This page intentionally left blank QUANTUM FINANCE Path Integrals and Hamiltonians for Options and Interest Rates This book applies the mathematics and concepts of quantum mechanics and quan- tum field theory to the modelling of interest rates and the theory of options. Particular emphasis is placed on path integrals and Hamiltonians. Financial mathematics at present is almost completely dominated by stochastic calculus. This book is unique in that it offers a formulation that is completely independent of that approach. As such many new results emerge from the ideas developed by the author. This pioneering work will be of interest to physicists and mathematicians work- ing in the field of finance, to quantitative analysts in banks and finance firms, and to practitioners in the field of fixed income securities and foreign exchange. The book can also be used as a graduate text for courses in financial physics and financial mathematics. B ELAL E. BAAQUIE earned his B.Sc. from Caltech and Ph.D. in theoretical physics from Cornell University. He has published over 50 papers in leading inter- national journals on quantum field theory and related topics, and since 1997 has regularly published papers on applying quantum field theory to both the theoretical and empirical aspects of finance. He helped to launch the International Journal of Theoretical and Applied Finance in 1998 and continues to be one of the editors. QUANTUM FINANCE Path Integrals and Hamiltonians for Options and Interest Rates BELAL E. BAAQUIE National University of Singapore CAMBRIDGE UNIVERSITY PRESS Cambridge, New York, Melbourne, Madrid, Cape Town, Singapore, São Paulo Cambridge University Press The Edinburgh Building, Cambridge CB2 8RU, UK First published in print format ISBN-13 978-0-521-84045-3 ISBN-13 978-0-511-26469-6 © B. E. Baaquie 2004 2004 Informationonthistitle:www.cambrid g e.or g /9780521840453 This publication is in copyright. Subject to statutory exception and to the provision of relevant collective licensing agreements, no reproduction of any part may take place without the written permission of Cambridge University Press. ISBN-10 0-511-26469-0 ISBN-10 0-521-84045-7 Cambridge University Press has no responsibility for the persistence or accuracy of urls for external or third-party internet websites referred to in this publication, and does not guarantee that any content on such websites is, or will remain, accurate or appropriate. Published in the United States of America by Cambridge University Press, New York www.cambridge.org hardback eBook (EBL) eBook (EBL) hardback I dedicate this book to my father Mohammad Abdul Baaquie and to the memory of my mother Begum Ajmeri Roanaq Ara Baaquie, for their precious lifelong support and encouragement. Contents Foreword page xi Preface xiii Acknowledgments xv 1 Synopsis 1 Part I Fundamental concepts of finance 2 Introduction to finance 7 2.1 Efficient market: random evolution of securities 9 2.2 Financial markets 11 2.3 Risk and return 13 2.4 Time value of money 15 2.5 No arbitrage, martingales and risk-neutral measure 16 2.6 Hedging 18 2.7 Forward interest rates: fixed-income securities 20 2.8 Summary 23 3 Derivative securities 25 3.1 Forward and futures contracts 25 3.2 Options 27 3.3 Stochastic differential equation 30 3.4 Ito calculus 31 3.5 Black–Scholes equation: hedged portfolio 34 3.6 Stock price with stochastic volatility 38 3.7 Merton–Garman equation 39 3.8 Summary 41 3.9 Appendix: Solution for stochastic volatility with ρ = 041 Part II Systems with finite number of degrees of freedom 4 Hamiltonians and stock options 45 4.1 Essentials of quantum mechanics 45 4.2 State space: completeness equation 47 vii viii Contents 4.3 Operators: Hamiltonian 49 4.4 Black–Scholes and Merton–Garman Hamiltonians 52 4.5 Pricing kernel for options 54 4.6 Eigenfunction solution of the pricing kernel 55 4.7 Hamiltonian formulation of the martingale condition 59 4.8 Potentials in option pricing 60 4.9 Hamiltonian and barrier options 62 4.10 Summary 66 4.11 Appendix: Two-state quantum system (qubit) 66 4.12 Appendix: Hamiltonian in quantum mechanics 68 4.13 Appendix: Down-and-out barrier option’s pricing kernel 69 4.14 Appendix: Double-knock-out barrier option’s pricing kernel 73 4.15 Appendix: Schrodinger and Black–Scholes equations 76 5 Path integrals and stock options 78 5.1 Lagrangian and action for the pricing kernel 78 5.2 Black–Scholes Lagrangian 80 5.3 Path integrals for path-dependent options 85 5.4 Action for option-pricing Hamiltonian 86 5.5 Path integral for the simple harmonic oscillator 86 5.6 Lagrangian for stock price with stochastic volatility 90 5.7 Pricing kernel for stock price with stochastic volatility 93 5.8 Summary 96 5.9 Appendix: Path-integral quantum mechanics 96 5.10 Appendix: Heisenberg’s uncertainty principle in finance 99 5.11 Appendix: Path integration over stock price 101 5.12 Appendix: Generating function for stochastic volatility 103 5.13 Appendix: Moments of stock price and stochastic volatility 105 5.14 Appendix: Lagrangian for arbitrary α 107 5.15 Appendix: Path integration over stock price for arbitrary α 108 5.16 Appendix: Monte Carlo algorithm for stochastic volatility 111 5.17 Appendix: Merton’s theorem for stochastic volatility 115 6 Stochastic interest rates’ Hamiltonians and path integrals 117 6.1 Spot interest rate Hamiltonian and Lagrangian 117 6.2 Vasicek model’s path integral 120 6.3 Heath–Jarrow–Morton (HJM) model’s path integral 123 6.4 Martingale condition in the HJM model 126 6.5 Pricing of Treasury Bond futures in the HJM model 130 6.6 Pricing of Treasury Bond option in the HJM model 131 6.7 Summary 133 6.8 Appendix: Spot interest rate Fokker–Planck Hamiltonian 134 [...]