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On ladder height densities and Laguerre series in the study of stochastic functionals. II. Exponential functionals of brownian motion and asian option values

Part of: Probabilistic methods, simulation and stochastic differential equations Markov processes

Published online by Cambridge University Press:  08 September 2016

Michael Schröder
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Abstract

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In this paper we develop a constructive structure theory for a class of exponential functionals of Brownian motion which includes Asian option values. This is done in two stages of differing natures. As a first step, the functionals are represented as Laguerre reduction series obtained from main results of Schröder (2006), this paper's companion paper. These reduction series are new and given in terms of the negative moments of the integral of geometric Brownian motion, whose structure theory is developed in a second step. Providing a new angle on these processes, this is done by establishing connections with theta functions. Integral representations and computable formulae for the negative moments are thus derived and then shown to furnish highly efficient ways for computing the negative moments. Application of this paper's Laguerre reduction series in numerical examples suggests that one of the most efficient methods for the explicit valuation of Asian options is obtained. The paper also provides mathematical background results referred to in Schröder (2005c).

Keywords

Laguerre series methods for stochastic functionalsladder height densitymoment formulae for the integral of geometric Brownian motionexplicit valuation of an Asian option

MSC classification

Primary: 60J65: Brownian motion
Secondary: 65C50: Other computational problems in probability 33C45: Orthogonal polynomials and functions of hypergeometric type (Jacobi, Laguerre, Hermite, Askey scheme, etc.)
Type
General Applied Probability
Information
Advances in Applied Probability , Volume 38 , Issue 4 , December 2006 , pp. 995 - 1027
Copyright
Copyright © Applied Probability Trust 2006 

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