Essays on pricing under l\'evy models and computing risk measures

Abstract

This thesis deals with questions regarding on pricing under exponential $L\'{e}vy$ models and computing risk measures and their sensitivities. $L\'{e}vy$ models form a tractable class of models incorporating jumps and are frequently used to describe asset dynamics in finance. While these models are capable of capturing the stylized features of financial asset dynamics, such advantages also produce various theoretical and computational burdens on users. In this work, we look at two fundamental questions arising in financial mathematics under exponential $L\'{e}vy$ models and try to answer how we can reduce such burdens in practical aspects.

In the first part of this thesis, we are concerned with a simulation technique for sampling from a $L\'{e}vy$ bridge, that is, a technique for sampling the values of a $L\'{e}vy$ process at a fixed time point conditioned on the initial and terminal values of the process. This bridge sampling method is applied to simulation of path-dependent options and American options, and it significantly increases simulation efficiency.

In Chapter 2, we consider stable and tempered stable $L\'{e}vy$ subordinators, which frequently appear in financial contexts, and develop a bridge sampling method. An approximate conditional PDF given the terminal values is derived with stable index less than one, using the double saddlepoint approximation. We then propose an acceptance-rejection algorithm based on the existing gamma bridge and the inverse Gaussian bridge as proposal densities. Its performance is comparable to existing sequential sampling methods such as [30] and [44] when generating a fixed number of observations. As applications, we consider option pricing problems in $L\'{e}vy$ models. First, we demonstrate the effectiveness of bridge sampling when combined with adaptive sampling under finite-variance CGMY processes. Further efficiency gain is achieved via stratified sampling. Under normal tempered processes, we propose a hybrid sampling method for efficient least square Monte Carlo for American options valuation which reduces large memory requirements.

The second part of this thesis aims to explore fast and accurate solutions to risk management problems that arise frequently in practice in the form of conditional expectations. By extending the existing literature on saddlepoint techniques to conditional expectations, we succeed in developing new approximations that are potentially useful and give stable and accurate answers for computing risk measures and their sensitivities.

Chapter 3 derives saddlepoint expansions for conditional expectations in the form of $\mathsf{E}[\overline{X}

\overline{\bY} = \ba]$ and $\mathsf{E}[\overline{X}

\overline{\bY} \geq \ba]$ for the sample mean of a continuous random vector $(X, \bY^\top)$ whose joint moment generating function is available. %Theses conditional expectations frequently appear in various applications, particularly in quantitative finance and risk management. Using the newly developed saddlepoint expansions, we propose fast and accurate methods to compute the sensitivities of risk measures such as value-at-risk and conditional value-at-risk, and the sensitivities of financial options with respect to a market parameter. Numerical studies are provided for the accuracy verification of the new approximations.