Karpathopoulos, Nikolaos
(2019)
Local risk minimization analysis for contingent claims using weak derivatives and running infimum processes.
*University of Southampton, Doctoral Thesis*, 184pp.

## Abstract

In this thesis, we examine the local risk minimization approach and the FöllmerSchweizer decomposition, in certain credit risk models. We start by extending the model proposed by Okhrati et al. (2014) to the non-smooth case for which the hedging strategies are based on. Assuming that the evolution of the price of the underlying asset is a Lévy process of finite variation, we investigate the local risk minimization for a defaultable claim, whose default time is given through a first passage time (structural framework) where the default barrier is constant. We derive the KunitaWatanabe (KW) decomposition through a solution of a partial-integro differential equation (PIDE) using non-smooth Itô’s formula of Okhrati and Schmock (2015). This allow us to obtain a solution of a PIDE which is continuous but not necessarily smooth.

We also investigate a structural credit risk model using the local risk minimization approach where the default is modelled via a random variable. In this model, the underlying asset is a spectrally positive Lévy process and the compensator technique are used to obtain the Föllmer-Schweizer decomposition for a contingent claim that is prone to default from an investor’s point of view. In our analysis, we use a progressive filtration G. We highlight that we do not assume that the H-hypothesis holds, which states that a local martingale under the initial filtration F remains a local martingale under the expanded filtration G.

Furthermore, we study the local risk minimization for a defaultable contingent claim where the default time is exogenously defined though a hazard rate model depending on both the underlying and its infimum. This allows us to introduce some particularly interesting cases for claims that are subject to both endogenous and exogenous defaults.

The endogenous default is determined in a structural framework depending on the infimum process with constant barrier. Similarly to the previous model, our construction is made under a progressive filtration expansion G. In this setup, the underlying asset is modelled through an exponential jump diffusion Lévy process. We aim at determining locally risk minimizing hedging strategies through solutions of either PDEs or PIDEs. We also provide some applications and examples in credit risk modelling for the diffusion and jump diffusion case.

Finally, under the setup of the above models, we provide some credit risk models examples and their associated numerical implementations through solutions of PDEs and PIDEs using finite differences.

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