Conditional distribution of the time-average of a geometric Brownian motion and option pricing in the SABR model

Dan Pirjol

doi:10.1016/j.amc.2026.130144

Conditional distribution of the time-average of a geometric Brownian motion and option pricing in the SABR model

Dan Pirjol

School of Business

Research output: Contribution to journal › Article › peer-review

Abstract

An analytical approximation for the distribution of the time-average of a geometric Brownian motion conditional on its terminal value is proposed, following from an asymptotic expansion for the Hartman-Watson distribution. An error bound is available for the approximation error. We study the performance of this approximation and show good agreement with exact results for the density and moments of this distribution, for sufficiently small maturity. As an application we discuss option pricing in the correlated log-normal SABR model, and demonstrate good numerical performance for cases of practical interest.

Original language	English
Article number	130144
Journal	Applied Mathematics and Computation
Volume	529
DOIs	https://doi.org/10.1016/j.amc.2026.130144
State	Published - 15 Nov 2026

Keywords

Asymptotic expansions
Option pricing
Stochastic processes

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10.1016/j.amc.2026.130144

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abstract = "An analytical approximation for the distribution of the time-average of a geometric Brownian motion conditional on its terminal value is proposed, following from an asymptotic expansion for the Hartman-Watson distribution. An error bound is available for the approximation error. We study the performance of this approximation and show good agreement with exact results for the density and moments of this distribution, for sufficiently small maturity. As an application we discuss option pricing in the correlated log-normal SABR model, and demonstrate good numerical performance for cases of practical interest.",

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AB - An analytical approximation for the distribution of the time-average of a geometric Brownian motion conditional on its terminal value is proposed, following from an asymptotic expansion for the Hartman-Watson distribution. An error bound is available for the approximation error. We study the performance of this approximation and show good agreement with exact results for the density and moments of this distribution, for sufficiently small maturity. As an application we discuss option pricing in the correlated log-normal SABR model, and demonstrate good numerical performance for cases of practical interest.

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KW - Stochastic processes

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Conditional distribution of the time-average of a geometric Brownian motion and option pricing in the SABR model

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Keywords

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