Choosing Dirichlet and Neumann Boundaries for Option PDEs
Summary
The document considers boundary conditions for a finite-difference solution of the Black–Scholes call pricing equation in log-price coordinates. It gives the usual call values at the lower and upper price limits, then notes that the upper-boundary payoff behavior can also imply a relationship between the first and second spatial derivatives. The question is whether that Neumann-style condition can replace or supplement the upper Dirichlet value.
The answer says Dirichlet and Neumann conditions can be mixed when they are consistent, such as applying a derivative condition at a high-price limit and a value condition near zero. It favors Neumann conditions in some settings because they can follow from the pricing equation and can help estimate a hedge. It suggests similar conditions may be useful at variance boundaries in a Heston PDE, but this is an expressed preference rather than a demonstrated comparison. No numerical evidence is provided, and boundary choices remain dependent on the grid and model setup.
Key ideas
- Dirichlet conditions specify option values at the computational domain boundaries.
- Neumann conditions specify derivative behavior and can be combined with Dirichlet conditions when consistent.
- At high underlying prices, asymptotic option behavior can motivate a derivative-based boundary condition.
- Boundary conditions affect both price approximation and the accuracy of hedge estimates.
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# Boundary conditions: Dirichlet vs Neumann
# Boundary conditions: Dirichlet vs Neumann
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.
- 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?
- can I use both types of BC at once? Would this yield a better approximation?
- 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?
## Answer by Brian B (score 1)
https://quant.stackexchange.com/a/21456
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:
- Neumann boundary conditions come from the SDE/PDE, so I don't need to do any work finding boundary values
- 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.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.