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Smart expansion and fast calibration for jump diffusions. (English) Zbl 1195.91153
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.

MSC:
91G20 Derivative securities (option pricing, hedging, etc.)
60J75 Jump processes (MSC2010)
60H07 Stochastic calculus of variations and the Malliavin calculus
91G80 Financial applications of other theories
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