Imperfect Hedging in Defaultable Markets and Insurance Applications

Abstract

In this thesis, we study the impact of random times to model and manage unpredictable risk events in the financial models. First, as a generalization of the classical Neyman-Pearson lemma, we show how to minimize the probabil- ity of type-II-error when the null hypothesis, alternative and the significance level all are revealed to us randomly. This randomness arises some measurabil- ity requirements that we have dealt with them by using a measurable selection argument. Then, we consider a regime-switching financial model which is sub- ject to a default time satisfying the so-called the density hypothesis. For this model, we present a Girsanov type result and an explicit representation for the problem of superhedging. In both cases, the desired representation is decom- posed into an after-default and a global before-default decomposition. Another problem consists in minimizing the expected shortfall risk for defaultable se- curities under initial capital constraint. The underlying model is exposed to multiple independent default times satisfying the intensity hypothesis. We il- lustrate the results by numerical examples and the applications to Guaranteed Minimum Maturity Benefit (GMMB) equity-linked life insurance contracts. Finally, we construct a framework to consider a Guaranteed Minimum Death Benefit (GMDB) equity-linked life insurance contract as a Bermudan option. Under an initial capital constraint, we provide closed-form solutions for the quantile hedging problem of a GMDB contract with a constant guarantee.