Assume I am solving numerically Black Scholes PDE $$u_t+0.5\sigma^2s^2u_{ss}+rsu_s-ru=0$$ and I decided to have boundary condition on the right boundary as $u_{ss}=0$. One way is to write the discrete approximation for $u_{ss}$, for example, in the case of Crank-Nicolson: $$0.5\frac{u^{n+1}_{i-1}-2u^{n+1}_{i}+u^{n+1}_{i+1}}{\delta x^2}+0.5\frac{u^{n}_{i-1}-2u^{n}_{i}+u^{n}_{i+1}}{\delta x^2}=0$$ and add this to the discretization system . Another way to to put this back into equation itself and get $$u_t+rsu_s-ru=0$$ on that boundary. I might be lucky to write the closed formula on that boundary so no discretization is needed. In the second approach I basically assume the equation is solved on that boundary, along with the fact the second derivative is zero. In the first approach, I only assume the second derivative is zero. They don't look equivalent to me. So what approach is correct and why?

  • $\begingroup$ But you still have to solve in the domain so you still need values on the discrete points at the boundary, so how would you escape the first approach? $\endgroup$ – Mats Lind Sep 12 '16 at 16:04
  • $\begingroup$ I don't, But the fact is that I can either have $u_{ss}$ on the boundary or $u_t+rsu_s-ru$ on it. In both cases I will discretize the appropriate equations but they look different to me, therefore providing different boundary conditions. $\endgroup$ – Medan Sep 12 '16 at 16:07
  • $\begingroup$ I mean how do they look after discretisation and insertion of bc? $\endgroup$ – Mats Lind Sep 12 '16 at 17:25
  • $\begingroup$ in the first case, the last line in the matrix $A$ of $Au^{n+1}=Bu^n$ is $1, -2, 1$, while in the second case the last line is $0.5rs_n\delta t/\delta s, 1+0.5rs_n\delta t/\delta s-0.5r, 0.5rs_n\delta t/\delta s$ where one sided difference have been used to approximate $u_s$ and Crank-Nicolson in time. The last line the equation at the last $n$th point on the grid. $\endgroup$ – Medan Sep 12 '16 at 17:58

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