Financial Engineering with Finite Elements - Buch.de · Financial Engineering with Finite Elements J¨urgen Topper v. ... 1 Introduction 3 2 Some Prototype Models 7. 2.1 Optimal price

WY061-FMdrv WY061-Topper January 10, 2005 22:21 Char Count= 0

Financial Engineeringwith Finite Elements

Jurgen Topper

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The rich knowledge of numerical analysis from engineering is beginning to merge withmathematical finance in the this new book by Jurgen Topper one of the first introducingFinite Element Methods (FEM) in financial engineering. Many differential equations relevantin finance are introduced quickly. A special focus is the Black-Scholes/Merton equation in oneand more dimensions. Detailed examples and case studies explain how to use FEM to solvethese equations easy to access for a large audience. The examples use Sewells PDE2Deasy-to-use, interactive, general-purpose partial differential equation solver which has beenin development for 30 years. We find a detailed discussion of boundary conditions, handlingof dividends and applications to exotic options including baskets with barriers and options ona trading account. Many useful mathematical tools are listed in the extended appendix.

Uwe Wystup, Commerzbank Securities and HfB,Business School of Finance and Management, Germany

Engineers have very successfully applied finite elements methods for decades. For the firsttime, Jurgen Topper now introduces this powerful technique to the financial community. Hegives a comprehensive overview of finite elements and demonstrates how this method can beused in elegantly solving derivatives pricing problems. This book fills a gap in the literaturefor financial modelling techniques and will be a very useful addition to the toolkit of financialengineers.

Christian T. Hille, Nomura International plc, London

Jurgen Topper provides the first textbook on the numerical solution of differential equationsarising in finance with finite elements (FE). Since most standard FE textbooks only coverself-adjoint PDEs, this book is very useful because it discusses FE for problems which are notself-adjoint like most problems in option pricing. Besides, it presents a methods (collocationFE) for problems which cannot be cast into divergence form, so that the popular Galerkinapproach cannot be applied. Altogether: A recommendable recource for quants and academicslooking for an alternative to finite differences.

Matthias Heurich, Capital Markets Rates, Quantitative Analyst,Dresdner Kleinwort Wasserstein

This book is to my knowledge the first one which covers the technique of finite elementsincluding all the practical important details in conjunction with applications to quantitativefinance. Throughout the book, detailed case studies and numerical examples most of themrelated to option pricing illustrate the methodology. This book is a must for every quantimplementing finite elements techniques in financial applications.

Wolfgang M. Schmidt, Professor for Quantitative Finance,Hochschule fur Bankwirtschaft, Frankfurt

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For other titles in the Wiley Finance seriesplease see www.wiley.com/Finance

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Jurgen Topper

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Copyright C 2005 John Wiley & Sons Ltd, The Atrium, Southern Gate, Chichester,West Sussex PO19 8SQ, England

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Library of Congress Cataloging-in-Publication Data

Topper, Jrgen.Financial engineering with finite elements / by Jrgen Topper.

p. cm. (Wiley finance series)Includes bibliographical references (p. ) and index.ISBN 0-471-48690-6 (cloth : alk. paper)1. Financial engineeringEconometric models. 2. Finite element method. I. Title. II. Series.HG176.7.T66 2005658.155015195dc22 2004022228

British Library Cataloguing in Publication Data

A catalogue record for this book is available from the British Library

ISBN 0-471-48690-6 (HB)

Typeset in 10/12pt Times and Helvetica by TechBooks, New Delhi, IndiaPrinted and bound in Great Britain by Antony Rowe Ltd, Chippenham, WiltshireThis book is printed on acid-free paper responsibly manufactured from sustainable forestryin which at least two trees are planted for each one used for paper production.

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Contents

Preface xv

List of symbols xvii

PART I PRELIMINARIES 1

1 Introduction 3

2 Some Prototype Models 72.1 Optimal price policy of a monopolist 72.2 The BlackScholes option pricing model 82.3 Pricing American options 102.4 Multi-asset options with stochastic correlation 122.5 The steady-state distribution of the Vasicek interest rate process 142.6 Notes 16

3 The Conventional Approach: Finite Differences 173.1 General considerations for numerical computations 17

3.1.1 Evaluation criteria 173.1.2 Turning unbounded domains into bounded domains 18

3.2 Ordinary initial value problems 223.2.1 Basic concepts 223.2.2 Eulers method 233.2.3 Taylor methods 283.2.4 RungeKutta methods 303.2.5 The backward Euler method 333.2.6 The CrankNicolson method 353.2.7 Predictorcorrector methods 363.2.8 Adaptive techniques 383.2.9 Methods for systems of equations 39

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3.3 Ordinary two-point boundary value problems 463.3.1 Introductory remarks 463.3.2 Finite difference methods 463.3.3 Shooting methods 49

3.4 Initial boundary value problems 493.4.1 The explicit scheme 493.4.2 The implicit scheme 513.4.3 The CrankNicolson method 523.4.4 Integrating early exercise 52

