Quantum finance: path integrals and Hamiltonians for options

Contents......Page 9
Foreword......Page 13
Preface......Page 15
Acknowledgments......Page 17
1 Synopsis......Page 19
Systems with finite number of degrees of freedom......Page 20
Quantum field theory of interest rates models......Page 21
2 Introduction to finance......Page 25
2.1 Efficient market: random evolution of securities......Page 27
2.2 Financial markets......Page 29
2.3 Risk and return......Page 31
2.4 Time value of money......Page 33
2.5 No arbitrage, martingales and risk-neutral measure......Page 34
2.6 Hedging......Page 36
2.7 Forward interest rates: fixed-income securities......Page 38
2.8 Summary......Page 41
3.1 Forward and futures contracts......Page 43
3.2 Options......Page 45
3.2.1 Path-independent options......Page 46
3.2.2 Path-dependent options......Page 47
3.3 Stochastic differential equation......Page 48
3.4 Ito calculus......Page 49
Geometric Mean of Stock Price......Page 51
3.5 Black–Scholes equation: hedged portfolio......Page 52
3.5.1 Assumptions in the derivation of Black–Scholes......Page 53
3.5.2 Risk-neutral solution of the Black–Scholes equation......Page 54
Black–Scholes price for the European call option......Page 55
3.6 Stock price with stochastic volatility......Page 56
3.7 Merton–Garman equation......Page 57
3.9 Appendix: Solution for stochastic volatility with ρ = 0......Page 59
4.1 Essentials of quantum mechanics......Page 63
4.2 State space: completeness equation......Page 65
4.3 Operators: Hamiltonian......Page 67
Hermitian adjoint of…......Page 68
4.4 Black–Scholes and Merton–Garman Hamiltonians......Page 70
4.5 Pricing kernel for options......Page 72
4.6 Eigenfunction solution of the pricing kernel......Page 73
Hamiltonian derivation of the Black–Scholes pricing kernel......Page 75
4.7 Hamiltonian formulation of the martingale condition......Page 77
4.8 Potentials in option pricing......Page 78
4.9 Hamiltonian and barrier options......Page 80
4.9.1 Down-and-out barrier option......Page 81
4.9.2 Double-knock-out barrier option......Page 82
4.11 Appendix: Two-state quantum system (qubit)......Page 84
4.12 Appendix: Hamiltonian in quantum mechanics......Page 86
4.13 Appendix: Down-and-out barrier option’s pricing kernel......Page 87
4.14 Appendix: Double-knock-out barrier option’s pricing kernel......Page 91
4.15 Appendix: Schrodinger and Black–Scholes equations......Page 94
5.1 Lagrangian and action for the pricing kernel......Page 96
5.2 Black–Scholes Lagrangian......Page 98
5.2.1 Black–Scholes path integral......Page 99
Black–Scholes velocity correlation functions......Page 101
5.3 Path integrals for path-dependent options......Page 103
5.5 Path integral for the simple harmonic oscillator......Page 104
The simple harmonic path integral: Fourier expansion......Page 106
5.6 Lagrangian for stock price with stochastic volatility......Page 108
Derivation of the Merton–Garman Lagrangian......Page 109
5.7 Pricing kernel for stock price with stochastic volatility......Page 111
5.9 Appendix: Path-integral quantum mechanics......Page 114
5.10 Appendix: Heisenberg’s uncertainty principle in finance......Page 117
5.11 Appendix: Path integration over stock price......Page 119
5.12 Appendix: Generating function for stochastic volatility......Page 121
5.13 Appendix: Moments of stock price and stochastic volatility......Page 123
5.14 Appendix: Lagrangian for arbitrary alpha......Page 125
5.15 Appendix: Path integration over stock price for arbitrary alpha......Page 126
5.16 Appendix: Monte Carlo algorithm for stochastic volatility......Page 129
5.17 Appendix: Merton’s theorem for stochastic volatility......Page 133
6.1 Spot interest rate Hamiltonian and Lagrangian......Page 135
6.1.1 Stochastic quantization......Page 137
6.2 Vasicek model’s path integral......Page 138
Path-integral solution for the Treasury Bond in Vasicek’s model......Page 140
6.3 Heath–Jarrow–Morton (HJM) model’s path integral......Page 141
White noise for the HJM model......Page 143
6.4 Martingale condition in the HJM model......Page 144
Domains of integration…......Page 145
6.5 Pricing of Treasury Bond futures in the HJM model......Page 148
6.6 Pricing of Treasury Bond option in the HJM model......Page 149
6.7 Summary......Page 151
6.8 Appendix: Spot interest rate Fokker–Planck Hamiltonian......Page 152
6.9 Appendix: Affine spot interest rate models......Page 156
6.10 Appendix: Black–Karasinski spot rate model......Page 157
6.11 Appendix: Black–Karasinski spot rate Hamiltonian......Page 158
6.12 Appendix: Quantum mechanical spot rate models......Page 161
