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For the original PDE, the positivity can be deduced from the maximum principle for a parabolic operator. There is also a discrete version of the maximum principle for the finite difference parabolic operator as for example stated in Hung-Ju Kuo and N. S. Trudinger, On the discrete maximum principle for parabolic difference operators which can be applied to ...


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In terms of the settings, we know the current stock price, we have assumed that the stock price dynamics follow Geometric Brownian motion (GBM), we know the parameters of this process (volatility etc), and we know the characteristics of the options (option type, maturity). In practice, we know the current price of the option as well, but we pretend we don’t, ...


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Hence how would we know the value of V(S,T)=max(S−T,0),for S∈R+? You know the value at time T as a function of S: it is simply the pay out, which is $\max(S-K,0)$, where $K$ is a strike. Moreover, why do we care to solve for V(S,t),for t<T if a European option may only be exercised at the maturity time t=T? No, we're not interested in value at time T. ...


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Not sure if the problem is down to boundary conditions - could be a lot of other things, but the boundary conditions in the simplest form for call options are: For very large S: $V\left(t,S\right) \approx S$ For very small S: $V\left(t,S\right) \approx 0$ For put option, the equivalent boundary conditions are: For very large S: $V\left(t,S\right) \...


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You may want to have a look on Alternating Direction Implicit for solving multi-dimension PDE on finite difference method. The linear system will still be tridiagonal matrix.


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