Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates,9780521840453

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

by
ISBN13: 9780521840453
ISBN10: 0521840457
Format: Hardcover
Pub. Date: 2004-11-15
Publisher(s): Cambridge University Press
List Price: $111.27

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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

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