On the distribution of the time-integral of the geometric Brownian motion

TL;DR

This paper develops series expansions and numerical methods to accurately evaluate the distribution of the time-integral of geometric Brownian motion and its joint distribution with the terminal value, aiding in stochastic volatility modeling and Asian option pricing.

Contribution

It introduces new series expansions and complex analysis techniques for precise numerical evaluation of these distributions, with applications to financial modeling.

Findings

Derived convergent series expansions for the distributions.

Determined convergence radius and asymptotics of expansion coefficients.

Constructed an efficient numerical approximation for joint distribution.

Abstract

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.