Finite Element Methods for Partial Differential Equations for Option Pricing

Abstract

This manuscript gives introduction to parabolic Partial Differential Equations (PDE) in one-dimensional case. It recalls the notion and main limitations of Black-Scholes model and establishes the Black-Scholes formula. It recalls the standard machinery of solving the parabolic PDE which is well-known in standard literature such as text-book of Bass (1999). Here we recall the notion of infinitesimal generator of the diffusion process and recall some basic well-known results on the infinitesimal generator of the Markov process. We avoid giving complete proofs of known results and provide reader with a short hint of proofs. Here, we recall again the theory of solving the parabolic PDE, we recall machinery of changes of variables and unknown functions. Then, we derive the Black-Scholes formula and give some intuition of application to some classes of options, such as a Vanilla European Call and Put. The next section is devoted to giving some notions of the classical solution to the parabolic PDE. It is immediately followed by the variational framework. The introduction to the reader of weighted Sobolev spaces on which the solution to the class of parabolic PDE under study lives is a subject to the next section in this Chapter. Straight after that we give a notion of weak formulation of the Black-Scholes equation followed by proving the regularity property of the weak solutions, application of the maximum principle for weak solutions which is a powerful tool to provide the solution to parabolic PDE and work further on derivation of various bounds. We give more insight into the localization procedure. We apply this methodology to option pricing, via Put-Call parity relation we derive a payoff function for a European Vanilla Put and Call. Among the further examples are General European Options, Barrier Options, European Options on a Basket of Asset, American Options, Asian Options. The chapter continues with giving some introduction to application of Finite Element methods where we recall the main steps of using this machinery. We recall as well more innovative Finite Element methods for the class of stochastic volatility models.