There is a trend in investment banking to unify pricing tools in a framework of partial differential equations. The Black-Scholes equation and its extensions are solved numerically with pde-based techniques (instead of rather heuristic techniques like Monte Carlo or Trees). The predominant numerical technique today is Finite Differences. This technique has been studied since L. Euler and is fairly easy to apply. A more modern technique, Finite Elements, can also be applied to solve the pdes. It offers some advantages: 1. The solution is a polynomial approximation to the entire domain. The method of Finite Differences supplies an approximate solution only to distinct points in the domain, so that interpolation becomes necessary. 2. There are several FE methods that supply Delta and Gamma as a by-product. The other Greeks can be computed in a slightly easier manner than with FD. 3. Irregular domains, which occur when knock-out barriers are applied to multi-asset options, can be solved easily since the elements can be adjusted to almost any domain. FD techniques, however, are designed to cover rectangular domains. 4. FD techniques for problems involving Neumann conditions are not straightforward to apply. FE methods, however, have no problems with Neumann conditions. Neumann conditions are easier to apply to financial problems since the behavior of the function at infinity has to be approximated. Approximating the function at infinity is normally harder than approximating its derivative. We will demonstrate these advantages with various exotic options (for instance, capped power calls, double barrier options on baskets, volatility models, and the like) and bond pricing models (Duffie/Kan, Sandmann/ Sondermann/ Miltersen). We also demonstrate how to use FE for nonlinear models arising through the incorporation of transactions costs and through passport options.
