A new Bismut-Elworthy-Li-formula for diffusions with singular coefficients driven by a pure jump Levy process and applications to life insurance

Abstract

The main result of my mine in the master thesis is a new Bismut-Elworthy-Li-formula with respect to a pure jump Levy noise driven stochastic differential equation (SDE), with non-Lipschitz continuous coefficients. This thesis consists of 5 chapters, where chapter 1 is an introduction to what Greeks are and why they are interesting in finance. In chapter 2 there is an overview and discussion of basic methods for the calculation of Greeks in the literature. In chapter 3 there is an implementation of what we refer to as Zhang s formula, namely a Bismut-Elworthy-Li type formula. This is a derivative free type formula for SDEs driven by pure jump process, namely an α-stable process. In the first part of chapter 3 simulations are conducted confirming that Zhang formula in numerical implementations works, then there is presented an application of this formula to life insurance, where we also conduct simulations. Chapter 4 is the highlight of this thesis, where we derive a BismutElworthy-Li type formula for the Greek Delta. This derivative free representation is obtained by using methods in [17] and [8]. The formula can be regarded as an extension of Zhang s formula in case of the Greek Delta, in the sense that we deal with Holder coefficients and don t demand that the coefficients have continuous first order derivative. Chapter 5 suggests possible extensions to this thesis.