量子金融

Foreword

Preface

Acknowledgments

1 Synopsis

Part I Fundamental concepts of finance

2 Introduction to finance

2.1 Efficient market： random evolution of securities

2.2 Financial markets

2.3 Risk and return

2.4 Time value of money

2.5 No arbitrage， martingales and risk-neutral measure

2.6 Hedging

2.7 Forward interest rates： fixed-income securities

2.8 Summary

3 Derivative securities

3.1 Forward and futures contracts

3.2 Options

3.3 Stochastic differential equation

3.4 Ito calculus

3.5 Black-Scholes equation： hedged portfolio

3.6 Stock price with stochastic volatility

3.7 Merton——Garman equation

3.8 Summary

3.9 Appendix： Solution for stochastic volatility with p = 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 Biack-Scholes and Merton-Garman Hamiltonians

4.5 Pricing kernel for options

4.6 Eigenfunction solution of the pricing kernel

4.7 Hamiltonian formulation of the martingale condition

4.8 Potentials in option pricing

4.9 Hamiltonian and barrier options

4.10 Summary

4.11 Appendix： Two-state quantum system （qubit）

4.12 Appendix： Hamiltonian in quantum mechanics

4.13 Appendix： Down-and-out barrier options pricing kernel

4.14 Appendix： Double-knock-out barrier options 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

5.3 Path integrals for path-dependent options

5.4 Action for option-pricing Hamiltonian

5.5 Path integral for the simple harmonic oscillator

5.6 Lagrangian for stock price with stochastic volatility

5.7 Pricing kernel for stock price with stochastic volatility

5.8 Summary

5.9 Appendix： Path-integral quantum mechanics

5.10 Appendix： Heisenbergs 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 at

5.15 Appendix： Path integration over stock price for arbitrary at

5.16 Appendix： Monte Carlo algorithm for stochastic volatility

5.17 Appendix： Mertons theorem for stochastic volatility

6 Stochastic interest rates Hamiltonians and path integrals

6.1 Spot interest rate Hamiltonian and Lagrangian

6.2 Vasicek models path integral

6.3 Heath-Jarrow-Morton （HJM） models path integral

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

6.7 Summary

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

7.1 Quantum field theory

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

7.7 Change of numeraire

7.8 Nonlinear forward interest rates

7.9 Lagrangian for nonlinear forward rates

7.10 Stochastic volatility： function of the forward rates

7.11 Stochastic volatility： an independent quantum field

7.12 Summary

7.13 Appendix： HJM limit of the field theory

7.14 Appendix： Variants of the rigid propagator

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

7.19 Appendix： Action S， for change of numeraire

8 Empirical forward interest rates and field theory models

8.1 Eurodollar market

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

8.5 Empirical properties of the forward rates

8.6 Constant rigidity field theory model and its variants

8.7 Stiff field theory model

8.8 Summary

8.9 Appendix： Curvature for stiff correlator

9 Field theory of Treasury Bonds derivatives and hedging

9.1 Futures for Treasury Bonds

9.2 Option pricing for Treasury Bonds

9.3 Greeks for the European bond option

9.4 Pricing an interest rate cap

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

9.9 Finite time hedging

9.10 Empirical results for finite time hedging

9.11 Summary

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

10.8 Martingale condition： forward rates with stochastic volatility

10.9 Nonlinear change of numeraire

10.10 Summary

10.11 Appendix： Propagator for stochastic volatility

10.12 Appendix： Effective linear Hamiltonian

10.13 Appendix： Hamiltonian derivation of European bond option

11 Conclusions

A Mathematical background

A.1 Probability distribution

A.2 Dirac Delta function

A.3 Gaussian integration

A.4 White noise

A.5 The Langevin Equation

A.6 Fundamental theorem of finance

A.7 Evaluation of the propagator

Brief glossary of financial terms

Brief glossary of physics terms

List of main symbols

References

Index

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