QUANTUM FINANCE
Path Integrals and Hamiltonians for Options and Interest Rates
BELAL E. BAAQUIE
National University of Singapore新加坡国立大学

Cambrige University Press



1 Synopsis 1
Part I Fundamental concepts of ?nance
2 Introduction to ?nance 7
2.1 Ef?cient 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: ?xed-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 = 0 41
Part II Systems with ?nite 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
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 ?nance 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
6.9 Appendix: Af?ne 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 ?eld theory of interest rates models
7 Quantum ?eld theory of forward interest rates 147
7.1 Quantum ?eld 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 ?eld 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 ?eld 169
7.12 Summary 172
7.13 Appendix: HJM limit of the ?eld 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 ?eld theory of forward rates 183
7.19 Appendix: Action S? for change of numeraire 188
8 Empirical forward interest rates and ?eld 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 ?eld theory model and its variants 205
8.7 Stiff ?eld 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
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 ?nite 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 ?nance 296
A.7 Evaluation of the propagator 298
Brief glossary of ?nancial terms 301
Brief glossary of physics terms 303
List of main symbols 305
References 310
Index 315