It has been long observed that asset returns follow a distribution which is far from normal, in particular one that exhibits a substantial degree of excess kurtosis [see for example Fama (1965, Journal of Business)]. Large (positive or negative) returns, that would be considered extreme outliers under the lognormality assumption appear relatively often. Therefore, distributions that exhibit kurtosis which is substantially higher are required. Since such distributions exhibit higher probability mass around the tails, leptokurtic distributions are also often coined fat-tailed. Merton (1976, Journal of Financial Economics), among others, notes that mixtures of normal distributions exhibit fat tails relative to the normal, and therefore models that result in such distributions can be used in order to improve on the Black-Merton-Scholes option pricing results.

Merton, in the first attempt to relax the assumption of a geometric Brownian motion, introduced random jumps in the asset price diffusion. In this fashion, the jump diffusion model creates mixtures of densities that have the potential to match the fat tails that are empirically observed. In the eighties, Garch and stochastic volatility models generate leptokurtosis by mixing a large number of normal distributions with different volatilities.

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