On certain integral functionals of squared Bessel processes

TL;DR

This paper analyzes integral functionals of squared Bessel processes using Feynman-Kac techniques, deriving Laplace transforms, solving small ball problems, and applying results to derivative pricing and option asymptotics.

Contribution

It introduces new Laplace transform formulas for joint laws of functionals of squared Bessel processes and applies these to financial mathematics and stochastic process theory.

Findings

Derived Laplace transforms for joint laws of functionals

Solved small ball probability problems for integral functionals

Established asymptotic behavior of certain option prices

Abstract

Let $X$ be a squared Bessel process. Following a Feynman-Kac approach, the Laplace transforms of joint laws of $(U, \int_0^{R_y}X_s^p\,ds)$ are studied where $R_y$ is the first hitting time of $y$ by $X$ and $U$ is a random variable measurable with respect to the history of $X$ until $R_y$. A subset of these results are then used to solve the associated small ball problems for $\int_0^{R_y}X_s^p\,ds$ and determine a Chung's law of iterated logarithm. $(\int_0^{R_y}X_s^p\,ds)$ is also considered as a purely discontinuous increasing Markov process and its infinitesimal generator is found. The findings are then used to price a class of exotic derivatives on interest rates and determine the asymptotics for the prices of some put options that are only slightly in-the-money.