Sensitivity estimates for Lévy-driven models in finance - Liu, Zongjian (ProQuest Dissertations & Theses, 2008)
Lévy-driven models have become increasingly popular in financial engineering in recent years, due to their capabilities of interpreting the observed features of financial markets in a more accurate way than models based on Brownian motion, such as jumps, fat tail and skewness, etc. In financial applications, sensitivity estimates are important in measuring and managing risk. Among simulation-based sensitivity estimation methods, the likelihood ration method (LRM) and the pathwise method (PM) are widely used both in academia and industry. This thesis focuses on how to apply LRM and PM for Lévy-driven models. In the first part, we investigate the application of the likelihood ratio method for sensitivity estimation in a more general setting, in which the relevant densities for the underlying model are known only through their characteristic functions or Laplace transforms. This includes, but not limited to, the Lévy process case. This problem arises in financial applications, where a substantial class of models have transition densities known only through their transforms. We quantify various sources of errors arising when numerical transform inversion is used to sample through the characteristic function and to evaluate the density and its derivative, as required in LRM. This analysis provides guidance for setting parameters in the method to accelerate convergence. In the second part, we explain in full detail the Laplace inversion algorithm used in the first part. This algorithm is an extension of the one-sided Euler inversion algorithm proposed by Abate and Whitt (1992) to the two-sided case. By introducing a shift parameter, we show that an exponential decay discretization error can be achieved so that the two-sided Euler inversion algorithm, like the one-sided one, is still accurate, efficient and concise. In addition, how to strike a balance between the discretization error reduction and the truncation error reduction is also discussed. Some practical examples in financial engineering are provided to illustrate the application of the inversion algorithm. This work is jointly done with Ning Cai and S.G. Kou. In the last part, we develop other methods to apply LRM and PM for Lévy-driven models, based on alternative approaches to approximating and simulating Lévy processes. We develop estimators based on exact sampling of increments, time-change representations of Lévy processes, saddlepoint approximations to the score functions of the increments, compound Poisson approximation to infinite activity processes, and compound Poisson approximations with Brownian approximations to small jumps. We discuss the relative merits of these various alternatives, both in theory and in practice, and we illustrate their use through examples.