Finite-Difference Boundaries for Option Pricing and Barrier Conditions
Summary
The document discusses boundary conditions in finite-difference option pricing, contrasting Dirichlet conditions, which prescribe the value at the boundary, with Neumann conditions, which prescribe a spatial derivative. It notes that Neumann conditions can be appropriate when the payoff behaves linearly at large values, but does not give examples across option types. It also describes a commonly used linear boundary treatment that sets the second spatial derivative to zero.
For a discretized pricing PDE, this assumption removes the second-derivative term at the edge, leaving the first spatial derivative and time derivative to be approximated there. For an up-and-out or down-and-out barrier option, the proposed setup places the grid boundary at the barrier and imposes a zero-value Dirichlet condition. The guidance is brief: it gives no convergence results or detailed treatment of rebates, discrete monitoring, or alternative barrier implementations.
Key ideas
- Dirichlet conditions set the option value at a grid boundary, while Neumann conditions specify its spatial derivative.
- A linear boundary approximation sets the second spatial derivative to zero at the edge of the grid.
- That approximation simplifies the pricing PDE boundary equation to first spatial and time derivatives.
- For an out barrier, the grid can terminate at the barrier with the option value set to zero.
- The discussion does not provide option-specific evidence for when Neumann conditions are suitable.
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# Question on boundary conditions when using Finite Difference
# Question on boundary conditions when using Finite Difference
I have two questions appearing to me (they are not related directly to each other).
- My first question is about boundary conditions when using Finite difference methods. There are two ways to do it: a) Dirichlet b) von Neumann (using always two values "inside" the grid/mesh to get the one on boundary). For me von Neumann conditions are more elegant. It is always written they hold as long as Payoff is linear for large values. But I am searching a bit for examples on that. In which types of options von Neumann works in which not?
- Barrier option pricing with FD methods (such as Crank Nicoloson). How would you approach there? Say you have an up and out call. Would use use the same technique as plain vanilla Call just that you say the upper boundary for the stock price is the value of the barrier (so grid in stock price direction goes up to barrier level) and the boundary condition for the value of the option there is zero?
Thanks a lot for sharing your knowledge and answer!
## Answer by Antoine Conze (score 3, accepted)
https://quant.stackexchange.com/a/63275
1 - The Neumann boundary condition is actually named after Carl Neumann, not John von Neumann.
There is another boundary condition not often mentioned but used very often in practice in Quant finance FD solvers, which is linear (zero second spatial derivative on the boundary). This means that on the boundary the PDE $a\frac{\partial U}{\partial x} + b \frac{\partial^2 U}{\partial x^2} + \frac{\partial U}{\partial t} = 0$ simplifies into $a\frac{\partial U}{\partial x} + \frac{\partial U}{\partial t} = 0$ which you discretize using the uncentered discrete difference for $\frac{\partial U}{\partial x}$.
2 - Barrier options: you do as you suggested, make the barrier the upper value (resp. lower value) for the grid when pricing an up and out (resp. down and out), and use a Dirichlet condition on the barrier.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.