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At this link I have asked what is the market standard when pricing options in different asset classes. Based on the answers, the standard for FX and equities seems to be the local-stochastic volatility models. My question is how can such a model be calibrated in practice for FX. The reason why I mention FX specifically is that I can see that only 3 strikes are quoted in the market (those embedded in the ATM straddle and in the 25 delta risk reversal and butterfly). If also the 10 delta are quoted, there are 5 quotes. But how can that be enough to successfully calibrate on a FX surface? Is there any example that shows how to calibrate, for example, Heston local volatility model?

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Have a look at this paper.

This is a rather exhaustive paper summarizing 9 models to be used as Local-Stochastic Volatility (LSV) models.

It describes various aspects of Calibration and Pricing of LSV models with the associated references, so that you can dig deeper in the topic should you find a suitable model for your needs.

The beginning of the abstract goes as follows:

We analyze in detail calibration and pricing performed within the framework of local stochastic volatility LSV models, which have become the industry market standard for FX and equity markets. We present the main arguments for the need of having such models, and address the question whether jumps have to be included. We include a comprehensive literature overview, and focus our exposition on important details related to calibration procedures and option pricing using PDEs or PIDEs derived from LSV models.

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  • $\begingroup$ Thanks, but is there something to show the application to FX surfaces where only few strikes are traded? $\endgroup$ – opt Mar 21 at 9:49
  • $\begingroup$ I think Uwe Wystup did some work on this. Have a look at The Simplified Formula paragraph on page 16 of mathfinance.com/wp-content/uploads/2017/06/…. If I remember, he uses only 3 anchor points to calibrate the volatility smile. $\endgroup$ – JejeBelfort Mar 21 at 10:02
  • $\begingroup$ But this is not in the context of local stochastic volatility right? $\endgroup$ – opt Mar 21 at 17:59
  • $\begingroup$ I am mostly curious how the actual calibration on FX does not suffer from the fact that only few strikes or deltas are available at a given time $\endgroup$ – opt Mar 21 at 18:00
  • $\begingroup$ I see. Then have a look at the robustness of Local Volatility (LV) and Stochastic Volatility (SV) (i.e. Heston) models against scarce data separately. Then, as the LSV can be seen as some "weighted-average" of LV and SV, you could introduce some weight factor in order to gear your model towards the one which performs best against scarce data. $\endgroup$ – JejeBelfort Mar 22 at 1:26

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