Finite Difference Method

The Problem

The Black-Scholes-Merton PDE for a derivative with value $V(S,t)$ is

$\frac{\partial V}{\partial t}+rS\frac{\partial V}{\partial S}+\frac{1}{2}{\sigma }^2{S}^2\frac{{\partial }^{2}V}{\partial S^2}=rV.$

You want to price a European put with strike $K$, maturity $T$, on a non-dividend stock. You decide to discretise on a grid $S_i=i\Delta S$, $t_j=j\Delta t$ and use $V_i^j\approx V(S_i,t_j)$.

1. Explicit scheme. Write down the explicit (forward Euler) finite-difference update for $V_i^{j+1}$ in terms of $V^j$ values. State the CFL stability condition.

2. Implicit scheme. Write down the implicit (backward Euler) update. Why is it unconditionally stable?

3. Crank-Nicolson. Write down the Crank-Nicolson update and state its key advantage over both (1) and (2).

Boundary conditions: $V(0,t)=K{e}^{-r(T-t)}$ (put is in-the-money at zero spot), $V(S_{\max },t)\approx 0$ for $S_{\max }\gg K$, and terminal condition $V(S,T)=\max (K-S,0)$. It's standard to switch to $\tau =T-t$ so we march forward in $\tau$.