Adaptive Space-Time FEM for Option Pricing

Summary

The document raises a numerical methods question about pricing vanilla European calls and puts with a finite element method discretized across both space and time. The author reports having applied an adaptive mesh to reduce error caused by nonsmooth option payoffs, and asks why this approach appears less common in finance than finite difference methods.

The text gives no answers, citations, comparisons, or numerical results explaining the relative use of FEM and FDM. Its contribution is identifying a research direction and a practical issue: adaptive meshing can target regions where payoff irregularities affect numerical accuracy. It does not establish that space-time FEM is superior, or explain implementation, stability, computational cost, or boundary treatment. Further evidence would be needed to decide whether the method offers advantages for particular pricing problems.

Key ideas

• The author describes pricing European vanilla options with a space-time finite element discretization.
• An adaptive mesh is reported as a way to reduce numerical error near nonsmooth payoffs.
• The document asks why finite difference methods appear more prevalent in financial applications.
• It provides no literature references or comparative performance evidence.

Tags

# Time discretisations, FDM vs FEM


# Time discretisations, FDM vs FEM












I am interested in adaptive mesh methods for numerical solution of PDEs with applications to finance. As part of a school project, I have been pricing vanilla European call and put options using 2D FEM (space+time) and successfully applied an adaptive mesh algorithm to reduce the numerical error introduced by non-smooth payoffs.

I have not been able to find a lot of existing work on 2D space-time FEM in finance, let alone adaptive mesh methods in this context. Hence the question: do people tend to use FDMs due to ease of implementation (separate discretisation of the dimensions is much easier to handle code-wise), or are there other reasons for the apparent absence of space-time FEM in the literature?

I am contemplating whether or not it makes sense to keep working in this direction, as the lack of similar papers/works in progress might suggest that it is not of any relevance. If I have missed any existing papers on this topic, please provide references if possible.

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