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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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  • Pirjol, Dan

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.

Suggested Citation

  • Pirjol, Dan, 2026. "Conditional distribution of the time-average of a geometric Brownian motion and option pricing in the SABR model," Applied Mathematics and Computation, Elsevier, vol. 529(C).
    • Handle: RePEc:eee:apmaco:v:529:y:2026:i:c:s0096300326001967
    DOI: 10.1016/j.amc.2026.130144
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