Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates

by Belal E. Baaquie

• ISBN13: 9780521840453
• ISBN10: 0521840457
• Format: Hardcover
• Copyright: 2004-11-15
• Publisher: Cambridge University Press

Summary

This book applies the mathematics and concepts of quantum mechanics and quantum field theory to the modelling of interest rates and the theory of options. Particular emphasis is placed on path integrals and Hamiltonians. Financial mathematics is currently almost completely dominated by stochastic calculus. The present 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 working 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.

Table of Contents

Foreword	xi
Preface	xiii
Acknowledgments	xv
Synopsis	1	(6)
Part I Fundamental concepts of finance
Introduction to finance	7	(18)
Efficient market: random evolution of securities	9	(2)
Financial markets	11	(2)
Risk and return	13	(2)
Time value of money	15	(1)
No arbitrage, martingales and risk-neutral measure	16	(2)
Hedging	18	(2)
Forward interest rates: fixed-income securities	20	(3)
Summary	23	(2)
Derivative securities	25	(20)
Forward and futures contracts	25	(2)
Options	27	(3)
Stochastic differential equation	30	(1)
Ito calculus	31	(3)
Black-Scholes equation: hedged portfolio	34	(4)
Stock price with stochastic volatility	38	(1)
Merton-Garman equation	39	(2)
Summary	41	(1)
Appendix: Solution for stochastic volatility with p = 0	41	(4)
Part II Systems with finite number of degrees of freedom
Hamiltonians and stock options	45	(33)
Essentials of quantum mechanics	45	(2)
State space: completeness equation	47	(2)
Operators: Hamiltonian	49	(3)
Black-Scholes and Merton-Garman Hamiltonians	52	(2)
Pricing kernel for options	54	(1)
Eigenfunction solution of the pricing kernel	55	(4)
Hamiltonian formulation of the martingale condition	59	(1)
Potentials in option pricing	60	(2)
Hamiltonian and barrier options	62	(4)
Summary	66	(1)
Appendix: Two-state quantum system (qubit)	66	(2)
Appendix: Hamiltonian in quantum mechanics	68	(1)
Appendix: Down-and-out barrier option's pricing kernel	69	(4)
Appendix: Double-knock-out barrier option's pricing kernel	73	(3)
Appendix: Schrodinger and Black-Scholes equations	76	(2)
Path integrals and stock options	78	(39)
Lagrangian and action for the pricing kernel	78	(2)
Black--Scholes Lagrangian	80	(5)
Path integrals for path-dependent options	85	(1)
Action for option-pricing Hamiltonian	86	(1)
Path integral for the simple harmonic oscillator	86	(4)
Lagrangian for stock price with stochastic volatility	90	(3)
Pricing kernel for stock price with stochastic volatility	93	(3)
Summary	96	(1)
Appendix: Path-integral quantum mechanics	96	(3)
Appendix: Heisenberg's uncertainty principle in finance	99	(2)
Appendix: Path integration over stock price	101	(2)
Appendix: Generating function for stochastic volatility	103	(2)
Appendix: Moments of stock price and stochastic volatility	105	(2)
Appendix: Lagrangian for arbitrary α	107	(1)
Appendix: Path integration over stock price for arbitrary α	108	(3)
Appendix: Monte Carlo algorithm for stochastic volatility	111	(4)
Appendix: Merton's theorem for stochastic volatility	115	(2)
Stochastic interest rates' Hamiltonians and path integrals	117	(30)
Spot interest rate Hamiltonian and Lagrangian	117	(3)
Vasicek model's path integral	120	(3)
Heath--Jarrow--Morton (HJM) model's path integral	123	(3)
Martingale condition in the HJM model	126	(4)
Pricing of Treasury Bond futures in the HJM model	130	(1)
Pricing of Treasury Bond option in the HJM model	131	(2)
Summary	133	(1)
Appendix: Spot interest rate Fokker--Planck Hamiltonian	134	(4)
Appendix: Affine spot interest rate models	138	(1)
