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Extreme value theory for geometric Brownian motion and pricing of short maturity barrier options

Ze-An Ng

TL;DR

The paper develops an extreme value framework for continuous-path processes by conditioning Brownian motion and geometric Brownian motion on large running maxima. It proves that Brownian paths converge deterministically to a linear path and geometric Brownian paths converge to a deterministic exponential curve under suitable normalization, with a rate of convergence $O(\sqrt{T})$. Building on this, it derives short-maturity barrier option price asymptotics with quantitative error bounds and exact results for European, Asian, and lookback payoffs. The results bridge extreme-value theory and financial mathematics, delivering practical closed-form asymptotics for barrier options in the short time horizon and offering precise probabilistic justifications for the deterministic limiting behavior of conditioned paths.

Abstract

We investigate the limiting distribution of geometric Brownian motion conditional on its running maximum taking large values. We show that the conditional distribution of the geometric Brownian motion converges after a suitable normalization to a deterministic exponential curve. We obtain quantitative bounds on the rate of convergence. Analogous results are shown for the Brownian motion, which converges to a straight line. As an application of our results to financial mathematics, we obtain 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 European, Asian and lookback style payoffs.

Extreme value theory for geometric Brownian motion and pricing of short maturity barrier options

TL;DR

The paper develops an extreme value framework for continuous-path processes by conditioning Brownian motion and geometric Brownian motion on large running maxima. It proves that Brownian paths converge deterministically to a linear path and geometric Brownian paths converge to a deterministic exponential curve under suitable normalization, with a rate of convergence . Building on this, it derives short-maturity barrier option price asymptotics with quantitative error bounds and exact results for European, Asian, and lookback payoffs. The results bridge extreme-value theory and financial mathematics, delivering practical closed-form asymptotics for barrier options in the short time horizon and offering precise probabilistic justifications for the deterministic limiting behavior of conditioned paths.

Abstract

We investigate the limiting distribution of geometric Brownian motion conditional on its running maximum taking large values. We show that the conditional distribution of the geometric Brownian motion converges after a suitable normalization to a deterministic exponential curve. We obtain quantitative bounds on the rate of convergence. Analogous results are shown for the Brownian motion, which converges to a straight line. As an application of our results to financial mathematics, we obtain 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 European, Asian and lookback style payoffs.
Paper Structure (6 sections, 8 theorems, 116 equations)

This paper contains 6 sections, 8 theorems, 116 equations.

Table of Contents

  1. Introduction
  2. Literature Review
  3. Heuristics
  4. Main Theorem
  5. Pricing of Short Maturity Barrier Options
  6. Additional Proofs

Key Result

Theorem 1

Let $X$ be the solution to the SDE with $W$ a standard one dimensional Brownian motion, and $\mu, \sigma>0$ constants. Let $B>1$ be arbitrary. For every $T>0$, let $A_{T}$ denote the event and let $\mathbb{P}_{T}$ be the probability measure given by for all events $E$. Denote by $\mathbb{E}_{\mathbb{P}_{T}}$ the expectation under $\mathbb{P}_{T}$. Then we have as $T \rightarrow 0^{+}$, where t

Theorems & Definitions (17)

  • Theorem 1: Extreme value theorem for geometric Brownian motion
  • Lemma 2
  • proof
  • Lemma 3
  • proof
  • Lemma 4
  • proof
  • Lemma 5: Hitting time bounds
  • proof
  • proof : Proof of Theorem 1
  • ...and 7 more