QUANTUM FINANCE: Path Integrals and Hamiltonians for Options And
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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. BELAL 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 Published in the United States of America by Cambridge University Press, New York www.cambridge.org Informationonthistitle:www.cambridge.org/9780521840453 © B. E. Baaquie 2004 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. First published in print format 2004 ISBN-13 978-0-511-26469-6 eBook (EBL) ISBN-10 0-511-26469-0 eBook (EBL) ISBN-13 978-0-521-84045-3 hardback ISBN-10 0-521-84045-7 hardback 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. 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 Synopsis1 Part I Fundamental concepts of finance 2 Introduction to finance7 2.1 Efficient market: random evolution of securities9 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 Contents ix 6.9 Appendix: Affine spot interest rate models 138 6.10 Appendix: Black–Karasinski spot rate model 139 6.11 Appendix: Black–Karasinski spot rate Hamiltonian 140 6.12 Appendix: Quantum mechanical spot rate models 143 Part III Quantum field theory of interest rates models 7 Quantum field theory of forward interest rates 147 7.1 Quantum field theory 148 7.2 Forward interest rates’ action 151 7.3 Field theory action for linear forward rates 153 7.4 Forward interest rates’ velocity quantum field A(t, x) 156 7.5 Propagator for linear forward rates 157 7.6 Martingale condition and risk-neutral measure 161 7.7 Change of numeraire 162 7.8 Nonlinear forward interest rates 164 7.9 Lagrangian for nonlinear forward rates 165 7.10 Stochastic volatility: function of the forward rates 168 7.11 Stochastic volatility: an independent quantum field 169 7.12 Summary 172 7.13 Appendix: HJM limit of the field theory 173 7.14 Appendix: Variants of the rigid propagator 174 7.15 Appendix: Stiff propagator 176 7.16 Appendix: Psychological future time 180 7.17 Appendix: Generating functional for forward rates 182 7.18 Appendix: Lattice field theory of forward rates 183 7.19 Appendix: Action S∗ for change of numeraire 188 8 Empirical forward interest rates and field theory models 191 8.1 Eurodollar market 192 8.2 Market data and assumptions used for the study 194 8.3 Correlation functions of the forward rates models 196 8.4 Empirical correlation structure of the forward rates 197 8.5 Empirical properties of the forward rates 201 8.6 Constant rigidity field theory model and its variants 205 8.7 Stiff field theory model 209 8.8 Summary 214 8.9 Appendix: Curvature for stiff correlator 215 9 Field theory of Treasury Bonds’ derivatives and hedging 217 9.1 Futures for Treasury Bonds 217 9.2 Option pricing for Treasury Bonds 218 9.3 ‘Greeks’ for the European bond option 220 9.4 Pricing an interest rate cap 222 x Contents 9.5 Field theory hedging of Treasury Bonds 225 9.6 Stochastic delta hedging of Treasury Bonds 226 9.7 Stochastic hedging of Treasury Bonds: minimizing variance 228 9.8 Empirical analysis of instantaneous hedging 231 9.9 Finite time hedging 235 9.10 Empirical results for finite time hedging 237 9.11 Summary 240 9.12 Appendix: Conditional probabilities 240 9.13 Appendix: Conditional probability of Treasury Bonds 242 9.14 Appendix: HJM limit of hedging functions 244 9.15 Appendix: Stochastic hedging with Treasury Bonds 245 9.16 Appendix: Stochastic hedging with futures contracts 248 9.17 Appendix: HJM limit of the hedge parameters 249 10 Field theory Hamiltonian of forward interest rates 251 10.1 Forward interest rates’ Hamiltonian 252 10.2 State space for the forward interest rates 253 10.3 Treasury Bond state vectors 260 10.4 Hamiltonian for linear and nonlinear forward rates 260 10.5 Hamiltonian for forward rates with stochastic volatility 263 10.6 Hamiltonian formulation of the martingale condition 265 10.7 Martingale condition: linear and nonlinear forward rates 268 10.8 Martingale condition: forward rates with stochastic volatility 271 10.9 Nonlinear change of numeraire 272 10.10 Summary 274 10.11 Appendix: Propagator for stochastic volatility 275 10.12 Appendix: Effective linear Hamiltonian 276 10.13 Appendix: Hamiltonian derivation of European bond option 277 11 Conclusions 282 A Mathematical background 284 A.1 Probability distribution 284 A.2 Dirac Delta function 286 A.3 Gaussian integration 288 A.4 White noise 292 A.5 The Langevin Equation 293 A.6 Fundamental theorem of finance 296 A.7 Evaluation of the propagator 298 Brief glossary of financial terms 301 Brief glossary of physics terms 303 List of main symbols 305 References 310 Index 315 Foreword After a few early isolated cases in the 1980s, since the mid-1990s hundreds of pa- pers dealing with economics and finance have invaded the physics preprint server xxx.lanl.gov/cond-mat, initially devoted to condensed matter physics, and now covering subjects as different as computer science, biology or probability theory.