A review of results pertaining to the pricing of Asian options

Marc Yor

Abstract: The original results concerning the price of Asian options were obtained with H. Geman (1991-92) by computing the price at an independent exponential time, which I shall explain by developing the underlying Laperti representation of geometric Brownian motion, and more generally the exponential of a Lévy process.

From the results, D. Dufresne derived an interesting relation between the reciprocals of exponential functionals of Brownian motions with two opposite drifts. I intend to explain how, from Dufresne's identity, it was possible to obtain (jointly with H. Matsumoto at Nagoya) some exponential versions of the famous L\'evy's and Pitman's theorem for $M-X$ and $2M-X$.