How Local and Jump Volatility Models Shape Option Skew

Summary

The document explains how local volatility and jump diffusion models can each produce skew and kurtosis in Black–Scholes implied volatilities. It clarifies that “jump volatility” can refer to randomly varying jump sizes, which are parameters within a general jump diffusion model, rather than a separate model class by itself.

The models have complementary strengths. Jump effects tend to average out over longer horizons, making jump diffusion less effective at reproducing long tenor skew. Continuous local volatility models, meanwhile, can struggle to generate the pronounced skew or implied distributions seen at short tenors. The answer therefore describes combining the approaches, or keeping both available, for pricing vanilla and exotic options. It offers a qualitative modeling distinction rather than a calibration procedure or empirical comparison, and it does not claim that either model uniquely captures all observed smiles.

Key ideas

• Jump volatility can describe randomness in jump sizes within a jump diffusion model.
• Both local volatility and jump diffusion can generate implied skew and kurtosis.
• Jump diffusion may have difficulty sustaining observed skew at longer tenors.
• Continuous local volatility may struggle to reproduce pronounced short tenor skew.
• Combining the models can provide complementary capabilities for option valuation.

Tags

# What are the main differences in Jump Volatility and Local Volatility


# What are the main differences in Jump Volatility and Local Volatility












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?

## Answer by Brian B (score 5, accepted)

https://quant.stackexchange.com/a/502

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.

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.