| When underlying financial variables follow a Markov diffusion process, the value function of a derivative security satisfies a partial differential equation (PDE) for European-style exercise or a partial differential variational inequality (PDVI) for American-style exercise. Unless the Markov process has a special structure, analytical solutions are generally not available, and it is necessary to solve the PDE or the PDVI numerically. In this dissertation we develop a general high-order method for solving multi-dimensional problems arising in valuation of European and American-style derivatives. The method is based on: (1) converting the PDE or PDVI to a variational (weak) form; (2) applying a smooth penalization to transform PDVI into a nonlinear variational equality; (3) discretizing the weak formulation spatially by the Galerkin finite element method to obtain a system of ODEs; and (4) integrating the resulting system of ODEs in time. We apply this method to obtain numerical solutions in up to 6 dimensions for the American options on multiple stocks and conclude that the high-order penalty method is computationally superior compared to the commonly used non-smooth penalization. We also consider a valuation of convertible bonds in a 4-factor model with stochastic interest rate, stock price, volatility, and default intensity. With the possibility of call from a bond issuer and conversion by a bond holder, we formulate the problem in terms of a stochastic differential game, transform it into a weak formulation and solve by the described method in a 4-dimensional setting. In both applications, we demonstrate a high-order convergence of the FEM scheme, and remarkable efficiency of the adaptive multi-step solver used for the time integration. The numerical solutions are used to analyze the dependence of value functions on the risk factors and the parameters, and to construct the multi-dimensional optimal exercise boundary for the American option, and the optimal conversion and call policies for the convertible bond. |
