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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?

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  • $\begingroup$ Add the equations for both stochastic processes to your question please. $\endgroup$
    – will
    Commented Apr 4, 2017 at 20:40

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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.

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