The Merton-Garman Hamiltonian

This week we will extend our quantum finance framework with stochastic volatility. The price of an option on an equity with stochastic volatility is given by the Merton-Garman Hamiltonian

Again we can define variables x and y as S and V are positive-valued random variables.

Using these definitions, we obtain

and again this equation can be written in the form of a Schrödinger equation

with the Merton-Garman Hamiltonian

.

We have now a system with two degrees of freedom. Except for α=1/2 and α=1 the system can only be solved numerically.