Bessel processes, the integral of geometric Brownian motion, and Asian options

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 [M. Yor, Adv. Appl. Probab., 24 (1992), pp. 509–531], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes using the Hartman–Watson theory of [M. Yor, Z. Wahrsch. Verw. Gebiete, 53 (1980), pp. 71–95]. Consequences of this approach for valuing Asian options proper have been spelled out in [H. Geman and M. Yor, Math. Finance, 3 (1993), pp. 349–375] whose Laplace transform results were in fact regarded as a significant advance. Unfortunately, a number of difficulties with the key results of this last paper 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 development 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.