I'm thinking about the interplay of Dirichlet and Neumann BCs in a FDM scheme.

Let's assume a simple Black-Scholes call option problem, with BS PDE with constant coefficients, i.e. instead of $S$, in terms of $x=\ln(S)$.

In that case, the Dirichlet BC's are: \begin{equation} \begin{array}{l} V(t,{x_ + }) = \exp ({x_ + }) - K{e^{ - r(T - t)}}\\ V(t,{x_ - }) = 0 \end{array}. \end{equation} This is normally sufficient for solving the PDE. However, if I consider that \begin{equation} \frac{{\partial V(t,{x_ + })}}{{\partial x}} = \frac{{{\partial ^2}V(t,{x_ + })}}{{\partial {x^2}}}, \end{equation} this is also a valid BC for the upper boundary because $V(t,x) \propto {e^x}$ at the boundary.

  1. can I drop the Dirichlet BC, if for that boundary I have also Neumann BC? I know the result won't be the same but is that approach correct?
  2. can I use both types of BC at once? Would this yield a better approximation?
  3. Is the role of the Neumann BCs important rather in the case of, say, defining the option's behaviour at the boundaries of the variance grid ($v_-,v_+$), e.g. in the Heston model, where no Dirichlet boundaries for $v$ exist?

You can certainly mix Dirichlet and Neumann boundary conditions, though the mixture has to be consistent. For example it is fine to use Neumann as $x \rightarrow \infty$ and Dirichlet as $x \rightarrow 0$. When pricing options on an $S$ grid rather than an $x$ grid this can make a lot of sense, because then you can put your bottom node right at zero.

I tend to use Neumann more than Dirichlet for two reasons:

  1. Neumann boundary conditions come from the SDE/PDE, so I don't need to do any work finding boundary values
  2. Once the option is in our portfolio, we care most about getting the hedge right, which is better done with Neumann.

I haven't used a PDE scheme for Heston but I would be inclined to go Neumann for the very reasons you cite.

  • $\begingroup$ Thanks! My question about mixing the Dirichlet and Neumann BC was more related to the idea whether both Dirichlet and Neumann can be applied for one boundary (say the upper spatial boundary). This would basically result into Robin boundary condition. Is this still a valid approach or not? In the most textbooks I can find only a classification of BCs and that in the BS model we specify the Dirichlet BCs but other related topics are usually not covered. $\endgroup$ – user2743931 Oct 27 '15 at 9:12
  • $\begingroup$ I had to look up Robin boundary conditions just now. I have never seen them used in finance (and I've seen a hell of a lot of PDE solvers in my day). It's definitely valid, but you would have to quantify what you mean by a "better" approximation before trying to judge its potential advantages. The disadvantages are obvious: (1) another degree of freedom (relative weight) to hassle with in the PDE solver and (2) combined complexity of both Dirichlet and Neumann terms. Overall, I'm skeptical about its value. $\endgroup$ – Brian B Oct 27 '15 at 13:25

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