In this chapter the concept of the Hamiltonian is introduced in the pricing of options. Hamiltonians occur naturally in finance; to demonstrate this the analysis of the Black–Scholes equation is recast in the formalism of quantum mechanics. It is then shown how the Hamiltonian plays a central role in the general theory of option pricing.

The Hamiltonian formulation provides new tools for obtaining solutions for option pricing; two key concepts related to the Hamiltonian are (a) eigenfunctions and (b) potentials. Knowledge of all the eigenfunctions of a Hamiltonian yields an exact solution for a large class of path-dependent and path-independent options. For example, barrier options can be modelled by placing constraints on the eigenfunctions of the Hamiltonian. The potentials are a means for defining new financial instruments, and for modelling path-dependent options.

Essentials of quantum mechanics

It is shown in this chapter that option pricing in finance has a mathematical description that is identical to a quantum system; hence the key features of quantum theory are briefly reviewed.

Quantum theory is a vast subject that forms the bedrock of contemporary physics, chemistry and biology. Only those aspects of quantum mechanics are reviewed that are relevant for the analysis of option pricing.

In classical mechanics the position of a particle at time t, denoted by xt, is a deterministic function of t, and is given by Newton's law of motion. Classical mechanics is analogous to the case of the evolution of a stock price with zero volatility (σ = 0) that yields a deterministic evolution of the stock price.