\(\large -\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2}+rS\frac{\partial V}{\partial S}-rV=0\)

With boundary conditions:

\(V\large(0,S\right) = max\large(E-S,0\right)\)

\(V\large(t,S^*\right) = E-S^*\large(t\right)\)

\(V\large(t,\infty\right) = 0\)

Where

\(\left. \frac{\partial V}{\partial S} \right|_{S=S^*} = -1\)

and for \(S \le S^*\), \(\frac{\partial V}{\partial t} = 0\)

Sigma, r and E are given. The goal is to find numerical solution for V(a, b), where a and b is some values.

Seems like it's a free boundary problem and numerical solution is the only way. So I tried to transform this PDE to heat equation in order to get rid of free boundary later (this described, for example, in "The Mathematics of Financial derivatives" by Wilmott p.166).

But, it seems, that resulting heat-like equation \(\frac{\partial v}{\partial t} =

-\frac{\partial^2 v}{\partial x^2}\) is typical ill-posed PDE. That minus is from original equation. I wonder if I skipped some transformation which would allow me to come to normal PDE or original equation is ill-posed from the beginning.

Since i have really minimal knowledge of options and Black-Scholes atm, maybe someone could clarify, is this equation makes any sense from financial point of view? Or someone would give any advice how to succeed in finding correct solution.

Thanks in advance.