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
.
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]](http://l.wordpress.com/latex.php?latex=V+%3D+W+%5Cexp%28%5Cint_%5Cgamma+A%29+%3D+W+%5Cexp%5B%28r%2B%5Cfrac%7Br%5E2%7D%7B2%5Csigma%5E2%7D%29+t+-+%28%5Cfrac%7Br%7D%7B%5Csigma%5E2%7D%29+x%5D&bg=ffffff&fg=000000&s=0)
into the Black-Scholes PDE results in
,
which is simply the heat equation from physics!
