ORIE Colloquium on 2/14/2012 - Pierre Patie: Asian Options and Wiener-Hopf Type Factorization

Tuesday, February 14 at 4:15pm Frank H.T. Rhodes Hall, 253 Asian options are path-dependent contingent claims, whose payoff is a function of the average of underlying prices over a prescribed time period. For several economical reasons, they are among the most attractive and popular derivative products. We start this talk by reviewing some well known results regarding the pricing of Asian options in the Black-Scholes model. In particular, we present the methodology developed by H. Geman and M. Yor which consists in reducing the problem to the characterization of the law of the exponential functional of a Brownian motion with drift. We proceed by discussing the extension of Geman-Yor's formula to models whose price dynamics are driven by a Lévy process. In particular, we show, under very mild conditions, that the exponential functional of a Lévy process satisfies a Wiener-Hopf type factorization. As by-product, we provide some simple analytical expression for the density of this variable for several classes of Lévy processes as well as some fine distributional properties.