Applications of Lévy processes in credit and volatility modelling

Market participants are faced with the problem of finding a good trade-off between the model adequacy and its tractability. The aim of this thesis is to develop tractable and intuitive models in the credit and equity areas and to assess their performance according to different criteria of interest for practitioners. Although sometimes criticized for its inability to reproduce quoted option prices which manifests itself in what is commonly referred to as the volatility smile, the Black-Scholes model and its implied volatility are widespread and their success is due to their very simple and intuitive concept. The first contribution of this thesis consists of the introduction of two kinds of alternative implied volatility, namely the implied Lévy space volatility and the implied Lévy time volatility, and of the investigation of the resulting skew adjustment. Moreover, we show that under the Lévy parameter settings, the model performs systematically better than the classical Black-Scholes model as regards the risk of the commonly used delta hedging strategy. This is illustrated by looking at the daily hedging error and P&L distributions and by noting that, for the Lévy models under investigation, the empirical variance is smaller and, for a wide range of in the money options, the empirical acceptability indices are higher than in the Black-Scholes setting. The second part of this thesis illustrates the impact of calibration risk under the He-ston model. In particular, we show that different plausible calibration procedures lead to different optimal parameter sets and hence to significantly different prices for a wide range of exotic products, emphasizing the necessity to take into account some additional safety margin for the pricing of these structured products, as it has been recommended in a recent directive of the Basel committee. The third contribution consists of the extension of the aVG model where the constraints on the subordinator parameters are relaxed such that the calibration does not require the existence of a liquid multivariate derivatives market which is nowadays pretty rare since the marginal characteristic functions become dependent on the whole parameter set. Moreover, the stocks log-return volatility, which is an indicator of the trading volume, becomes dependent on both the idiosyncratic and common subordinator settings, making the generalized model more coherent with the empirical evidence of the presence of both an idiosyncratic and a common component in the business clock. Furthermore we emphasize the presence of model risk inside a particular class of multivariate models by pricing standard multivariate options. Finally, we compare the exponential Lévy model with the classical Gaussian copula model for the pricing of CDO-squared tranches, using several approximations of the recursive approach, namely a full Monte Carlo approximation and a Monte Carlo approximation which rests either on the multivariate Normal approximation of the joint inner CDO loss distribution or on the multivariate Poisson approximation of the joint number of defaults affecting the inner CDOs. The numerical study shows in particular that the multivariate Poisson approximation method outperforms the multivariate Normal approximation for CDOs-squared made up of inner equity tranches.