
Natural sciences
 Probability theory ^{ }
Markets where interest rate derivatives such as swaptions are traded are among the largest and most liquid options markets. These options are widely used both for hedging against risk (and speculation against changes in interest rates. Because of the size of these markets, they have the accurate and efficient pricing and hedging of swaptions have enormous practical significance.
Theoretical research of interest rate derivatives has produced a variety of models and techniques to price options, and often the literature focuses on assessing the "pricing performance" of the models. But beyond pricing, hedging is a necessary and complementary test to evaluate the performance of a model to evaluate. The issues of pricing and hedging of swaptions usually appear in the literature in a continuous setting. This involves testing the effectiveness of hedging for different hedging strategies and the hedging performance is sometimes also compared. Empirical studies show that incorrect pricing using continuous models can be overcome by using models based on (timeinhomogeneous) Lévy processes. In contrast to hedging in the context of Lévy asset price models, the study of hedging problems proposed in this project is in the case of interest rate derivatives in Lévy interest rate models is novel. Two main issues will be addressed. First, given a Lévy model, examine and compare the effectiveness of different hedging strategies. Second, the question arises as to how a change in the model affects the performance of a given hedging strategy. To this end, we want to study the influence of model risk on a hedging strategy. The different Lévy models under consideration are the Lévy driven HeathJarrowMorton model, the Lévy Libor market model, the Lévy forward pricing model and the
Lévy swap market model. The hedging strategies to be compared are the (selffunding) delta hedge, the deltagamma hedge and the quadratic hedging strategies 'mean variance hedging' and 'local risk minimization. We will consider the hedging problem for swaptions based on investments in zerocoupon bonds because a swaption can be expressed as an option on a basket of zerocoupon bonds.
This project will also focus on two complementary parts, namely the theoretical foundation and development of hedging strategies for swaptions in Lévy interest rate models and the numerical validation of the results found. This theoretical part aims to reduce the gap between theory and practice. This includes the establishment of pricing formulas and formulas for the strategies in the different Lévy models for swaptions. For the simulations, the models will calibrated to recent market data.