Transform-based evaluation of prices and Greeks of lookback options driven by Lévy processes

Naser M. Asghari and Michel Mandjes

Journal of Computational Finance

Abstract: ABSTRACT In this paper, we develop a technique, based on numerical inversion, to compute the;prices and Greeks of lookback options driven by Lévy processes. In this setup, the risk neutral;evolution of the stock price, say St , is given by S0eΧt;, with S0 the initial price;and Χt a Lévy process. Lookback option prices are functions of the stock prices ST at;maturity time T and the running maximum;ST;:= sup0≤t≤T;St. As a consequence,;the Wiener-Hopf decomposition provides us with all the probabilistic information;needed to evaluate these prices. To overcome the complication that, in general, only;an implicit form of the Wiener-Hopf factor is available, we approximate the Lévy;process under consideration by an appropriately chosen other Lévy process, for which;the double transform ?e−αXt(q) is known, where t(q) is an exponentially distributed;random variable with mean q−1. The second step is to write the transform of the;lookback option prices in terms of this double transform. Finally, we use state-of-the-art;numerical inversion techniques to compute the prices and Greeks (ie, sensitivities;with respect to initial price S0 and maturity time T). We test our procedure for a;broad range of relevant Lévy processes, including a number of "traditional" models;(Black-Scholes, Merton) and more recently proposed models (Carr-Geman-Madan-Yor (CGMY) and Beta processes), and show excellent performance in terms of speed;and accuracy.

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