  We study the numerical evaluation of several functions appearing in the small
time expansion of the distribution of the time-integral of the geometric
Brownian motion as well as its joint distribution with the terminal value of
the underlying Brownian motion. A precise evaluation of these distributions is
relevant for the simulation of stochastic volatility models with log-normally
distributed volatility, and Asian option pricing in the Black-Scholes model. We
derive series expansions for these distributions, which can be used for
numerical evaluations. Using tools from complex analysis, we determine the
convergence radius and large order asymptotics of the coefficients in these
expansions. We construct an efficient numerical approximation of the joint
distribution of the time-integral of the gBM and its terminal value, and
illustrate its application to Asian option pricing in the Black-Scholes model.
