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Applications of Lie Group Analysis in Financial Mathematics Ljudmila A. Bordag University of Applied Sciences Zittau/Görlitz, Germany World Scientific NEW JERSEY LONDON SINGAPORE BEIJING SHANGHAI HONG KONG TAIPEI CHENNAI
Published by World Scientific Publishing Co. Pte. Ltd. 5 Toh Tuck Link, Singapore 596224 USA office: 27 Warren Street, Suite 401-402, Hackensack, NJ 07601 UK office: 57 Shelton Street, Covent Garden, London WC2H 9HE Library of Congress Cataloging-in-Publication Data Bordag, Ljudmila A. Geometrical properties of differential equations : applications of Lie group analysis in financial mathematics / by Ljudmila A. Bordag (University of Applied Sciences, Zittau/Görlitz, Germany). pages cm Includes bibliographical references and index. ISBN 978-9814667241 (hardcover : alk. paper) 1. Lie groups. 2. Differential equations, Partial. 3. Geometry, Differential. 4. Business mathematics. I. Title. QA387.B625 2015 512'.482--dc23 2015002797 British Library Cataloguing-in-Publication Data A catalogue record for this book is available from the British Library. Copyright 2015 by World Scientific Publishing Co. Pte. Ltd. All rights reserved. This book, or parts thereof, may not be reproduced in any form or by any means, electronic or mechanical, 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, MA 01923, USA. In this case permission to photocopy is not required from the publisher. Printed in Singapore
Preface This textbook is devoted to Lie group analysis of differential equations with applications to financial mathematics. The new models arising in financial mathematics are often presented in the form of nonlinear partial differential equations (PDEs). Lie group analysis was earlier successfully applied to the studies of algebraic and group structures of linear and nonlinear PDEs which describe physical or chemical processes. It is to be expected that this method will also be widely applied to new types of PDEs appearing in financial mathematics. The content of this book was developed for the lecture courses Geometrical properties of differential equations given in the framework of the master program Master in financial mathematics at Halmstad University, Sweden for the academic years 2006/2007 till 2010/2011. The one-year program was oriented towards mathematicians, physicists and computer sciences engineers who wanted to learn actual methods and modern models in financial mathematics. The goal of this and other lecture courses in the master program was to provide a set of practical skills which allowed the students to use the studied methods in industry. So the main focus of the program was on a clear understanding of the main ideas and self-confident use of learned tools instead of deep theoretical proofs and theorems. We had about 20-30 students per year. The program was very international and the students came from Armenia, China, Germany, Lithuania, Pakistan, Poland, Russia, Serbia, Singapore, Slovakia, Sweden, Tunisia and Turkey. The atmosphere in the master program was very friendly and students supported each other. As far as I know many students stayed in touch years after finishing their studies, working in finance institutions in different countries all over the world. Despite different skills and educational backgrounds the students in my lecture course were really enthusiastic to learn new ideas and apply them in practice. At Halmstad University the complete academic year was divided into four v
vi Geometrical Properties of DE periods with eight teaching weeks each and one or two examination weeks in between. During the first period I provided four academic hours of lectures and four academic hours of classroom tutorials every week. Students were also given assignments with problems to solve every week. These assignments and classroom tutorials, as well as exams are included in this book accompanied by detailed solutions. In the last, fourth, teaching period of the study year all students wrote a master project (15 ECTS points) devoted to one of the actual problems in financial mathematics. These projects are published on the homepage of the library of Halmstad University. Part of the results of the master projects were later published as journal papers. Some students used Lie group analysis and studied new financial models with this method. Later in Germany I used these lecture notes during the Compact course on Lie group analysis that took place in Zittau, 2013, in the framework of FP 7 Marie Curie Initial Training Network (ITN) STRIKE (see www.itnstrike.eu). The aim of the project ITN STRIKE is to understand complex (mostly nonlinear) financial models and to develop effective and robust numerical schemes for solving linear and nonlinear problems arising from the mathematical theory of pricing financial derivatives and related financial products. This two-week intensive course was conducted for PhD students of different European countries working in the ITN STRIKE. The lectures were presented by me, while the practical exercises and assignments were led by Ivan P. Yamshchikov. Because the participants were interested in applications of the new knowledge to numerical schemes we emphasized this part. We discussed how the admitted Lie algebraic structure of studied equations can be used to provide improved numerical schemes, how to get and use the invariant solutions and to get in touch with new models. In two weeks the participants learned a tool which they can use practically. In my opinion this book with its large number of problems with detailed solutions can be used as a textbook for a regular lecture course, or for a compact lecture course of a few weeks or as a book for convenient self-study. The lecture notes were prepared with the strong help of my previous students: Tony Huschto, Anna Mikaelyan, Ivan P. Yamshchikov and Dmitry Zhelezov. I am very grateful for their help and efforts to make this text better. Zittau, Germany Ljudmila A. Bordag May 11, 2015
