Lognormal-Mixture Dynamics and Calibration to Volatility Smiles and Skews
21 Pages Posted: 11 Jul 2001
We introduce a general class of analytically tractable models for the dynamics of an asset price based on the assumption that the asset-price density is given by the mixture of known basic densities. We consider the lognormal-mixture model as a fundamental example, and for the first time we derive the related explicit dynamics and show that it leads to a stochastic differential equation admitting a unique strong solution. We also provide closed form formulas for option prices and analytical approximations for the implied volatility function. We then introduce the asset-price model that is obtained by shifting the previous lognormal-mixture dynamics and investigate its analytical tractability. We finally consider a specific example of calibration to real market option data.
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