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Yes it can be done. However, bear in mind that a naive explicit FD scheme is not unconditionally stable (see CFL stability condition). As far as your initial/boundary conditions issue is concerned: [Time domain] Use terminal condition $V(S,T)=h(T)$ where $h(T)$ figures the payoff of the target derivative claim at maturity $T$ (e.g. $h(T)=(S-K)^+$ for a ...

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