Appendix: Black-Karasinski spot rate model	139	(1)
Appendix: Black-Karasinski spot rate Hamiltonian	140	(3)
Appendix: Quantum mechanical spot rate models	143	(4)
Part III Quantum field theory of interest rates models
Quantum field theory of forward interest rates	147	(44)
Quantum field theory	148	(3)
Forward interest rates' action	151	(2)
Field theory action for linear forward rates	153	(3)
Forward interest rates' velocity quantum field A (t, x)	156	(1)
Propagator for linear forward rates	157	(4)
Martingale condition and risk-neutral measure	161	(1)
Change of numeraire	162	(2)
Nonlinear forward interest rates	164	(1)
Lagrangian for nonlinear forward rates	165	(3)
Stochastic volatility: function of the forward rates	168	(1)
Stochastic volatility: an independent quantum field	169	(3)
Summary	172	(1)
Appendix: HJM limit of the field theory	173	(1)
Appendix: Variants of the rigid propagator	174	(2)
Appendix: Stiff propagator	176	(4)
Appendix: Psychological future time	180	(2)
Appendix: Generating functional for forward rates	182	(1)
Appendix: Lattice field theory of forward rates	183	(5)
Appendix: Action S* for change of numeraire	188	(3)
Empirical forward interest rates and field theory models	191	(26)
Eurodollar market	192	(2)
Market data and assumptions used for the study	194	(2)
Correlation functions of the forward rates models	196	(1)
Empirical correlation structure of the forward rates	197	(4)
Empirical properties of the forward rates	201	(4)
Constant rigidity field theory model and its variants	205	(4)
Stiff field theory model	209	(5)
Summary	214	(1)
Appendix: Curvature for stiff correlator	215	(2)
Field theory of Treasury Bonds' derivatives and hedging	217	(34)
Futures for Treasury Bonds	217	(1)
Option pricing for Treasury Bonds	218	(2)
`Greeks' for the European bond option	220	(2)
Pricing an interest rate cap	222	(3)
Field theory hedging of Treasury Bonds	225	(1)
Stochastic delta hedging of Treasury Bonds	226	(2)
Stochastic hedging of Treasury Bonds: minimizing variance	228	(3)
Empirical analysis of instantaneous hedging	231	(4)
Finite time hedging	235	(2)
Empirical results for finite time hedging	237	(3)
Summary	240	(1)
Appendix: Conditional probabilities	240	(2)
Appendix: Conditional probability of Treasury Bonds	242	(2)
Appendix: HJM limit of hedging functions	244	(1)
Appendix: Stochastic hedging with Treasury Bonds	245	(3)
Appendix: Stochastic hedging with futures contracts	248	(1)
Appendix: HJM limit of the hedge parameters	249	(2)
Field theory Hamiltonian of forward interest rates	251	(31)
Forward interest rates' Hamiltonian	252	(1)
State space for the forward interest rates	253	(7)
Treasury Bond state vectors	260	(1)
Hamiltonian for linear and nonlinear forward rates	260	(3)
Hamiltonian for forward rates with stochastic volatility	263	(2)
Hamiltonian formulation of the martingale condition	265	(3)
Martingale condition: linear and nonlinear forward rates	268	(3)
Martingale condition: forward rates with stochastic volatility	271	(1)
Nonlinear change of numeraire	272	(2)
Summary	274	(1)
Appendix: Propagator for stochastic volatility	275	(1)
Appendix: Effective linear Hamiltonian	276	(1)
Appendix: Hamiltonian derivation of European bond option	277	(5)
Conclusions	282	(2)
A Mathematical background	284	(17)
Probability distribution	284	(2)
Dirac Delta function	286	(2)
Gaussian integration	288	(4)
White noise	292	(1)
The Langevin Equation	293	(3)
Fundamental theorem of finance	296	(2)
Evaluation of the propagator	298	(3)
Brief glossary of financial terms	301	(2)
Brief glossary of physics terms	303	(2)
List of main symbols	305	(5)
References	310	(5)
Index	315