . . . . . . . . . . . . . . . . . 445
B.4.4 Confidence Ellipsoids . . . . . . . . . . . . . . . . . . . 445
B.4.5 Confidence Intervals II . . . . . . . . . . . . . . . . . . 445
B.5 Precision of Maximum Likelihood Estimator for Serially Independent Observations . . . . . . . . . . . . . . . . . . . . . . 446
B.5.1 Estimation of Fisher Information Matrix . . . . . . . . 446

B.6 Convergence in Probability and the Central Limit Theorem for
Serially Dependent Observations . . . . . . . . . . . . . . . . 448
B.7 Precision of Maximum Likelihood Estimator for Serially Dependent Observations . . . . . . . . . . . . . . . . . . . . . . 448
B.8 Method of Moments . . . . . . . . . . . . . . . . . . . . . . . 450
B.9 Combining the Maximum Likelihood Method and the Method
of Moments . . . . . . . . . . . . . . . . . . . . . . . . . . . . 452
B.10 M-estimators . . . . . . . . . . . . . . . . . . . . . . . . . . . 453
B.11 Suggested Reading . . . . . . . . . . . . . . . . . . . . . . . . 454
B.12 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
Bibliography . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 454
Index

455



Preface

The aim of this book is to guide existing and future practitioners through
the implementation of the most useful stochastic models used in financial
engineering. There is a plethora of books on financial engineering but the statistical aspect of the implementation of these models, where lie many of the
challenges, is often overlooked or restricted to a few well-known cases like the
Black-Scholes and GARCH models. So in addition to a basic presentation of
the models, my objective in writing this book was also to include the relevant
questions related to their implementation. For example, the chapter on the
modeling of interest rates includes the estimation of the parameters of the
proposed models, which is essential from an implementation point-of-view,
but is usually ignored. Other such important topics, including the effect of
estimation errors on the value of options, hedging in discrete time, dependence modeling through copulas and hedge fund replication, are also covered.
Overall, I believe this book fills an important gap in the financial engineering

literature since I faced many of these implementation issues in my own work
as a part-time consultant in the financial industry. Another aspect covered
here that is largely ignored in most textbooks pertains to the validation of
the models. Throughout the chapters, in addition to showing how to estimate
parameters efficiently, I also demonstrate, whenever possible, how one can test
the validity of the proposed models. Many techniques I used in this book appeared in research papers (many I have authored myself), and while powerful,
are too rarely applied in practice. A companion website also offers MATLAB R
and R programs that are likely to help practitioners with the implementation
of these tools in the context of real-life financial problems.
The content of this book has been developed in the last ten years for a
graduate course on statistical methods for students in finance and financial engineering. Since the course can be taken by first-year graduate students, I try
to avoid as much as possible any reference to stochastic calculus. The book
is also self-contained in the sense that no financial background is required,
although it would definitely help. Rigor is shown by proving most results.
However, for the sake of readability, the proofs are presented in a series of
appendices, together with more advanced topics, including two appendices
on probability distributions and parameter estimation, which both provide
theoretical support for the results presented in the book. In every chapter, I
try to introduce the statistical tools required to implement the models taken
from the cornerstone articles in financial engineering. The use of each tool is
xxi


xxii

Preface

facilitated by examples of application using MATLAB programs that are available on my website at www.brunoremillard.com.
Starting with the pioneering contribution of Black & Scholes, properties of
univariate and multivariate models for asset dynamics are studied in Chapters

1 and 2, together with estimation techniques which are valid for independent
observations. The effect of parameter estimation on the value of options is also
covered. Furthermore, using techniques developed by Broadie-Glasserman, I
show how Monte Carlo simulations can be used to estimate option prices and
sensitivity parameters known as “greeks.”
In Chapter 3, the limits of the Black-Scholes model are discussed, statistical
tests are introduced to verify some of its assumptions, and a section discusses
the challenges of dynamic hedging in discrete time.
Next, in Chapter 4, the estimation of risk and performance measures is covered, starting with a discussion on the axioms for coherent risk measures. The
main tools used in this chapter are Monte Carlo methods, and other statistical
tools such as nonparametric estimation of distribution and quantile functions,
Edgeworth and Cornish-Fisher expansions, saddlepoint approximations, and
the inversion of characteristic functions.
In Chapter 5, I present the foundations of the spot interest rate modeling
literature using the article of Vasicek, and I show especially how to estimate
parameters of the so-called Vasicek and Cox-Ingersoll-Ross models, including
the market price of risk parameters. To do so, maximum likelihood techniques
for dependent observations are used, along with a method proposed by Duan
for dealing with the unobservable nature of the spot rates.
The article of Merton on jump-diffusion processes and option pricing is
covered in Chapter 6 and is used as a pretext for the introduction of L´evy
processes and their financial applications, including path properties, change
of measure, option pricing, and parameter estimation.
Using the famous article of Duan on GARCH models and option pricing,
the properties and parameter estimation of GARCH models are presented in
Chapter 7. The chapter also covers the goodness-of-fit tests, using the Khmaladze transform and parametric bootstrap. I show that as a limiting case, one
obtains stochastic volatility models, in particular the model studied by Hull
& White. The well-known Heston model is also discussed.
In Chapter 8, weak replication of hedge funds and simple credit risk models are used to illustrate the tremendous importance of dependence models.
This issue is discussed at great length with the use of copulas. All aspects

pertaining to these models are covered: properties, simulation, dependence
measures, estimation, and goodness-of-fit. Because of the inherent serial dependence observed in financial time series, I also show how to deal with residuals of stochastic volatility models.
Finally, in Chapters 9 and 10, I cover the topic of filtering and its financial
applications, when unobservable factors have to be predicted. Following the
insights of Schwartz on filtering in a commodities context, the famous Kalman
filter is introduced. Two other methods, the IMM and particle filters are then


Preface

xxiii

studied. The latter is a class of Monte Carlo methods for solving the general
filtering problem. Estimation of the parameters of the underlying models is
also discussed. Then, filtering is applied in three contexts, namely estimation
of ARMA models, estimation and prediction of Hidden Markov models using
the powerful EM algorithm, and hedge funds replication.
This book, written over such a long period of time, has benefited from
the valuable help and feedback of many people. I first wish to thank Matt
Davidson, professor at University of Western Ontario, and Hugues Langlois,
a Ph.D. student at McGill University, for their helpful comments and suggestions on an earlier version of this book. I would also like to thank my colleague
at HEC Jean-Fran¸cois Plante, for his detailed and valuable comments on the
chapter on copulas, and HEC Montr´eal for their financial support. Finally,
I would like to thank the students in the Financial Engineering program at
HEC Montr´eal, especially Fr´ed´eric Godin, for their comments and suggestions
along with their understanding throughout the years when they often found
themselves in the position to be the first to test all the new material. Finally,
a special thanks to Alexis Constantineau for his help in the preparation of
exercises and to David-Shaun Guay for converting my MATLAB programs
into R programs.

Bruno R´emillard
Montr´eal, October 3rd , 2012.
MATLAB and Simulink are registered trademark of The MathWorks, Inc. For
product information, please contact:
The Mathworks, Inc.
3 Apple Hill Drive
Natick, MA 01760-2098 USA
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Web: www.mathworks.com