Why Stochastic Volatility Alone May Miss Short-Dated Skew

Summary

The document summarizes a distinction between local volatility and stochastic volatility models when representing equity option markets. It reports that local volatility can fit observed vanilla option prices but may produce unrealistic forward volatility behavior, including a smile that flattens over time. Stochastic volatility is described as giving more realistic volatility dynamics while struggling to reproduce the skew in short-dated options.

The proposed ways to combine these strengths are local-stochastic volatility models or models with correlated jumps in the stock and volatility. The response notes that jump models can be difficult to calibrate. It points readers to material in Gatheral’s book, which illustrates the stochastic-volatility limitation using a historical SPX data example; that example is not recent, and the document offers no new empirical study or broader test of how the conclusion varies across markets and model specifications.

Key ideas

• Local volatility models can fit vanilla option prices while producing unrealistic future smile dynamics.
• Stochastic volatility models can give more realistic volatility dynamics but may lack enough short-dated skew.
• Combining local and stochastic volatility is presented as one way to address both limitations.
• Correlated price and volatility jumps are another proposed extension, though calibration can be difficult.
• The cited illustration uses historical SPX data and is not a recent empirical study.

Tags

# stochastic vol modelling not enough for smile


# stochastic vol modelling not enough for smile












It seems in practice models that include Stochastic Volatility alone do not have enough power to produce actual observed implied vol surfaces. Is there recent empirical literature documenting this?

## Answer by byouness (score 3)

https://quant.stackexchange.com/a/40323

For diffusion models (i.e. no jumps):

- Local volatility models: match vanilla options market prices; give unrealistic volatility dynamic (smile flattens when we move forward in time);

- Stochastic volatility models: don't match vanilla options market prices (not enough skew for short dated expiries); give realistic volatility dynamics.

You might want to have the best of both worlds, and in this case you have to combine both (local stochastic volatility models), or include correlated stock and volatility jumps (but it's not easy to calibrate this kind of models).

For the doc supporting point 2.1. above, you can refer to chapters 3 and 7 of Gatheral's the Volatility Surface. It's not that recent (uses SPX data of Sep 15th, 2005) but it does illustrate the point and the results are still valid.

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.