# Black-Scholes equation (a type of diffusion equation)

## Homework Statement

The equation for the probability distribution of the price of a call option is

$$\frac{\partial P}{\partial t} = \frac{1}{2}\sigma^2 S^2 \frac{\partial^2 P}{\partial S^2} + rS\frac{\partial P}{\partial S} - rP$$

with the conditions $P(0,t) = 0, P(S,0) = \max(S-K,0)$, and the goal is to find the current price of the option, which is given by

$$\int_K^{\infty}\ (S-K)P(S,T)\ dS$$

## The Attempt at a Solution

The obvious thing to try is separation of variables: $P(S,t) = u(S)v(t)$. The time ODE is not homogeneous, so the eigenvalue problem is the spatial problem:

$\frac{\sigma^2}{2}S^2 u'' + rSu' - (r-\lambda)u = 0, u(0) = 0$

(with the understanding that u is finite as S goes to infinity)

This is an equidimensional equation, so its solutions are of the form $S^n$. Unfortunately, the form of n is not so nice:

$$n = \frac{(1/2)\sigma^2-r \pm \sqrt{r^2+\sigma^2 r + (1/4)\sigma^4 - 2\sigma^2\lambda}}{\sigma^2}$$

Without knowing anything about the constants σ and r, I don't see how to proceed from here. I don't think you can tell a priori what you'll get: two positive exponents, one of each sign (though with the boundary conditions, you would only get the trivial solution in these cases), two negative exponents, complex exponents... it's hard to say.

The other thing that occurs to me to try is a Laplace transform. Doing that doesn't get me any farther, since it results in the same equation, except instead of the equation being homogeneous, it has that weird boundary condition on the right hand side. Since we have a semi-infinite interval with Dirichlet boundary conditions, you would expect to use a Fourier sine transform somewhere in here, but it's not directly applicable.

I've seen in different places that you can make a change of variables to condense this into a typical diffusion equation. However, I have no idea what motivates the particular change of variables they use, so I wouldn't feel right approaching it that way unless I could figure out why people make that transformation (no resource I've found has adequately explained this. For example, they change the independent variables to $S = Ke^x,t = T-\tau/(\sigma^2/2)$. I can sort of understand the S change, since that is how you solve an equidimensional equation. I think the t change is an artifact from this equation normally taking place in finite time)

Any thoughts? Thank you so much! :)