# Phorgy Phynance

## Gauge Transforming Black-Scholes

In the last article, I showed how the Black-Scholes PDE is equivalent to a Wick-rotated Schrodinger equation describing a charged particle in an electromagnetic field. Here, I will expand a little bit on that.

In quantum mechanics, the “momentum operator” for a point particle is

$p = -i\hbar\partial_x$.

To account for the interaction of the (charged) particle with an electomagnetic field, the momentum operator is augmented by the vector potential. The effect of this vector potential is to deform the wave function by a “gauge” factor

$\psi = \phi \exp(\int_\gamma A)$

for some curve $\gamma$.

The gauge connection for the Black-Scholes PDE is given by

$A = (r+\frac{r^2}{2\sigma^2}) dt - (\frac{r}{\sigma^2}) dx$.

Inserting the corresponding gauge factor

$V = W \exp(\int_\gamma A) = W \exp[(r+\frac{r^2}{2\sigma^2}) t - (\frac{r}{\sigma^2}) x]$

into the Black-Scholes PDE results in

$\partial_t W = -\frac{\sigma^2}{2} \partial_x^2 W$,

which is simply the heat equation from physics!

Therefore, the quest to categorify Black-Scholes is effectively a quest to categorify the heat equation.

What we want to do I think is to contruct a “yield curve space” in analogy to “loop space” where a point in yield curve space corresponds to a yield curve in some base space. We want to study Brownian motion on yield curve space under constraints of no arbitrage, which can hopefully be formulated as a statement about curvature.

Update:

In a comment, Blake Stacey points out this book. Digging a little bit turns up these arxiv papers and the author’s web page. It now appears likely that the author has done what I set out to do, but perhaps did not quite make the connection to “categorification” which is just a minor point. I’m tempted to order the book, but will definitely have a look at the arxiv papers in the meantime.