Black-Scholes and Schrodinger
In this post, I will perform some computations to demonstrate a relationship between the Black-Scholes PDE and the Schrodinger equation of quantum mechanics. This relationship is not new. You can, for example, find something similar discussed here. The purpose for writing the gory details here is that I hope we (that’s right.. you too!) can “categorify” the Black-Scholes framework.
The Black-Scholes PDE is derived in a million places. One of those places is a paper I wrote in May 2002 and published on the quantitative finance web site Wilmott.com.
I derived it in a slightly unusual way via noncommutative geometry. Regardless of how it is derived, the Black-Scholes PDE is given by
One day, for some reason that I forget, I was curious whether this could be written in a form that looked like Schrodinger’s equation
so I started chugging away with some algebraic gymnastics. The first thing is to rewrite the equation with the time derivative on one side and everything else on the other.
Comparing this to the Schrodinger equation
tempted me to complete the square in the BS PDE resulting in
we find that the Black-Scholes PDE can be written as
The Black-Scholes PDE is a “Wick rotated” Schrodinger equation for a charged particle in an electromagnetic field, where the risk-free rate plays the role of a gauge connection.
The volatility plays the role of Planck’s constant while satisfies the commutative relation
Now let the games begin!
Note: Thanks to naumovic who pointed out in the comments below an algebraic mistake in an earlier version of this article.