Abstract This textbook is a short comprehensive and intuitive introduction to Lie group analysis of ordinary and partial differential equations. This practical oriented material contains a large number of examples and problems accompanied by detailed solutions and figures. In comparison with known beginner guides to Lie group analysis, this book is oriented towards students who are interested in financial mathematics and mathematical economics. Avoiding reference to physical intuition, ideas, such as the following, are developed step-by-step: point transformation, meaning of a continuous one-parameter group, infinitesimal action, invariants, Lie algebra, symmetry reductions of differential equations. In this book we look at differential equations from a geometrical point of view, and explain the idea of invariant solutions and symmetries. The admitted symmetries will be used to find the reductions of given differential equations and invariant solutions accompanied by a large number of examples. The book contains nine chapters. The first few chapters are devoted to the development of the ideas and tools of Lie group analysis. All notations, ideas and methods are explained first with very simple examples of ordinary differential equations and only then with partial differential equations. This structure allows the students to get familiar with the main tools of Lie group analysis as quickly and as easily as possible. In Chapter 8 we study the famous Black-Scholes model for option pricing. Its algebraic structure is compared with the structure of the heat equation. The explicit analytic solution for a call option is derived simply as an invariant solution of Black-Scholes equation. In Chapter 9 we provide the results of the Lie group analysis of actual models in financial mathematics using recent publications. These models are usually formulated as nonlinear partial differential equations and are rather difficult for investigations. With the help of Lie group analysis it is possible to describe some important properties of these models and get some interesting reductions in a clear and understandable algorithmic way. The material in this book has been used for the academic years 2006/2007 vii
viii Geometrical Properties of DE till 2010/2011 as part of the Master in Financial Mathematics program at Halmstad University, Sweden and in a two-week intensive compact course given 2013 in Zittau in the framework of FP 7 Marie Curie Initial Training Network (ITN) STRIKE (www.itn-strike.eu). The participants of the compact course were PhD students primarily interested in new effective numeric schemes for advanced models in financial mathematics. As prerequisites for this textbook one needs to know some basics of the theory of differential equations which corresponds usually to the bachelor level in mathematics or applied mathematics. This book can be a short introductory for a further study of modern geometrical analysis applied to models in financial mathematics. It can also be used as a textbook in a master program, in an intensive compact course or for self-study.
Contents Preface Abstract 1 Introduction 1 2 Point transformations on R 2 7 2.1 Groups of transformations.................... 8 2.2 The general form of a one-parameter group of point transformations in R 2........................... 15 2.3 Problems with solutions..................... 22 2.4 Exercises.............................. 32 3 Invariants of a one-parameter group of point transformations 33 3.1 Group invariants......................... 33 3.2 An infinitesimal generator U.................. 37 3.3 Invariant curves and points.................... 41 3.4 Invariant families of curves.................... 42 3.5 Canonical variables, the normal form of an infinitesimal generator............................... 45 3.6 A geometrical meaning of canonical coordinates........ 48 3.7 Problems with solutions..................... 50 3.8 Exercises.............................. 56 v vii 4 First order ODEs 59 4.1 Geometrical image of first order ODEs............. 59 4.2 A linear first order differential operator A related to a first order ODE............................. 64 4.3 Solution of a first order ODE using an integrating factor... 66 4.4 Solution of a first order ODE using canonical variables of the admitted symmetry........................ 69 ix
x Geometrical Properties of DE 4.5 Relation between the infinitesimal generator U and the operator A............................... 71 4.6 Problems with solutions..................... 72 4.7 Exercises.............................. 92 5 Prolongation procedure 95 5.1 Differential invariants and prolongation of one-parameter groups of point transformations..................... 95 5.2 Differential invariants of n-th order............... 105 5.3 Problems with solutions..................... 114 5.4 Exercises.............................. 135 6 Short overview of the Lie algebra properties 137 6.1 Linear space of differential operators.............. 138 6.2 Introduction to Lie algebras................... 140 6.3 An optimal system of subalgebras................ 153 6.4 Problems with solutions..................... 155 7 High order ODEs 161 7.1 Operator A for a high order ordinary differential equation.. 162 7.2 Second order ODEs with two symmetries............ 171 7.3 A case of n-th order ODE which admits a high-dimensional solvable Lie algebra........................ 173 7.4 A case of n-th order ODEs which are invariant under a nonsolvable Lie algebra with a special property........... 178 7.5 A case of a n-th order ODE with a general type of the admitted symmetry algebra......................... 181 7.6 Problems with solutions..................... 184 7.7 Exercises.............................. 207 8 Lie group analysis of PDEs 209 8.1 Notations and terminology in the case of many independent and dependent variables..................... 210 8.2 The heat equation........................ 221 8.3 Some important types of group-invariant solutions of the heat equation.............................. 231 8.4 The Black-Scholes equation................... 235 8.5 Problems with solutions..................... 239 8.6 Exercises.............................. 244
Contents xi 9 Study of new models in financial mathematics 249 9.1 Ideal financial market. Black-Scholes model for option pricing 250 9.2 Pricing options in illiquid markets................ 255 9.2.1 Frey, Frey-Patie, Frey-Stremme models of risk management for derivatives.................. 255 9.2.2 Sircar Papanicolaou model................ 277 9.3 Equilibrium or reaction-function models............ 282 9.3.1 Schönbucher-Wilmott model for self-financing trading strategies......................... 284 9.4 Pricing and hedging in incomplete markets........... 295 9.4.1 Musiela-Zariphopoulou model.............. 296 9.4.2 Problems of optimal consumption with random income 299 9.4.3 Transaction costs models................. 305 Bibliography 319 Index 327