Discrete extrema of Brownian motion and pricing of exotic options

Colin Atkinson · Gianluca Fusai

19 citations0/0 files storedThe Journal of Computational Finance

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Stochastic processes and financial applications Probability and Statistical Research Probability and Risk Models

Abstract

ABSTRACT 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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References

1973The Pricing of Options and Corporate Liabilities 2005Theory of rational option pricing 1996Handbook of Brownian Motion — Facts and Formulae 1994Option pricing: Mathematical models and computation 2002Non-Gaussian Merton-Black-Scholes Theory 1979Path Dependent Options: "Buy at the Low, Sell at the High"1997A Continuity Correction for Discrete Barrier Options 1991Path Dependent Options: The Case of Lookback Options

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