Speaker: Alexey Kuznetsov, York University
ABSTRACT
Functionals of a stochastic process, such as extrema, first/last passage time and overshoot, can be used to describe the payoff of a financial derivative, and their distribution is the key to evaluating the option price and hedging portfolio. In a Levy driven model the theoretical framework for studying these functionals is based on the Wiener-Hopf factorization and related fluctuation identities. The well known examples of Levy processes which have explicit expressions for Wiener-Hopf factors include processes having phase-type jumps, one-sided jumps and a subclass of stable processes. In this talk we will introduce several new classes of Levy processes: a ten-parameter beta-family (similar to generalized tempered stable processes) and a family related to theta functions. We show that these processes enjoy many properties which make them attractive in financial modeling: they can have a Gaussian component and an arbitrary behavior of small jumps; the Levy measure has exponential tails; the characteristic exponent is given in closed form and, most importantly, there exist semi-explicit expressions for Wiener-Hopf factors and the density of extrema. We will also discuss several techniques which help us to implement efficient numerical algorithms.