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.