  This paper is motivated by questions about averages of stochastic processes
which originate in mathematical finance, originally in connection with valuing
the so-called Asian options. Starting with research of Yor's in 1992, these
questions about exponential functionals of Brownian motion have been studied in
terms of Bessel processes using Yor's 1980 Hartman-Watson theory. Consequences
of this approach for valuing Asian options proper have been spelled out by
Geman and Yor in 1993 whose Laplace transform results were in fact regarded as
a noted advance. Unfortunately, a number of difficulties with the key results
of this last contribution have surfaced which are now addressed in this paper.
One of them in particular is of a principal nature and originates with the
Hartman-Watson approach itself: this approach is in general applicable without
modifications only if it does not involve Bessel processes of negative indices.
The main mathematical contribution of this paper is the developement of three
principal ways to overcome these restrictions, in particular by merging
stochastics and complex analysis in what seems a novel way, and the discussion
of their consequences for the valuation of Asian options proper.
