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math.PR updates on arXiv.org

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Extreme value theory for geometric Brownian motion and pricing of short maturity options
[Submitted on 12 May 2025 (v1), last revised 16 Jun 2026 (this v · 2026-06-17 · via math.PR updates on arXiv.org

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Abstract:We investigate the limiting distribution of geometric Brownian motion conditional on its running maximum taking large values. The Freidlin-Wentzell large deviations theory predicts that the conditional distribution of the sample paths converge weakly to a deterministic exponential curve. We complement this result by showing that the conditional sample paths in fact converge in strong sense, and obtain quantitative bounds on the rate of convergence. As an application of our results to financial mathematics, we obtain new closed form asymptotic formulae for the fair price of barrier options with general path dependent payoff in the short maturity limit, with quantitative error estimates. We provide exact formulae for Asian and lookback style payoffs.

Submission history

From: Ze-An Ng [view email]
[v1] Mon, 12 May 2025 20:08:04 UTC (14 KB)
[v2] Tue, 16 Jun 2026 06:59:20 UTC (14 KB)