Journal of Computational Finance

Discrete extrema of Brownian motion and pricing of exotic options

Colin Atkinson, Gianluca Fusai


We provide a closed-form expression for the distribution of the extrema of a Brownian motion observed at discrete times. We reduce the evaluation problem to a Wiener–Hopf integral equation that we solve analytically. Then, we apply the result to price in closed-form discrete monitored exotic options (lookback, quantile and barrier) in the Black–Scholes setting. Numerical results from our formulae are then compared with those from other numerical methods available in the literature. Finally, we discuss the relationship of our result with the well-known Spitzer identity.

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