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

Naser M. Asghari and Michel Mandjes

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 S_t , is given by S₀e^Χ_t , with S₀ the initial price and Χ_t a Lévy process. Lookback option prices are functions of the stock prices S_T at maturity time T and the running maximum S_T := sup_0≤t≤T S_t. 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^−αX_t(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.