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Zbl 1195.91153
Benhamou, Eric; Gobet, Emmanuel; Miri, Mohammed
Smart expansion and fast calibration for jump diffusions. (English)
[J] Finance Stoch. 13, No. 4, 563-589 (2009). ISSN 0949-2984; ISSN 1432-1122/e

The authors use Malliavin calculus techniques to derive an analytic formula for the price of European options, for any model including local volatility and Poisson jump processes. To perform a rigorous analysis, they use a suitable parameterization that is just a tool to derive convenient representations. By using an asymptotic expansion in the context of small diffusions and small jumps (relative to the frequency or to the size), estimates for the derivatives are established. This allows making an explicit contribution at given order and to control the error. It is proved that the accuracy depends on the smoothness of the payoff function. It is also demonstrated that under realistic parameters, the accuracy is good enough, and model calibration becomes very rapid. It is observed that one may use the approximation price and obtain a volatility smile for short maturities and a volatility skew for long maturities.
[Yuliya Mishura (Ky\"iv)]

MSC 2000:: *91G20
60J75 Jump processes
60H07 Stochastic calculus of variations and the Malliavin calculus
91G80

Keywords: Asymptotic expansion; Malliavin calculus; volatility skew and smile; small diffusion processes; small jump frequency/size

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