I have implemented a finite difference scheme for pricing options using a Black-Scholes-like model. I tested my implementation on a call option, and found that it gave extremely inaccurate results. I investigated intermediate values in my computations, and I suspect that my inaccurate results are caused by the discontinuity in the payoff function. The payoff function: Payoff(S)=max⁡(S−K,0)for some strike price K.\text{Payoff}(S) = \max(S-K, 0) \hspace{.4em} \text{for some strike price } K \text{.} My scheme requires that I calculate ∂U∂S\frac{\partial{U}}{\partial{S}} and ∂2U∂S2\frac{\partial^2{U}}{\partial{S}^2} at two points near KK . At these points, rs∂U∂Srs\frac{\partial{U}}{\partial{S}} causes a minor issue, but 12s2v∂2U∂S2\frac{1}{2}s^2v\frac{\partial^2{U}}{\partial{S}^2} explodes when I use a first order central approximation. What kinds of solutions exist for this problem? I'm trying to keep my derivative matrix AA (meaning U′=AU+xU' = A U+x ) "oblivious" to the type of derivative (or, in the case of a call option, the strike price) for simplicity of implementation. "Hack"-ish solutions are very welcome. An example: I set the strike to K=110. Three nearby asset prices are 107.336, 109.983, and 112.732. If you compute first order central approximations at the point 109.983, you get \begin{align} \frac{1}{2}s^2v\frac{\partial^2{U}}{\partial{S}^2} &= \frac{1}{2}(109.983^2)v\Big( 0.1400 + (-0.274) * 0 + 0.134858(112.732-110) \Big) \ &=2,228.32*v \end{align} At all other points, for the first time step, ∂2U∂S2=0\frac{\partial^2{U}}{\partial{S}^2}=0 .