# When do Finite Element method provide considerable advantage over Finite Differences for option pricing?

I'm looking for concrete examples where a Finite Element method (FEM) provides a considerable advantages (e.g. in convergence rate, accuracy, stability, etc.) over the Finite Difference method (FDM) in option pricing and/or calculating sensitivities.

I know that in general Finite Element is more powerful numerical method for solving PDE's (e.g. it allows to work on complex geometries), but I'm specifically interested if we can fully utilize it for option pricing. I've skimmed through the book Financial Engineering with Finite Elements, but it only states some general considerations about benefits of FEM in the introductory section, without going into details.

Currently I'm aware about only one case, when FEM outperforms FDM in convergence rate, namely in pricing under a Variance gamma model. But this advantage can be neglected by introducing certain enhancements in the FDM.

As far as PDEs (deterministic) are concerned we have the notion of a "strong solution" (directly solving the differential operator in the strong formulation of the problem) and the "weak solution" that deals with a weak formulation of the problem.

For the strong formulation, finite differences are the way to go since they are the natural discretization of the differential operator.

For the weak formulation, finite element methods are the way to go since they directly tackle the weak formulation of the problem by restricting it to a finite dimensional space (depending on which type of "element-functions" or basis-functions you choose for this space).

The finite difference method has problems with complex geometries and adaptive meshes - the geometry will not be a problem in option pricing since you always consider the rectangle $[0,T]\times[S_\text{min}, S_\text{max}]$. Local refinement can be a problem - but it depends on the equation and the initial/boundary condition. Further more, for some (nonlinear) differential equations, problems with disconutities occur and you can end up with oscillation effects. There are schemes that circumvent problems like this but you have to invest into the algorithm.

The finite element method is a more general notion. Since SPDEs are defined by their (Ito-) integral formulation an approach that approximates an integral-formulation will feel more natural. Thats because the payoff of a European Call (S-Variable) and the Brownian motion paths (t-Variable) are both not differentiable. Using finite difference methods for SPDEs most natural discretizations for "differential operators" does not give you the right scheme as far as I know. That would be another hint that points into the finite element direction a little bit.

One problem with comparing the performance of the two is definitely that there are so many different schemes (implicit/explicit of different ordes for finite differences and choices of basis functions and local refinement techniques for finite elements) that it is impossible to say. Maybe for some choice of model and initial/boundary condition one method will outperform the other but I think its hard to generalize.

FDMs represent PDEs over a simple grid shape; the different implementations are just different recurrence relations to approximate the solutions to the PDE between boundary values (e.g., for options pricing, $T=[t_\mathrm{now},t_\mathrm{maturity}]$ and $S=[\mathrm{deep\_itm},\mathrm{deep\_otm}])$.

FEM is a general name for a lot of different implementations of "adaptable" grids, where approximations in different regions are not necessarily the same over the entire (e.g., T,S) space. One (of many) examples where such approaches are important is path-dependent option valuations (e.g., Asian Options, see Zhang, 2001/3 which is FDM, imagine if we wanted to better model the smoothing effect of the averaging over time, or jump processes as described by Merton).

I have no personal experience comparing the two methods. However, I have heard that FEM might be preferable to FDM in connection with degenerate PDEs such as the ones that occur in the Hobson-Rogers model.