Pricing, Hedging and Assessing Risk in a General Lévy Context

The goal of this paper is to suggest a general approach for risk management by allowing jumps occurring in the underlying assets dynamics. This methodology is based on the generalized Fourier transform in line with the works of Lewis (2001), Boyarchenko and Levendorski i (2000) as well as Eberlein et al. (2010) and Raible (2000). It leads to a simple integration formula, particularly suited to the choice of calculation by fast Fourier transform, joining numerous works including Carr and Madan (1998). In addition, among the many approaches used in the literature for the hedging in a L évy context and for practical purpose, our choice is based on the local risk minimization used in Boyarchenko and Levendorski i (2000) and suggested by Föllmer and Schweizer (1991). We show that this hedging approach coincides with the optimal hedging ratio calculated by Tankov and Voltchkova (2009) with the main advantage of staying in a very large class of L évy processes and bringing a very short computation time. The same methodology is then extended to the computations of the standard risk measures, leading to a unifi ed approach for practitioners. Furthermore, this article shows that such useful quantities for professional investors as option prices, hedging ratios and risk indicators in exponential L évy models can be obtained in the same way by simply changing the integrand in a Fourier integral