Geometric Brownian motion with affine drift and its time-integral

TL;DR

This paper derives the joint distribution of a geometric Brownian motion with affine drift and its time-integral using Heun equations, extending previous results and applying them to Asian option pricing with demonstrated accuracy.

Contribution

It introduces a novel approach using the double-confluent Heun equation to analyze geometric Brownian motion with affine drift and its time-integral, expanding on classic methods.

Findings

Derived joint distribution using Heun equations

Extended methodology to affine drift case

Validated approach with numerical Asian option pricing

Abstract

The joint distribution of a geometric Brownian motion and its time-integral was derived in a seminal paper by Yor (1992) using Lamperti's transformation, leading to explicit solutions in terms of modified Bessel functions. In this paper, we revisit this classic result using the simple Laplace transform approach in connection to the Heun differential equation. We extend the methodology to the geometric Brownian motion with affine drift and show that the joint distribution of this process and its time-integral can be determined by a doubly-confluent Heun equation. Furthermore, the joint Laplace transform of the process and its time-integral is derived from the asymptotics of the solutions. In addition, we provide an application by using the results for the asymptotics of the double-confluent Heun equation in pricing Asian options. Numerical results show the accuracy and efficiency of this new method.