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

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.