## Categorified Option Pricing Theory

One of the bedrocks of mathematical finance is the Black-Scholes equation. This equation helps to evaluate the fair price of stock options and involves stochastic calculus.

The Black-Scholes equation can be mapped to the Schrodinger equation. I have a writeup somewhere (or maybe on some forum somewhere) showing the details, but it is fairly straightforward to work it out. The analogy I want to point out is that the Black-Scholes equation can be thought of as modeling the dynamics of “point prices” just as the Schrodinger equation models the dynamics of “point particles”.

There are two primary financial instruments that populate any traditional portfolio: stocks and bonds. Stocks are described by a “point price” and hence stock options are governed by the Black-Scholes equation. Bonds are more complicated because there is no 0-dimensional “point price” for bonds. Bonds depends on a 1-dimensional “price curve”. There are models to describe the dynamics of 1-d “price curves”, but nothing has had quite the impact that the original Black-Scholes model did.

It might sound silly, but just as string dynamics seems to relate to a categorification of point particle dynamics as described here, I suspect one could develop a bond option pricing theory (for “price curves”) based on a categorification of the Black-Scholes equation (for “point prices”).

The above content was first posted as a comment over on the n-Category Cafe, but wanted to reproduce it here as well.

[...] Phorgy Phynance « Categorified Option Pricing Theory [...]

Black-Scholes and Schrodinger « Phorgy PhynanceJune 3, 2008 at 9:48 pm

[...] the quest to categorify Black-Scholes is effectively a quest to categorify the heat [...]

Gauge Transforming Black-Scholes « Phorgy PhynanceJune 4, 2008 at 9:21 pm

I can’t believe it. How can stocks be *less* complicated than bonds?

quantum probabilityNovember 18, 2010 at 9:19 pm

Also I don’t get the categorification. What are the morphisms supposed to be?

quantum probabilityNovember 18, 2010 at 9:21 pm

On the other hand — it does sound a lot more tantalizing to have price curves instead of point prices. Fuller, richer, perhaps even more robust.

quantum probabilityDecember 13, 2010 at 11:17 am