Numerical Approximation of Valuation Equations Incorporating Stochastic Volatility Models

This dissertation studies the problem of controlling far field boundary errors arising in partial differential equation approaches for pricing financial contracts written on stochastic volatility models. Feynman-Kac type results are obtained by relating finite domain Dirichlet problems to options bearing barrier features. We then adopt a probabilistic framework to show convergence for strictly sublinear contracts even when the underlying process is a local martingale, and for linear contracts when it is a proper martingale. By restricting the stochastic volatility models to a smaller class, upper bounds for the far field boundary errors are derived for linear contracts. Convergence does not hold for linear contracts dependent on strict local martingales. While rigorous results for this case are unavailable, we conjecture inverse second order convergence in the far boundary distance when appropriate Neumann boundary conditions are imposed. Effective use of a finite difference alternating direction implicit algorithm is discussed. This scheme is implemented to test convergence theories and conjectures on well known models, such as the Bessel model and the Heston model.