This dissertation considers three topics. The first part discusses the pricing of American options under a local volatility model and two jump diffusion models: Kou's jump diffusion model and the Dupire system. In Chapter 2, we establish partial differential complementarity systems for pricing American options under the aforementioned three models. We also introduce two different discretization schemes, a finite difference method and a finite element method, for the discretization of the complementarity systems into a collection of linear complementarity problems (LCPs). In Chapter 3, we discuss four popular existing numerical algorithms---a PSOR method, a two phase active-set method, a semi-smooth Newton method and a pivoting method---for solving LCPs that arise under Kou's jump diffusion model and the Dupire system. The numerical results presented in the thesis summarize the effectiveness of each approach for solving the corresponding LCPs.
%In particular, we are interested in the numerical evaluation of four algorithms pricing these options: a PSOR method, a two-phase active-set method, a semi-smooth Newton method, and a parametric pivoting method.
In the second part, we consider the calibration problems of computing an implied volatility parameter for American options under the Dupire system and the local volatility model. In Chapter 4, we formulate the calibration problem as an inverse problem of the forward pricing problem, which is modeled as a discretized partial differential linear complementarity system
in Chapter 2. The resulting inverse problem then becomes an instance of a mathematical program with complementarity constraints (MPCC). Two methods for solving MPCCs, an implicit programming algorithm (IMPA) and a new hybrid algorithm, are studied in this dissertation. We test both algorithms and report their numerical performance for solving MPCCs derived under the Dupire system and the local volatility model with synthetic and market data.
In the third part of this thesis, we investigate a new class of MPCCs, a doubly uni-parametric MPCC, for which the calibration of American options under the Black-Scholes-Merton (BSM) model is a special case. In particular, we consider one new algorithm for solving this problem when the problem matrices are positive definite, and a second algorithm for the more general case when the matrices are merely positive semi-definite. We study the convergence of both algorithms based on the local stability of the solutions as well as the numerical performance of both algorithms for solving doubly uni-parametric MPCCs with tridiagonal matrices, which are applicable for the calibration problems under the BSM model