3.5 Notes 53

PART II FINITE ELEMENTS 55

4 Static 1D Problems 574.1 Basic features of finite element methods 574.2 The method of weighted residuals one-element solutions 574.3 The Ritz variational method 724.4 The method of weighted residuals a more general view 744.5 Multi-element solutions 75

4.5.1 The Galerkin method with linear elements 764.5.2 The Galerkin method with quadratic trial functions 894.5.3 The collocation method with cubic Hermite trial functions 93

4.6 Case studies 994.6.1 The Evans model of a monopolist 994.6.2 First exit time of a geometric Brownian motion 994.6.3 The steady-state distribution of the OrnsteinUhlenbeck

process 1014.6.4 Convection-dominated problems 102

4.7 Convergence 1064.8 Notes 107

5 Dynamic 1D Problems 1095.1 Derivation of element equations 109

5.1.1 The Galerkin method 1095.1.2 The collocation method 114

5.2 Case studies 1155.2.1 Plain vanilla options 1155.2.2 Hedging parameters 1235.2.3 Various exotic options 1325.2.4 The CEV model 1425.2.5 Some practicalities: Dividends and settlement 150

6 Static 2D Problems 1616.1 Introduction and overview 1616.2 Construction of a mesh 1626.3 The Galerkin method 165

6.3.1 The Galerkin method with linear elements (triangles) 1656.3.2 The Galerkin method with linear elements (rectangular elements) 187


Contents xi

6.4 Case studies 1876.4.1 Brownian motion leaving a disk 1876.4.2 Ritz revisited 1886.4.3 First exit time in a two-asset pricing problem 191

6.5 Notes 194

7 Dynamic 2D Problems 1957.1 Derivation of element equations 1957.2 Case studies 197

7.2.1 Various rainbow options 1977.2.2 Modeling volatility as a risk factor 203

8 Static 3D Problems 2078.1 Derivation of element equations: The collocation method 2078.2 Case studies 209

8.2.1 First exit time of purely Brownian motion 2098.2.2 First exit time of geometric Brownian motion 211

8.3 Notes 213

9 Dynamic 3D Problems 2159.1 Derivation of element equations: The collocation method 2159.2 Case studies 216

9.2.1 Pricing and hedging a basket option 2169.2.2 Basket options with barriers 218

10 Nonlinear Problems 22110.1 Introduction 22110.2 Case studies 223

10.2.1 Penalty methods 22310.2.2 American options 22310.2.3 Passport options 22710.2.4 Uncertain volatility: Best and worst cases 24010.2.5 Worst-case pricing of rainbow options 248

10.3 Notes 252

PART III OUTLOOK 253

11 Future Directions of Research 255

PART IV APPENDICES 257

A Some Useful Results from Analysis 259A.1 Important theorems from calculus 259

A.1.1 Various concepts of continuity 259A.1.2 Taylors theorem 260A.1.3 Mean value theorems 262A.1.4 Various theorems 263


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A.2 Basic numerical tools 264A.2.1 Quadrature 264A.2.2 Solving nonlinear equations 268

A.3 Differential equations 270A.3.1 Definition and classification 270A.3.2 Ordinary initial value problems 272A.3.3 Ordinary boundary value problems 279A.3.4 Partial differential equations of second order 285A.3.5 Parabolic problems 287A.3.6 Elliptic PDEs 295A.3.7 Hyperbolic PDEs 296A.3.8 Hyperbolic conservation laws 297

A.4 Calculus of variations 299

B Some Useful Results from Stochastics 305B.1 Some important distributions 305

B.1.1 The univariate normal distribution 305B.1.2 The bivariate normal distribution 305B.1.3 The multivariate normal distribution 307B.1.4 The lognormal distribution 307B.1.5 The distribution 307B.1.6 The central 2 distribution 309B.1.7 The noncentral 2 distribution 310

B.2 Some important processes 310B.2.1 Basic concepts 310B.2.2 Wiener process 312B.2.3 Brownian motion with drift 313B.2.4 Geometric Brownian motion 313B.2.5 Ito process 314B.2.6 OrnsteinUhlenbeck process 314B.2.7 A process for commodities 314

B.3 Results 314B.3.1 The transition probability density function 314B.3.2 The backward Kolmogorov equation 315B.3.3 The forward Kolmogorov equation 316B.3.4 Steady-state distributions 321B.3.5 First exit times 322B.3.6 Itos lemma 325

B.4 Notes 326

C Some Useful Results from Linear Algebra 329C.1 Some basic facts 329C.2 Errors and norms 331C.3 Ill-conditioning 333


Contents xiii

C.4 Solving linear algebraic systems 333C.5 Notes 339

D A Quick Introduction to PDE2D 341

References 343

Index 351

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Financial Engineering with Finite Elements - Buch.de · Financial Engineering with Finite Elements J¨urgen Topper v. ... 1 Introduction 3 2 Some Prototype Models 7. 2.1 Optimal price