Journal of Computational Finance

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

Naser M. Asghari and Michel Mandjes

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 S:= sup0tT 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 𝔼eαXt(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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