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

Abstract

Keywords Complex analysis; Asymptotic expansions; Numerical approximation 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. Author Affiliation: (a) Department of Mathematical Sciences, Yeshiva University, New York, NY 10016, United States of America (b) School of Business, Stevens Institute of Technology, Hoboken, NJ 07030, United States of America * Corresponding author. Article History: Received 17 October 2020; Revised 2 September 2021; Byline: Péter Nándori [peter.nandori@yu.edu] (a), Dan Pirjol [dpirjol@stevens.com] (b,*)