Financial Mathematics: A Comprehensive Treatment in Continuous Time Volume II

511 Pages·2022·8.007 MB·English

by  Giuseppe Campolieti, Roman N. Makarov

Financial Mathematics
A Comprehensive Treatment in
Continuous Time
The book has been tested and refined through years of classroom teaching experience.
With an abundance of examples, problems, and fully worked-out solutions, the text in-
troduces the financial theory and relevant mathematical methods in a mathematically
rigorous yet engaging way.
This textbook provides complete coverage of continuous-time financial models that
form the cornerstones of financial derivative pricing theory. Unlike similar texts in the
field, this one presents multiple problem-solving approaches, linking related compre-
hensive techniques for pricing different types of financial derivatives.
Key features:
• In-depth coverage of continuous-time theory and methodology.
• Numerous, fully worked out examples and exercises in every chapter.
• Mathematically rigorous and consistent yet bridging various basic and more ad-
vanced concepts.
• Judicious balance of financial theory and mathematical methods.
• Guide to material.
This revision contains:
• Almost 150 pages worth of new material in all chapters.
• An expanded set of solved problems and additional exercises.
• Answers to all exercises.
This book is a comprehensive, self-contained, and unified treatment of the main theory
and application of mathematical methods behind modern-day financial mathematics.
The text complements Financial Mathematics: A Comprehensive Treatment in Dis-
crete Time, by the same authors, also published by CRC Press.
Chapman & Hall/CRC Financial Mathematics Series
Series Editors
M.A.H. Dempster Rama Cont
Centre for Financial Research Department of Mathematics
Department of Pure Mathematics and Statistics Imperial College, UK
University of Cambridge, UK
Robert A. Jarrow
Dilip B. Madan Lynch Professor of Investment Management
Robert H. Smith School of Business Johnson Graduate School of Management
University of Maryland, USA Cornell University, USA
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Financial Mathematics
A Comprehensive Treatment in
Continuous Time
Volume II
Giuseppe Campolieti
Roman N. Makarov
First edition published 2023
by CRC Press
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and by CRC Press
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© 2023 Giuseppe Campolieti and Roman N Makarov
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Library of Congress Cataloging-in-Publication Data
Names: Campolieti, Giuseppe (Mathematics professor), author. | Makarov,
Roman N., author.
Title: Financial mathematics Volume II : a comprehensive treatment in
continuous time / Giuseppe Campolieti and Roman N. Makarov.
Description: First Edition. | Boca Raton, FL : CRC Press, an imprint of
Taylor and Francis, 2023. | Series: Textbooks in mathematics | Includes
bibliographical references and index.
Identifiers: LCCN 2022032096 (print) | LCCN 2022032097 (ebook) | ISBN
9781138603639 (hardback) | ISBN 9781032392592 (paperback) | ISBN
9780429468889 (ebook)
Subjects: LCSH: Finance--Mathematical models.
Classification: LCC HG106 .C35 2023 (print) | LCC HG106 (ebook) | DDC
332.01/5195--dc23/eng/20221104
LC record available at https://lccn.loc.gov/2022032096
LC ebook record available at https://lccn.loc.gov/2022032097
ISBN: 978-1-138-60363-9 (hbk)
ISBN: 978-1-032-39259-2 (pbk)
ISBN: 978-0-429-46888-9 (ebk)
DOI: 10.1201/9780429468889
Typeset in CMR10 font
by KnowledgeWorks Global Ltd.
Publisher’s note: This book has been prepared from camera-ready copy provided by the authors.
