A Black-Scholes-Schrödinger Equation

In Quantum Physics and Options I promised to discuss the essentials of quantum mechanics that are relevant for option pricing. In classical mechanics Newton's law of motion determines the position of a particle at a given time by a deterministic function. So classical mechanics can be seen as the (deterministic) evolution of a stock price with zero volatility. In contrast, in quantum mechanics the particle's evolution is random, like it is the case for a stock price with non zero volatility. In the one dimensional case the random numbers indicating the position of the particle can be all points on the real line. At some fixed time the probability of finding the quantum particle at position x  is given by the product of the wave function times its conjugate complex.

The wave function more generally spoken the state vector of the quantum system is an element of a linear vector space. In quantum mechanics physically measurable quantities such as energy, position and so on are represented by Hermitian operators that map the linear vector space on itself. The Hamilton operator evolves the system in time. Whereas the Hamilton operator in quantum mechanics is hermitian (Schrödinger equation) in finance it is in general not. Let us take a look at the Black Scholes equation (used for example for option pricing with constant volatility)

Changing of the variable

leads us to a  "Black-Scholes-Schrödinger" equation

with the Black Scholes Hamiltonian given by

Seen as a quantum mechanical system, the Black-Scholes equation has one degree of freedom (x). Next week we will compare properties of the Schrödinger equation with properties of our Black-Scholes Schrödinger equation.