Large deviations, transforms, and simulations for some affine processes

Abstract

This dissertation is devoted to the study of computational aspects for some affine processes. In particular, we deal with affine diffusion processes on the canonical state space, general affine processes on positive semidefinite matrices, and a multifactor stochastic volatility model with affine property.

For affine diffusion processes, we establish the sample path and finite dimensional large deviation principles. The large deviation principle is a crucial tool in asymptotic computations of probabilities of rare events, and it is recently applied to various problems in mathematical finance. Large deviation principles for some specific affine diffusion processes have been studied by some authors. We provide in this thesis a unified treatment of their large deviation results under the standard uniform topology and a mild technical condition.

As the financial derivatives get complicated, the correlation and covariance between assets have become one of the most prominent sources of risks. Affine processes on positive semidefinite matrices provide a flexible and tractable family of stochastic processes for modeling realistic stochastic covariance matrices. The affine transform formula lies at the center of the computational tractability of such processes. We extend the affine transform formula from marginal distributions to linear functionals of affine processes. Moreover, we find that the transforms of brides of affine processes are closely related to marginal distributions under equivalent probability measures as well as unconditional transforms.

Many empirical studies have documented that multifactor stochastic volatility models outperform single factor models in fitting term structure of implied volatilities. Among such multifactor models, the Wishart multidimensional stochastic volatility model provides most flexible features by incorporating matrix-valued Wishart process as the volatility factor. The analytic aspects of the model are well studied, but ...