Chapter
2.5 No arbitrage, martingales and risk-neutral measure
2.7 Forward interest rates: fixed-income securities
3.1 Forward and futures contracts
3.2.1 Path-independent options
3.2.2 Path-dependent options
3.3 Stochastic differential equation
Geometric Mean of Stock Price
3.5 Black–Scholes equation: hedged portfolio
3.5.1 Assumptions in the derivation of Black–Scholes
3.5.2 Risk-neutral solution of the Black–Scholes equation
Black–Scholes price for the European call option
3.6 Stock price with stochastic volatility
3.7 Merton–Garman equation
3.9 Appendix: Solution for stochastic volatility with ρ = 0
Part II Systems with finite number of degrees of freedom
4 Hamiltonians and stock options
4.1 Essentials of quantum mechanics
4.2 State space: completeness equation
4.3 Operators: Hamiltonian
4.4 Black–Scholes and Merton–Garman Hamiltonians
4.5 Pricing kernel for options
4.6 Eigenfunction solution of the pricing kernel
4.6.1 Black–Scholes pricing kernel
Hamiltonian derivation of the Black–Scholes pricing kernel
4.7 Hamiltonian formulation of the martingale condition
4.8 Potentials in option pricing
4.9 Hamiltonian and barrier options
4.9.1 Down-and-out barrier option
4.9.2 Double-knock-out barrier option
4.11 Appendix: Two-state quantum system (qubit)
4.12 Appendix: Hamiltonian in quantum mechanics
4.13 Appendix: Down-and-out barrier option’s pricing kernel
4.14 Appendix: Double-knock-out barrier option’s pricing kernel
4.15 Appendix: Schrodinger and Black–Scholes equations
5 Path integrals and stock options
5.1 Lagrangian and action for the pricing kernel
5.2 Black–Scholes Lagrangian
5.2.1 Black–Scholes path integral
Black–Scholes velocity correlation functions
5.3 Path integrals for path-dependent options
5.4 Action for option-pricing Hamiltonian
5.5 Path integral for the simple harmonic oscillator
The simple harmonic path integral: Fourier expansion
5.6 Lagrangian for stock price with stochastic volatility
Derivation of the Merton–Garman Lagrangian
5.7 Pricing kernel for stock price with stochastic volatility
5.9 Appendix: Path-integral quantum mechanics
5.10 Appendix: Heisenberg’s uncertainty principle in finance
5.11 Appendix: Path integration over stock price
5.12 Appendix: Generating function for stochastic volatility
5.13 Appendix: Moments of stock price and stochastic volatility
5.14 Appendix: Lagrangian for arbitrary alpha
5.15 Appendix: Path integration over stock price for arbitrary alpha
5.16 Appendix: Monte Carlo algorithm for stochastic volatility
5.17 Appendix: Merton’s theorem for stochastic volatility
6 Stochastic interest rates’ Hamiltoniansand path integrals
6.1 Spot interest rate Hamiltonian and Lagrangian
Black–Scholes Hamiltonian
6.1.1 Stochastic quantization
6.2 Vasicek model’s path integral
Path-integral solution for the Treasury Bond in Vasicek’s model
6.3 Heath–Jarrow–Morton (HJM) model’s path integral
White noise for the HJM model
6.4 Martingale condition in the HJM model
6.5 Pricing of Treasury Bond futures in the HJM model
6.6 Pricing of Treasury Bond option in the HJM model
6.8 Appendix: Spot interest rate Fokker–Planck Hamiltonian
6.9 Appendix: Affine spot interest rate models
6.10 Appendix: Black–Karasinski spot rate model
6.11 Appendix: Black–Karasinski spot rate Hamiltonian
6.12 Appendix: Quantum mechanical spot rate models
Part III Quantum field theory of interest rates models
7 Quantum field theory of forward interest rates
7.2 Forward interest rates’ action
7.3 Field theory action for linear forward 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
7.8 Nonlinear forward interest rates
7.9 Lagrangian for nonlinear forward rates
7.9.1 Fermion path integral
7.10 Stochastic volatility: function of the forward rates
7.11 Stochastic volatility: an independent quantum field
7.13 Appendix: HJM limit of the field theory
7.14 Appendix: Variants of the rigid propagator
7.14.1 Constrained spot rate
7.14.2 Non-constant rigidity
7.15 Appendix: Stiff propagator
7.16 Appendix: Psychological future time
7.17 Appendix: Generating functional for forward rates
7.18 Appendix: Lattice field theory of forward rates
8 Empirical forward interest rates and field theory models
8.1.1 Libor forward rates curve
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
8.4.1 Gaussian correlation functions
HJM correlation functions
8.5 Empirical properties of the forward rates
8.5.1 The Volatility of volatility of the forward rates
8.6 Constant rigidity field theory model and its variants
8.7 Stiff field theory model
8.7.1 Psychological future time
8.9 Appendix: Curvature for stiff correlator
9 Field theory of Treasury Bonds’ derivativesand hedging
9.1 Futures for Treasury Bonds
9.2 Option pricing for Treasury Bonds
Field theory derivation of European bond option price
9.3 ‘Greeks’ for the European bond option
9.4 Pricing an interest rate cap
9.4.1 Black’s formula for interest rate caps
9.5 Field theory hedging of Treasury Bonds
9.6 Stochastic delta hedging of Treasury Bonds
9.7 Stochastic hedging of Treasury Bonds: minimizing variance
9.8 Empirical analysis of instantaneous hedging
9.10 Empirical results for finite time hedging
9.12 Appendix: Conditional probabilities
9.13 Appendix: Conditional probability of Treasury Bonds
9.14 Appendix: HJM limit of hedging functions
9.15 Appendix: Stochastic hedging with Treasury Bonds
9.16 Appendix: Stochastic hedging with futures contracts
9.17 Appendix: HJM limit of the hedge parameters
10 Field theory Hamiltonian of forward interest rates
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
10.7.1 General form of the action
10.8 Martingale condition: forward rates with stochastic volatility
10.9 Nonlinear change of numeraire
10.11 Appendix: Propagator for stochastic volatility
10.12 Appendix: Effective linear Hamiltonian
10.13 Appendix: Hamiltonian derivation of European bond option
10.13.1 Forward interest rates’ pricing kernel
Appendix A Mathematical background
A.1 Probability distribution
A.2.1 Functional derivatives
A.3.1 One-dimensional Gaussian integral
A.3.2 Higher-dimensional Gaussian integral
A.3.3 Infinite-dimensional Gaussian integration
A.3.4 Normal random variable
A.4.1 Integral of white noise
A.5 The Langevin equation
A.5.1 Martingale condition
A.6 Fundamental theorem of finance
A.7 Evaluation of the propagator
A.7.1 Eigenfunction expansion
Brief Glossary of Financial Terms
Brief Glossary of Physics Terms
Quantum Finance
：Path Integrals and Hamiltonians for Options and Interest Rates
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