This talk pursues that interplay between stochastics and analysis initiated by the work of Yor on the valuation of exotic options. Using the Laplace transform technique the focus is on the so-called Asian option. These are options on the arithmetic average of the price of the underlying. The main result is the closed form solution for its Black-Scholes price I have derived in 1997.
Our result is obtained using 1993 results of Geman-Yor. Unfortunately, the valuation problem they consider is from a finance perspective unrelated to valuing Asian options. Using a crucial insight of Peter Carr (Banc of America Securities, New York) into the working of Asian options and with the kind support of Yor we have been able to adapt this approach to now valuing Asian options. The modified results, however, are valid only under restrictions. These have their origin in the still limited knowledge of Bessel processes. At the current state of affairs, they are unfortunately such that the pertinent Laplace transforms do not seem to be available in those typical situations where Asian options are desirable to use.
For our valuation formula we have been able to lift these restrictions using analytic techniques. This illustrates a typical application of integral representations. They are good for establishing results of a structural nature about the special function they represent. In another typical application we have used our integral representation to derive explicit series and asymptotic expansions for computing the value of the Asian option. Since these are seemingly the first and only such results ever derived, an example is briefly discussed at the end of the talk.