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Schröder Abstract. 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 [Y], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes using the Hartman-Watson theory of [Y80]. Consequences of this approach for valuing Asian options proper have been spelled out in [GY] whose Laplace transform results were in fact regarded as a noted 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 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.

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 [Y], these questions about exponential functionals of Brownian motion have been studied in terms of Bessel processes us- ing the Hartman-Watson theory of [Y80]. Consequences of this approach for valuing Asian options proper have been spelled out in [GY] whose Laplace transform results were in fact regarded as a noted 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 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.

There are two types of Asian options in the financial markets which differ according to the role of the average price. We give a symmetry result between the floating and fixed-strike Asian options. The proof involves a change of numéraire and time reversal of Brownian motion. Symmetries are very useful in option valuation and in this case, the result allows the use of more established fixed-strike pricing methods to price floating-strike Asian options.

Averaging problems are ubiquitous in Finance with the valuation of the so–called Asian options on arithmetic averages as their most conspicuous form. There is an abundance of numerical work on them, and their stochastic structure has been extensively studied by Yor and his school. However, the analytical structure of these problems is largely unstudied. Our philosophy now is that such valuation problems should be considered as an extension of the theory of special functions: they lead to new problems about new classes of special functions which should be studied in terms of and using of the methods of special functions and their theory. This is exemplified by deriving integral representations for the Black–Scholes prices based on Yor’s Laplace transform ansatz to their valuation. They are obtained by analytic Laplace inversion using complex analytic methods. The analysis ultimately rests on the gamma function which in this sense is found to be at the base of Asian options. The results improve on those of Yor and have served us a as starting point for deriving first time benchmark prices for these options. 1. Introduction: This