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Jun 13 2008

Black-Scholes and Heat Equation

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 –

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!

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