Smile Dynamics and Further Topics in Stochastic Volatility

Summary

The document asks what to study beyond prominent stochastic-volatility models such as Hull–White, Heston, and SABR. It identifies incompleteness as one important issue: stochastic-volatility models may admit multiple equivalent martingale measures, so pricing requires a way to choose among them. The questioner also mentions selecting a measure that is close to the real-world measure, and seeks further topics that would deepen understanding of the field.

The answer recommends Lorenzo Bergomi’s series on smile dynamics. It frames the distinction as moving from models that fit the current implied-volatility surface, including its term structure and skew, to models that also describe how that vanilla-options market evolves. The document does not summarize the articles or compare their methods, and offers no derivations or empirical evidence. Its useful contribution is therefore a reading direction and a conceptual distinction, rather than a comprehensive map of open problems in stochastic-volatility modeling.

Key ideas

• Stochastic-volatility models can be incomplete, leaving multiple equivalent martingale measures available for pricing.
• Choosing a pricing measure, including one close to the real-world measure, is raised as a modeling issue.
• Smile models describe the current implied-volatility surface, including its term structure and skew.
• Smile-dynamics models also express a view on how the vanilla options market evolves.
• The answer recommends further reading on smile dynamics but does not explain the articles’ methods.

Tags

# Extended Areas on Stochastic Volatility Modelling


# Extended Areas on Stochastic Volatility Modelling












I'm interested in the areas surrounding Stochastic Volatility Modelling. I've read up on the main models that are prominent in the literature (Hull White, Heston, SABR) but I was wondering what the other issues facing this field are.

Incorporating Stochastic Volatility makes an incomplete model and I've read a few papers on how one may potentially counteract this incompleteness. This also leads from the fact that there could be more than one EMM which turns the asset price process into a martingale, so I have also read up on choosing an optimal measure which is closest to the real world measure.

Does anyone know about other areas like the two I have mentioned which would be good to read up on to solidify my understanding of Stochastic Volatility?

Thanks!

## Answer by Quantuple (score 4)

https://quant.stackexchange.com/a/27464

Great reads to further explore and better understand stochastic volatility models are the series of articles "Smile Dynamics" by Lorenzo Bergomi.

As the name indicates the idea is to study stochastic volatility models not only as "smile models" (in the sense that SV models can be used to capture the state of the vanilla market by correctly accounting for implied volatility term structure and skew = static picture) but also as "smile dynamics models" (in the sense that they embed a view on the evolution of that vanilla market).

Smile Dynamics I

Smile Dynamics II

Smile Dynamics III

Smile Dynamics IV

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.