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Contents
List of Figures xi
Preface xiii
Authors xvii
I Stochastic Calculus with Brownian Motion 1
1 One-Dimensional Brownian Motion and Related Processes 3
1.1 Multivariate Normal Distributions . . . . . . . . . . . . . . . . . . . . . . . 3
1.1.1 Multivariate Normal Distribution . . . . . . . . . . . . . . . . . . . . 3
1.1.2 Conditional Normal Distributions. . . . . . . . . . . . . . . . . . . . 4
1.2 Standard Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1.2.1 One-Dimensional Symmetric Random Walk . . . . . . . . . . . . . . 5
1.2.2 Formal Definition and Basic Properties of Brownian Motion . . . . . 11
1.2.3 Multivariate Distribution of Brownian Motion. . . . . . . . . . . . . 14
1.2.4 The Markov Property and the Transition PDF . . . . . . . . . . . . 17
1.2.5 Quadratic Variation and Nondifferentiability of Paths . . . . . . . . 25
1.3 Some Processes Derived from Brownian Motion . . . . . . . . . . . . . . . 28
1.3.1 Drifted Brownian Motion . . . . . . . . . . . . . . . . . . . . . . . . 28
1.3.2 Geometric Brownian Motion . . . . . . . . . . . . . . . . . . . . . . 29
1.3.3 Processes related by a monotonic mapping . . . . . . . . . . . . . . 31
1.3.4 Brownian Bridge . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 32
1.3.5 Gaussian Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . 35
1.4 First Hitting Times and Maximum and Minimum of Brownian Motion . . 37
1.4.1 The Reflection Principle: Standard Brownian Motion . . . . . . . . . 37
1.4.2 Translated and Scaled Driftless Brownian Motion . . . . . . . . . . . 45
1.4.3 Brownian Motion with Drift. . . . . . . . . . . . . . . . . . . . . . . 47
1.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2 Introduction to Continuous-Time Stochastic Calculus 59
2.1 The Riemann Integral of Brownian Motion . . . . . . . . . . . . . . . . . . 59
2.1.1 The Riemann Integral . . . . . . . . . . . . . . . . . . . . . . . . . . 59
2.1.2 The Integral of a Brownian Path . . . . . . . . . . . . . . . . . . . . 59
2.2 The Riemann–Stieltjes Integral of Brownian Motion . . . . . . . . . . . . . 62
2.2.1 The Riemann–Stieltjes Integral . . . . . . . . . . . . . . . . . . . . . 62
2.2.2 Integrals w.r.t. Brownian Motion: Preliminary Discussion . . . . . . 64
2.3 The Itˆo Integral and Its Basic Properties . . . . . . . . . . . . . . . . . . . 66
2.3.1 The Itˆo Integral for Simple Processes. . . . . . . . . . . . . . . . . . 66
2.3.2 The Itˆo Integral for General Processes . . . . . . . . . . . . . . . . . 74
2.4 Itˆo Processes and Their Properties . . . . . . . . . . . . . . . . . . . . . . . 83
2.4.1 Gaussian Processes Generated by Itˆo Integrals . . . . . . . . . . . . 83
vii
viii Contents
2.4.2 Itˆo Processes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 84
2.4.3 Quadratic and Co-Variation of Itˆo Processes . . . . . . . . . . . . . 86
2.5 Itˆo’s Formula for Functions of BM and Itˆo Processes . . . . . . . . . . . . . 90
2.5.1 Itˆo’s Formula for Functions of BM . . . . . . . . . . . . . . . . . . . 90
2.5.2 An “Antiderivative” Formula for Evaluating Itˆo Integrals . . . . . . 93
2.5.3 Itˆo’s Formula for Itˆo Processes . . . . . . . . . . . . . . . . . . . . . 95
2.6 Stochastic Differential Equations . . . . . . . . . . . . . . . . . . . . . . . . 98
2.6.1 Solutions to Linear SDEs . . . . . . . . . . . . . . . . . . . . . . . . 99
2.6.2 Existence and Uniqueness of a Strong Solution to an SDE . . . . . . 106
2.7 TheMarkovProperty,Martingales,Feynman–KacFormulae,andTransition
CDFs and PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 107
2.7.1 Forward Kolmogorov PDE . . . . . . . . . . . . . . . . . . . . . . . 121
2.7.2 Transition CDF/PDF for Time-Homogeneous Diffusions . . . . . . . 123
2.8 Radon–Nikodym Derivative Process and Girsanov’s Theorem . . . . . . . . 124
2.8.1 Some Applications of Girsanov’s Theorem . . . . . . . . . . . . . . . 130
2.9 Brownian Martingale Representation Theorem . . . . . . . . . . . . . . . . 134
2.10 Stochastic Calculus for Multidimensional BM . . . . . . . . . . . . . . . . . 135
2.10.1 The Itˆo Integral and Itˆo’s Formula for Multiple Processes on
Multidimensional BM . . . . . . . . . . . . . . . . . . . . . . . . . . 135
2.10.2 Multidimensional SDEs, Feynman–Kac Formulae, and Transition
CDFs and PDFs . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 147
2.10.3 Girsanov’s Theorem for Multidimensional BM. . . . . . . . . . . . . 157
2.10.4 Martingale Representation Theorem for Multidimensional BM . . . 160
2.11 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 160
II Continuous-Time Modelling 169
3 Risk-Neutral Pricing in the (B,S) Economy: One Underlying Stock 171
3.1 From the CRR Model to the BSM Model . . . . . . . . . . . . . . . . . . . 172
3.1.1 Portfolio Strategies in the Binomial Tree Model . . . . . . . . . . . . 174
3.1.2 The Cox–Ross–Rubinstein Model and its Continuous-Time Limit . . 177
3.2 Replication (Hedging) and Derivative Pricing in the Simplest Black–Scholes
