Abstracts

Asset price models based on Lévy processes, e.g. exponential Lévy, are well established in literature. Several methods are available to price and hedge related options. One of these methods is based on Fourier transforms. This leads to formulas for the option price and the Greeks in terms of the characteristic function of the driving Lévy process and of the Fourier transform of the payoff function for European vanilla options, see [3]. On the other hand since Lévy models imply incomplete markets, perfect hedging is impossible. However partial hedging can be reached via quadratic hedging strategies. For markets observed in a martingale, resp. semimartingale, setting the quadratic hedging strategies are computed by Fourier transform techniques in [5], resp. [4].
To compute the option price, the Greeks, or the position in the quadratic hedging strategies at any date before time of maturity, simulated prices of the underlying asset are required. However simulation of Lévy processes with infinite activity is hard. The approximation introduced in [1], based on replacing the jumps with absolute size smaller than s of a Lévy process by a scaled Brownian motion, facilitates this simulation issue. One could also look at this approximation from a modelling point of view, since one can choose to consider infinitely small variations coming either from a Brownian motion, or from a Lévy process with infinite activity. For s tending to zero, the approximation clearly converges in distribution to the original Lévy process. Even though convergence of asset prices does not necessarily imply the convergence of option prices, it was proved in [2] that for the considered models the related option prices and the deltas are robust. The question remained whether quadratic hedging is also robust.
We reconsidered the robustness conditions in [2] and proved the convergence of the quadratic hedging strategies. Moreover we computed convergence rates. In other words, it is justified to use the approximation and facilitate simulations with it. Besides, we discuss the integrability conditions appearing in the Fourier based approach more carefully.

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[4] Hubalek, F., Kallsen, J., Krawczyk, L.: Variance-optimal hedging for processes with stationary independent increments. Ann. Appl. Probab. 16, 853-885 (2006)
[5] Tankov, P.: Pricing and hedging in exponential Lévy models: review of recent results. Paris-Princeton Lecture Notes in Mathematical Finance (2010)