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I haven't inspect everything in detail but I think I have a useful remark. Since you know the analytical solution, you can compute from there the boundary conditions. For $t = T$, you have $\tau = 0$. You get $F(t = T; \tau = 0) = S(T)$ when replacing in the analytical solution. Then, F(end) = S(end), as you stated in your code. On the other hand, when $t = ...

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