Abstract: This is the second part of our survey on exponential functionals of Brownian motion. We focus on the applications of the results about the distributions of the exponential functionals, which have been discussed in the first part. Pricing formula for call options for the Asian options, explicit expressions for the heat kernels on hyperbolic spaces, diffusion processes in random environments and extensions of Lévy’s and Pitman’s theorems are discussed.

In a recent significant advance, using Laguerre series, the valuation of Asian options has been reduced in [D] to computing the negative moments of Yor’s accumulation processes for which functional recursion rules are given. Stressing the role of Theta functions, this paper now solves these recursion rules and expresses these negative moments as linear combinations of certain Theta integrals. Using the Jacobi transformation formula, very rapidly and very stably convergent series for them are derived. In this way a computable series for Black–Scholes price of the Asian option results which is numerically illustrated. Moreover, the Laguerre series approach of [D] is made rigorous, and extensions and modifications are discussed. The key for this is the analysis of the integrability and growth properties of the Asia density in [Y], basic problems which seem to be addressed here for the first time. 1. Introduction: Asian

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 article is an introduction to Malliavin Calculus for practitioners.

Abstract. Laplace transforms for integrals of stochastic processes have been known in analytically closed form for just a handful of Markov processes: namely, the Ornstein-Uhlenbeck, the Cox-Ingerssol-Ross (CIR) process and the exponential of Brownian motion. In virtue of their analytical tractability, these processes are extensively used in modelling applications. In this paper, we construct broad extensions of these process classes. We show how the known models fit into a classification scheme for diffusion processes for which Laplace transforms for integrals of the diffusion processes and transitional probability densities can be evaluated as integrals of hypergeometric functions against the spectral measure for certain self-adjoint operators. We also extend this scheme to a class of finite-state Markov processes related to hypergeometric polynomials in the discrete series of the Askey classification tree. 1.

An option contract now commonly encountered is the Asian quanto-basket option. This contract is useful for risk managers willing to participate to the return of an industrial sector with an international exposure without the foreign exchange risk exposition. Although the price of such contracts can be obtained very accurately using Monte Carlo simulation, market participants prefer faster but less accurate analytical approximations. This paper thus examines the precision of three different analytical approximations available to price Asian quanto-basket options. The results of a comprehensive simulation experiment performed on a large test pool of option contracts reveal that the approximations based on the reciprocal gamma and Johnson-type densities are in general the most accurate.

In this note we extend and clarify some identities in law for Brownian motion proved by V. Seshadri [8] using a new identity in law obtained by H. Matsumoto and M. Yor [6].

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.

This short paper introduces the concept of Asian options in the capacity choice literature. We develop a simple model for optimal capacity setting under average demand uncertainty for a single firm. When the firm faces moderate or significant stochastic demands in its current product line, expanding capacity is beneficial. If the demand is extremely stochastic, a capacity lag or reduction is more profitable.

We would like to thank Arturo Kohatsu, Mamen Aranda, and seminar participants