Efficient Calibration of Time-Changed Lévy Models to Forward Implied Volatility Surfaces

S. Kassberger and H. Schmidt (Germany)


FFT; L´evy processes; changes of time; forward start op tions


Time-changed L´evy models are capable of accurately cal ibrating implied volatilities of plain vanilla options across strikes and maturities at a fixed point in time, a feature that distinguishes them from most other classes of option pric ing models. However, the quality of a pricing model is not only determined by its static fitting capabilities, but also by its dynamic properties, in particular if it is to be applied to the pricing of exotic derivatives. In this paper, we in vestigate the dynamic properties of a popular time-changed L´evy model by first calibrating it to a set of S&P 500 index options and then studying the forward implied volatilities it gives rise to. The main tools in our endeavor are forward character istic functions in conjunction with the Fast Fourier Trans form (FFT) approach to option pricing. After showing how to adapt the FFT-approach to the pricing of forward start options and to the efficient computation of forward im plied volatility surfaces, we derive the forward character istic functions for our model. We find that forward implied volatility surfaces are largely undeterminedfor a model that is calibrated to vanilla options only, and show how a trader can take advantage of this indeterminacy by incorporating his personal view of the future in the calibration. Our theo retical discussion is supplemented by numerical and graph ical illustrations.

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