Analytical ramifications of derivatives valuation: Asian options and special functions

TL;DR

This paper explores the analytical structure of Asian options valuation, linking it to special functions and deriving integral representations that improve existing results and establish benchmarks.

Contribution

It introduces a novel analytical approach connecting Asian options valuation to special functions, providing new integral representations and benchmark prices.

Findings

Derived integral representations for Asian options prices.

Connected Asian options valuation to the gamma function and special functions.

Provided first-time benchmark prices for Asian options.

Abstract

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.