7 Quantum field theory of forward interest rates......Page 165
7.1 Quantum field theory......Page 166
7.2 Forward interest rates’ action......Page 169
7.3 Field theory action for linear forward rates......Page 171
7.4 Forward interest rates’ velocity quantum field A(t, x)......Page 174
7.5 Propagator for linear forward rates......Page 175
7.6 Martingale condition and risk-neutral measure......Page 179
7.7 Change of numeraire......Page 180
7.8 Nonlinear forward interest rates......Page 182
7.9 Lagrangian for nonlinear forward rates......Page 183
7.9.1 Fermion path integral......Page 185
7.10 Stochastic volatility: function of the forward rates......Page 186
7.11 Stochastic volatility: an independent quantum field......Page 187
7.12 Summary......Page 190
7.13 Appendix: HJM limit of the field theory......Page 191
7.14.1 Constrained spot rate......Page 192
7.14.2 Non-constant rigidity......Page 193
7.15 Appendix: Stiff propagator......Page 194
7.16 Appendix: Psychological future time......Page 198
7.17 Appendix: Generating functional for forward rates......Page 200
7.18 Appendix: Lattice field theory of forward rates......Page 201
8 Empirical forward interest rates and field theory models......Page 209
8.1 Eurodollar market......Page 210
8.2 Market data and assumptions used for the study......Page 212
8.3 Correlation functions of the forward rates’ models......Page 214
8.4 Empirical correlation structure of the forward rates......Page 215
8.4.1 Gaussian correlation functions......Page 218
8.5 Empirical properties of the forward rates......Page 219
8.5.1 The Volatility of volatility of the forward rates......Page 222
8.6 Constant rigidity field theory model and its variants......Page 223
8.7 Stiff field theory model......Page 227
8.7.1 Psychological future time......Page 230
8.8 Summary......Page 232
8.9 Appendix: Curvature for stiff correlator......Page 233
9.1 Futures for Treasury Bonds......Page 235
9.2 Option pricing for Treasury Bonds......Page 236
Field theory derivation of European bond option price......Page 237
9.3 ‘Greeks’ for the European bond option......Page 238
9.4 Pricing an interest rate cap......Page 240
9.4.1 Black’s formula for interest rate caps......Page 242
9.5 Field theory hedging of Treasury Bonds......Page 243
9.6 Stochastic delta hedging of Treasury Bonds......Page 244
9.7 Stochastic hedging of Treasury Bonds: minimizing variance......Page 246
9.8 Empirical analysis of instantaneous hedging......Page 249
9.9 Finite time hedging......Page 253
9.10 Empirical results for finite time hedging......Page 255
9.12 Appendix: Conditional probabilities......Page 258
9.13 Appendix: Conditional probability of Treasury Bonds......Page 260
9.14 Appendix: HJM limit of hedging functions......Page 262
9.15 Appendix: Stochastic hedging with Treasury Bonds......Page 263
9.16 Appendix: Stochastic hedging with futures contracts......Page 266
9.17 Appendix: HJM limit of the hedge parameters......Page 267
10 Field theory Hamiltonian of forward interest rates......Page 269
10.1 Forward interest rates’ Hamiltonian......Page 270
10.2 State space for the forward interest rates......Page 271
10.4 Hamiltonian for linear and nonlinear forward rates......Page 278
10.5 Hamiltonian for forward rates with stochastic volatility......Page 281
10.6 Hamiltonian formulation of the martingale condition......Page 283
10.7 Martingale condition: linear and nonlinear forward rates......Page 286
10.8 Martingale condition: forward rates with stochastic volatility......Page 289
10.9 Nonlinear change of numeraire......Page 290
10.10 Summary......Page 292
10.11 Appendix: Propagator for stochastic volatility......Page 293
10.12 Appendix: Effective linear Hamiltonian......Page 294
10.13 Appendix: Hamiltonian derivation of European bond option......Page 295
10.13.1 Forward interest rates’ pricing kernel......Page 299
11 Conclusions......Page 300
A.1 Probability distribution......Page 302
A.1.1 Martingale......Page 303
A.2 Dirac Delta function......Page 304
A.3.1 One-dimensional Gaussian integral......Page 306
A.3.2 Higher-dimensional Gaussian integral......Page 307
A.3.3 Infinite-dimensional Gaussian integration......Page 308
A.3.4 Normal random variable......Page 309
A.4 White noise......Page 310
A.5 The Langevin equation......Page 311
A.5.1 Martingale condition......Page 313
A.6 Fundamental theorem of finance......Page 314
A.7.1 Eigenfunction expansion......Page 316
A.7.2 Greens function......Page 317
Brief Glossary of Financial Terms......Page 319
Brief Glossary of Physics Terms......Page 321
Main symbols......Page 323
References......Page 328
Index......Page 333