Let $C(t,S)$ be the value function of a call option. I want to price that option using (explicit) finite differences and the Black Scholes PDE. I consider the grid $0=t_0<t_1<...<t_{N-1}<t_N=T$ and $S_0<S_1<...<S_{M-1}<S_M$.
I impose the boundary conditions
- Payoff: $C(t_N,S_j)=(S_j-K)^+$ for all $j=0,...,M$,
- Low stock price: $C(t_i,S_0)=0$ for all $i=0,...,N-1$,
- High stock price: $C(t_i,S_M)=S_M$ for all $i=0,...,N-1$.
But isn't there a jump in the option value in the top right corner? At expiry, we use the payoff $C(t_N,S_M)=S_M-K$ but then we use $C(t_{N-1},S_M)=S_M$ as upper stock price boundary condition for all other time points? But that means that over $\Delta t$, the option price jumps by \$$K$.
The conditions for $S=0$ and $t=T$ match in the $(t_N,S_0)$ point but there seems to be a mismatch for $(t_N,S_M)$?