Book contents
- Frontmatter
- Contents
- Prologue
- Acknowledgements
- 1 Synopsis
- 2 Interest rates and coupon bonds
- 3 Options and option theory
- 4 Interest rate and coupon bond options
- 5 Quantum field theory of bond forward interest rates
- 6 Libor Market Model of interest rates
- 7 Empirical analysis of forward interest rates
- 8 Libor Market Model of interest rate options
- 9 Numeraires for bond forward interest rates
- 10 Empirical analysis of interest rate caps
- 11 Coupon bond European and Asian options
- 12 Empirical analysis of interest rate swaptions
- 13 Correlation of coupon bond options
- 14 Hedging interest rate options
- 15 Interest rate Hamiltonian and option theory
- 16 American options for coupon bonds and interest rates
- 17 Hamiltonian derivation of coupon bond options
- Epilogue
- A Mathematical background
- B US debt markets
- Glossary of physics terms
- Glossary of finance terms
- List of symbols
- References
- Index
15 - Interest rate Hamiltonian and option theory
Published online by Cambridge University Press: 11 April 2011
- Frontmatter
- Contents
- Prologue
- Acknowledgements
- 1 Synopsis
- 2 Interest rates and coupon bonds
- 3 Options and option theory
- 4 Interest rate and coupon bond options
- 5 Quantum field theory of bond forward interest rates
- 6 Libor Market Model of interest rates
- 7 Empirical analysis of forward interest rates
- 8 Libor Market Model of interest rate options
- 9 Numeraires for bond forward interest rates
- 10 Empirical analysis of interest rate caps
- 11 Coupon bond European and Asian options
- 12 Empirical analysis of interest rate swaptions
- 13 Correlation of coupon bond options
- 14 Hedging interest rate options
- 15 Interest rate Hamiltonian and option theory
- 16 American options for coupon bonds and interest rates
- 17 Hamiltonian derivation of coupon bond options
- Epilogue
- A Mathematical background
- B US debt markets
- Glossary of physics terms
- Glossary of finance terms
- List of symbols
- References
- Index
Summary
The Hamiltonian is a differential operator that acts on an underlying state space. A Hamiltonian formulation of option theory is discussed and shown to be equivalent to the Black–Scholes approach. In particular, it is shown that the Black–Scholes equation is mathematically identical to the (imaginary time) Schrodinger equation of quantum mechanics.
The Hamiltonian formulation of quantum field theory is equivalent to, and independent of, the framework based on the Feynman path integral and the Lagrangian discussed in Chapter 5. A Hamiltonian formulation of interest rates provides another perspective on option theory and interest rates. There are many advantages of having multiple formulations, since for some problems calculations based on the Hamiltonian are more transparent and tractable than using the Lagrangian approach. In particular, the Hamiltonian formulation is useful for exactly solving nonlinear martingale conditions as well as for studying a specific class of debt instruments options, which includes American and barrier options.
Introduction
The Hamiltonian is introduced by considering option theory for a single equity. Option theory is shown to have a Hamiltonian formulation in which the option price is a function of the matrix elements of the exponential of the Hamiltonian. The Black–Scholes option price is given a Hamiltonian derivation starting from first principles that are reasonable and intuitive.
The interest rate state space and Hamiltonian are derived from the forward interest rates Lagrangian and are a natural generalization of a similar Black–Scholes analysis for equities.
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- Information
- Interest Rates and Coupon Bonds in Quantum Finance , pp. 329 - 364Publisher: Cambridge University PressPrint publication year: 2009