... the path integral In Chapter 6 on ‘Stochastic interest rates Hamiltonians and path integrals , some of the important existing stochastic models for the spot and forward interest rates are reviewed The Fokker–Planck Hamiltonian and path integral are obtained for the spot interest rate, and a path- integral solution of the Vasicek model is presented The Heath–Jarrow–Morton (HJM) model for the forward interest. .. 10.1 Forward interest rates Hamiltonian 10.2 State space for the forward interest rates 10.3 Treasury Bond state vectors 10.4 Hamiltonian for linear and nonlinear forward rates 10.5 Hamiltonian for forward rates with stochastic volatility 10.6 Hamiltonian formulation of the martingale condition 10.7 Martingale condition: linear and nonlinear forward rates 10.8 Martingale condition: forward rates with... interest rate models Appendix: Black–Karasinski spot rate model Appendix: Black–Karasinski spot rate Hamiltonian Appendix: Quantum mechanical spot rate models ix 138 139 140 143 Quantum field theory of interest rates models 7 Quantum field theory of forward interest rates 7.1 Quantum field theory 7.2 Forward interest rates action 7.3 Field theory action for linear forward rates 7.4 Forward interest rates ... functional for forward rates 7.18 Appendix: Lattice field theory of forward rates 7.19 Appendix: Action S∗ for change of numeraire 8 Empirical forward interest rates and field theory models 8.1 Eurodollar market 8.2 Market data and assumptions used for the study 8.3 Correlation functions of the forward rates models 8.4 Empirical correlation structure of the forward rates 8.5 Empirical properties of the forward... emerge for such systems that are beyond the formalism of stochastic calculus, the most important being the concept of renormalization for nonlinear field theories All the chapters in this part treat the forward interest rates as a quantum field 4 Quantum Finance In Chapter 7 on Quantum field theory of forward interest rates , the formalism of path integration is applied to a randomly evolving curve: the forward... these two chapters is standard, and defines the framework and context for the next two chapters Systems with finite number of degrees of freedom In this part Hamiltonians and path integrals are applied to the study of stock options and stochastic interest rates models These models are characterized by having 1 The path- integral formulation of problems in finance opens the way for applying powerful computational... solution of the martingale condition for the nonlinear forward rates, as well as for forward rates with stochastic volatility A Hamiltonian derivation is given of the change of numeraire for nonlinear theories, of bond option price, and of the pricing kernel for the forward interest rates All chapters focus on the conceptual and theoretical aspects of the quantum formalism as applied to finance, with... derivation is given for the change of numeraire Nonlinear field theories are shown to arise naturally in modelling positive-valued forward interest rates as well as forward rates with stochastic volatility In Chapter 8 on ‘Empirical forward interest rates and field theory models’, the empirical aspects of the forward rates are discussed in some detail, and it is shown how to calibrate and test field theory... rates 7.4 Forward interest rates velocity quantum field A(t, x) 7.5 Propagator for linear forward rates 7.6 Martingale condition and risk-neutral measure 7.7 Change of numeraire 7.8 Nonlinear forward interest rates 7.9 Lagrangian for nonlinear forward rates 7.10 Stochastic volatility: function of the forward rates 7.11 Stochastic volatility: an independent quantum field 7.12 Summary 7.13 Appendix: HJM... forward interest rates is recast as a problem of path integration, and well-known results of the HJM model are re-derived using the path integral Chapter 6 is a preparation for the main thrust of this book, namely the application of quantum field theory to the modelling of the interest rates Quantum field theory of interest rates models Quantum field theory is a mathematical structure for studying systems . blank QUANTUM FINANCE Path Integrals and Hamiltonians for Options and Interest Rates This book applies the mathematics and concepts of quantum mechanics and. of Theoretical and Applied Finance in 1998 and continues to be one of the editors. QUANTUM FINANCE Path Integrals and Hamiltonians for Options and Interest Rates BELAL