*Wissenschaftliches Programm* *Liste der Vortragenden*

**Sektion 5**

Donnerstag, 21.09.2000,
17.00–17.20 Uhr, POT 361

#### Michael Schröder, Universität Mannheim

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.