How Heston Parameters Shape Implied Volatility Smiles

Summary

The discussion explains how several Heston model parameters affect the shape and level of the implied volatility curve. It focuses on producing a symmetric smile rather than a skewed smirk. The answer identifies correlation between the asset and its variance as a driver of skew, suggesting that a zero correlation can yield a more symmetric shape. It also describes volatility of variance and mean reversion as parameters that influence the strength of the smile.

The initial variance and its long-run level are characterized as affecting the overall level of the curve. The suggested parameter adjustments are qualitative examples rather than a calibration recipe, and the post provides no fitted market data, plots, or validation. The answer also does not specify all implementation details needed to reproduce a particular smile. These relationships are useful as intuition for calibration, but actual outcomes depend on the model setup and the interaction among parameters.

Key ideas

• Asset-variance correlation influences implied volatility skew; setting it to zero can support a symmetric smile.
• Volatility of variance affects the strength of the smile.
• Mean reversion also shapes smile intensity, with higher values described as reducing it.
• Initial variance and its long-run level shift the overall level of the smile.
• The parameter suggestions are qualitative and do not replace calibration to market data.

Tags

# What parameters give a smile (not smirk) in Heston?


# What parameters give a smile (not smirk) in Heston?












I am trying to create a smile in Heston model, however, as of yet, I have only been able to get smirks (i.e., big negative slope ITM that flattens out ATM, and then a very small positive slope OTM).

What parameters ($\kappa, \xi, \rho$ etc) should one use in Heston in order for the prices to actually show a proper smile? --> \_/

## Answer by JejeBelfort (score 2)

https://quant.stackexchange.com/a/33536

The impact of all the 5 parameters to be calibrated are the following:

- $\rho$ impacts the skewness: try $\rho = 0$ to get a symmetric smile

- $\sigma$ impacts the smile effect: you may want to move this parameter in your case. Increasing it (try 0.5 or more for instance) will amplify the smile effect you want.

- $v_0$ and $\theta$: the initial volatility and its long term level. This will only impact the level of your smile

- $\kappa$: as for $\sigma$, this will also modify the smile effect. Increasing $\kappa$ will decrease the smile effect, so you may want to decrease it (to 1 for instance).

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