I have the PDE $$ -\frac{\partial V}{\partial t} + \frac{1}{2}\sigma^2S^2\frac{\partial^2 V}{\partial S^2} + rS\frac{\partial V}{\partial S} - rV = 0$$ with initial and boundary conditions:
- $V(0,S)=max(E-S,0) $
- $V(t,S^*)=E-S^*(t) $
- $V(t,\infty)=0 $
$S^*$ finding from this condition - $\frac{\partial V(t,S)}{\partial S}\bigg|_{S=S^*}=-1$, and for $S\le S^*$ must be performed $\frac{\partial V(t,S)}{\partial t}=0$. Where $\sigma=0.2$, $r=0.08$, $E=100$.
I have to find $V(t,S)$ with $t=0.25$ and $S=100$.
So, I think that this PDE I can solve using finite difference method. Now I did approximation for PDE using forward and backward explicity methods. But I don't know how I can solve boundary condition. What I need to learn for solving this PDE? Or you can solve it, and tell me how you did it. :) Thanx!