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Schrodingers Option

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by Felix Salmon
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Jun 13 2008 12:00am EDT

Schrodinger's Option

Have you ever wondered why hedge funds love to hire mathematics and physics PhDs? Maybe this post, and this follow-up, might help explain. It turns out that the Black-Scholes PDE (that's partial differential equation, of course) is basically - um, actually, I have no idea what it is in English. But here it is in fluent mathematical Physics:

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.

What's more - and you might want to be sitting down for this -

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!

So next time someone tells you that activity in the options market is heating up, you can just tell them... something very clever. If you understand the first word of this. Which I don't.

Incidentally, once you go down this road, it won't be long before you'll want to buy Quantum Finance: Path Integrals and Hamiltonians for Options and Interest Rates, by Belal Baaquie. Amazon has it at 24% off, and has an enticing special offer:

Buy this book with Physics of Finance: Gauge Modelling in Non-Equilibrium Pricing by Kirill Ilinski today!

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