Advisor: Dr. Wolfgang Bangerth

Committee:  Dr. David Aristoff, Dr. Tianyang Wang

Title: Numerical Solution of the Black-Scholes Equation Using Finite Element Methods

Abstract: The Black-Scholes model is a well-known model for pricing financial options. This model takes the form of a partial differential equation (PDE) that, surprisingly, is deterministic. In the special case where the option only has one single underlying asset, what is called the one-dimensional version of the Black-Scholes model, there exists an analytical solution. In higher dimensions, however, there is no such analytical solution. This higher dimensional version refers to what is called a Basket-Case Option. This means that to get a solution to this Basket-Case Option PDE, one must employ numerical methods. This thesis will first discuss the stochastic calculus theory necessary to derive the Black-Scholes model, then will explain in detail the time and space discretization used to solve the PDE using a Finite Element Method (FEM). Finally, this thesis will explain some of the results and convergence of this numerical solution.

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