On some Brownian functionals and their applications to moments in lognormal and Stein stochastic volatility models

Abstract

The aim of this paper is to present the new results concerning some functionals of Brownian motion with drift and present their applications in financial mathematics. We find a probabilistic representation of the Laplace transform of special functional of geometric Brownian motion using the squared Bessel and radial Ornstein-Uhlenbeck processes. Knowing the transition density functions of the above we obtain computable formulas for certain expectations of the concerned functional. As an example we find the moments of processes representing an asset price in the lognormal volatility ans Stein models. We also present links among the geometric Brownian motion, the Markov processes studied by Matsumoto and Yor and the hyperbolic Bessel processes.