  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.
