How Heston and SABR Parameters Shape Implied Volatility Smiles

Summary

The document asks how the parameters of the SABR and Heston stochastic-volatility models relate, with particular interest in their effects on implied-volatility smile features such as skewness and kurtosis. It also asks how Heston parameters would vary when calibrated to a SABR smile whose parameters are held fixed, and seeks relevant literature.

The response points to work by Hagan and coauthors on implied-volatility formulas for Heston models. It says that an expansion of Heston implied volatility can be matched to a SABR expansion, yielding an explicit correspondence between model parameters. However, the discussion provides no equations, parameter mapping, calibration example, or detail about the assumptions and approximation range behind that correspondence. It therefore directs readers toward a relevant analytical framework but does not itself explain how to map individual parameters or predict calibrated values. The claimed relationship should be consulted in the cited model treatment before being used for quantitative calibration.

Key ideas

• The question concerns correspondence between SABR and Heston parameters through their implied-volatility smiles.
• Both models' parameters influence smile shape, including skew and curvature-related features.
• The response points to matching expansions of the models' implied volatilities as a way to derive parameter correspondence.
• No explicit mapping, equations, or calibration demonstration appears in the discussion.
• Any practical parameter conversion depends on the assumptions and scope of the approximation used.

Tags

# Relationship between SABR and Heston


# Relationship between SABR and Heston












What is the relationship between SABR parameters $\sigma, \alpha, \beta, \rho$ and heston parameters $\nu, \kappa, \theta, \xi, \rho$?

How do they influence the smile; skewness, kurtosis, etc? And how are they related to one another, i.e., if I calibrate Heston to SABR for some fixed SABR-parameters, how will the estimated Heston parameters depend on the fixed SABR parameters?

Is there some good literature on the relationship between these two?

## Answer by jherek (score 1)

https://quant.stackexchange.com/a/47336

The relationship between the two models is described in details in Implied Volatility Formulas for Heston Models by Hagan et al. In particular an expansion of the implied volatility under the Heston model that matches the one of a SABR model is described. It gives an explicit correspondence between the parameters of each model.

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.