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Numerical solution of the Black-Scholes equation using finite element methods
(Colorado State University. Libraries, 2023) Anderson, Tyler, author; Bangerth, Wolfgang, advisor; Aristoff, David, committee member; Wang, Tianyang, committee member
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.