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Is a JV model simply Local Vol + Jump Diffusion?

If so, it seems logical that an existing JV model be able to be used for valuation of both Vanilla and Exotic options. Is this true? Does a Local Vol model not model the smile parameters (Skew and Kurtosis) at all, and hence the use of JVol in equity option calculation?

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up vote 4 down vote accepted

Jump volatility is a term sometimes used to describe randomly varying jump sizes in a model with asset value jumps. So strictly speaking it is merely a parameter in generic jump diffusion.

Both local volatility models and jump diffusions end up resulting in skew and kurtosis (of Black-Scholes volatilities). However, they are complementary in practice, at least with sane parameters.

Because jumps tend to "average out" over time, jump-diffusion models have trouble reproducing skew at long tenors. At the same time, it is difficult for a continuous process to achieve the kinds of skews (or equivalently implied probability distributions) observed at short tenors.

Hence, you often see exotics desks combine the two, or at least have both available.

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