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Archive for June 2008

Gauge Transforming Black-Scholes

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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.

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Written by Eric

June 4, 2008 at 9:19 pm

Black-Scholes and Schrodinger

with 28 comments

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.

Noncommutative Geometry and Stochastic Calculus
Applications in Mathematical Physics

I derived it in a slightly unusual way via noncommutative geometry. Regardless of how it is derived, the Black-Scholes PDE is given by

\partial_t V + \frac{\sigma^2 S^2}{2} \partial_S^2 V + r S \partial_S V - r V = 0

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

i\hbar\partial_t \psi = H \psi = \frac{p^2}{2m} \psi + U \psi = -\frac{\hbar^2}{2m} \partial_x\psi + U\psi

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.

\partial_t V = -\frac{\sigma^2 S^2}{2} \partial_S^2 V - r S \partial_S V + r V

Comparing this to the Schrodinger equation

i\hbar\partial_t\psi = -\frac{\hbar^2}{2m} \partial_x^2\psi + U\psi

tempted me to complete the square in the BS PDE resulting in

\partial_t V = -\frac{\sigma^2}{2} \left[S\partial_S -\frac{1}{2}\left(1-\frac{2r}{\sigma^2}\right)\right]^2 V + \frac{\sigma^2}{8} \left(1 + \frac{2r}{\sigma^2}\right)^2 V.

Now letting

p = -i\sigma\left[\partial_x -\frac{1}{2}\left(1-\frac{2r}{\sigma^2}\right)\right],

where x = \log{S} and

U = \frac{\sigma^2}{8} \left(1 + \frac{2r}{\sigma^2}\right)^2

we find that the Black-Scholes PDE can be written as

\partial_t V = \frac{p^2}{2} V + U V.

Observation 1

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.

Observation 2

The volatility \sigma plays the role of Planck’s constant \hbar while p satisfies the commutative relation

{}[x,p] = i\sigma.

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.

Written by Eric

June 3, 2008 at 9:48 pm

Categorified Option Pricing Theory

with 5 comments

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.

Written by Eric

June 2, 2008 at 9:38 pm