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Volume 10/Number 3, Spring 2007 Research Papers Discrete extrema of Brownian motion and pricing of exotic options Colin Atkinson
Department of Mathematics, Imperial College, London SW7 2AZ, UK; email: c.atkinson@imperial.ac.uk Gianluca Fusai
Dipartimento SEMEQ, Università degli Studi del Piemonte Orientale, 28100 Novara, Italy and Associate Fellow, Financial Options Research Centre, Warwick Business School, The University of Warwick, Coventry CV4 7AL, UK; email: gianluca.fusai@eco.unipmn.it 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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