Economy . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 180
3.2.1 Pricing Standard European Calls and Puts . . . . . . . . . . . . . . 187
3.2.2 Hedging Standard European Calls and Puts . . . . . . . . . . . . . . 190
3.2.3 Europeans with Piecewise Linear Payoffs . . . . . . . . . . . . . . . 196
3.2.4 Power Options . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 201
3.2.5 Dividend Paying Stock . . . . . . . . . . . . . . . . . . . . . . . . . . 204
3.2.6 Option Pricing with the Stock Num´eraire . . . . . . . . . . . . . . . 211
3.3 Forward Starting, Chooser, and Compound Options . . . . . . . . . . . . . 216
3.4 Some European-Style Path-Dependent Derivatives . . . . . . . . . . . . . . 225
3.4.1 Risk-Neutral Pricing under GBM . . . . . . . . . . . . . . . . . . . . 228
3.4.2 Pricing Single Barrier Options . . . . . . . . . . . . . . . . . . . . . 232
3.4.3 Pricing Lookback Options . . . . . . . . . . . . . . . . . . . . . . . . 238
3.5 Structural Credit Risk Models . . . . . . . . . . . . . . . . . . . . . . . . . 245
3.5.1 The Merton Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 246
3.5.2 The Black–Cox Model . . . . . . . . . . . . . . . . . . . . . . . . . . 247
3.6 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 249
Contents ix
4 Risk-Neutral Pricing in a Multi-Asset Economy 261
4.1 General Multi-Asset Market Model: Replication and Risk-Neutral Pricing . 262
4.2 EquivalentMartingaleMeasures:DerivativePricingwithGeneralNum´eraire
Assets . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 269
4.3 Black–Scholes PDE and Delta Hedging for Standard Multi-Asset Derivatives 274
4.3.1 Standard European Option Pricing for Multi-Stock GBM . . . . . . 277
4.3.2 Explicit Pricing Formulae for the GBM Model . . . . . . . . . . . . 280
4.3.3 Cross-Currency Option Valuation . . . . . . . . . . . . . . . . . . . 289
4.3.4 Option Valuation with General Num´eraire Assets . . . . . . . . . . . 295
4.4 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 300
5 American Options 307
5.1 Basic Properties of Early-Exercise Options . . . . . . . . . . . . . . . . . . 307
5.2 Arbitrage-Free Pricing of American Options . . . . . . . . . . . . . . . . . 311
5.2.1 Optimal Stopping Formulation and Early-Exercise Boundary . . . . 311
5.2.2 The Smooth Pasting Condition . . . . . . . . . . . . . . . . . . . . . 314
5.2.3 Put-Call Symmetry Relation . . . . . . . . . . . . . . . . . . . . . . 316
5.2.4 Dynamic Programming Approach for Bermudan Options . . . . . . 317
5.3 Perpetual American Options . . . . . . . . . . . . . . . . . . . . . . . . . . 318
5.3.1 Pricing a Perpetual Put Option . . . . . . . . . . . . . . . . . . . . . 319
5.3.2 Pricing a Perpetual Call Option . . . . . . . . . . . . . . . . . . . . 321
5.4 Finite-Expiration American Options . . . . . . . . . . . . . . . . . . . . . . 322
5.4.1 The PDE Formulation . . . . . . . . . . . . . . . . . . . . . . . . . . 322
5.4.2 The Integral Equation Formulation . . . . . . . . . . . . . . . . . . . 325
5.5 Exercises . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 327
6 Interest-Rate Modelling and Derivative Pricing 331
6.1 Basic Fixed Income Instruments . . . . . . . . . . . . . . . . . . . . . . . . 331
6.1.1 Bonds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 331
6.1.2 Forward Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 332
6.1.3 Arbitrage-Free Pricing . . . . . . . . . . . . . . . . . . . . . . . . . . 333
6.1.4 Fixed Income Derivatives . . . . . . . . . . . . . . . . . . . . . . . . 334
6.2 Single-Factor Models . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 336
6.2.1 Diffusion Models for the Short Rate Process . . . . . . . . . . . . . . 337
6.2.2 PDE for the Zero-Coupon Bond Value . . . . . . . . . . . . . . . . . 338
6.2.3 Affine Term Structure Models . . . . . . . . . . . . . . . . . . . . . . 340
6.2.4 The Ho–Lee Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 341
6.2.5 The Vasiˇcek Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . 342
6.2.6 The Cox–Ingersoll–Ross Model . . . . . . . . . . . . . . . . . . . . . 344
6.3 Heath–Jarrow–Morton Formulation . . . . . . . . . . . . . . . . . . . . . . 346
6.3.1 HJM under Risk-Neutral Measure . . . . . . . . . . . . . . . . . . . 347
6.3.2 Relationship between HJM and Affine Yield Models . . . . . . . . . 350
6.4 Multifactor Affine Term Structure Models . . . . . . . . . . . . . . . . . . . 353
6.4.1 Gaussian Multifactor Models . . . . . . . . . . . . . . . . . . . . . . 354
6.4.2 Equivalent Classes of Affine Models . . . . . . . . . . . . . . . . . . 355
6.5 Pricing Derivatives under Forward Measures . . . . . . . . . . . . . . . . . 356
6.5.1 Forward Measures . . . . . . . . . . . . . . . . . . . . . . . . . . . . 356
6.5.2 Pricing Stock Options under Stochastic Interest Rates . . . . . . . . 358
6.5.3 Pricing Options on Zero-Coupon Bonds . . . . . . . . . . . . . . . . 360
6.6 LIBOR Model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363
6.6.1 LIBOR Rates . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 363