How Heston Parameters Shape the Implied Volatility Smile and Skew

Summary

The document asks what the Heston stochastic volatility parameters mean and how practitioners estimate them. It identifies long-run variance, the speed of mean reversion, and volatility of variance, then relates these parameters to features of the implied volatility curve. In the answer, greater volatility of variance increases the smile's curvature; mean reversion and the correlation between the asset and variance processes both affect at-the-money skew; and long-run variance shifts the overall skew level.

For practical calibration, the response describes fitting the model to market prices of vanilla options using Heston's semi-analytic pricing formula and a numerical optimizer. This is a qualitative guide, not a full parameter estimation recipe: it provides no objective function, constraints, optimizer details, or treatment of the correlation parameter in the original parameter list. The stated effects are also presented as broad relationships, not isolated sensitivities under every market condition.

Key ideas

• The long-run variance parameter influences the overall level of the implied volatility skew.
• Higher volatility of variance produces greater curvature in the volatility smile.
• Mean reversion and correlation between the asset and variance processes both affect at-the-money skew.
• Practitioners can calibrate Heston parameters to vanilla option quotes using its semi-analytic formula and numerical optimization.

Tags

# Please explain Heston Model parameters meaning


# Please explain Heston Model parameters meaning












The Heston Model is given by:

$$ dS_t = \mu S_t dt + \sqrt{v_t}S_tdB_{1t}$$ $$ dv_t = \kappa(\theta - v_t)dt + \xi \sqrt{v_t}dB_{2t}$$.

The parameters are:

$\theta$ is the long term variance

$\kappa$ is the rate at which $v_t$ reverts to $\theta$

$\xi$ is the volatility of the volatility

Can anyone explain what these 3 parameters mean in simple terms? Also, how are these parameters calculated in practice?

## Answer by alexprice (score 2, accepted)

https://quant.stackexchange.com/a/45398

- volatility of the volatility controls convexity of the skew/smile => more vol of vol generates more convex function ( = more smile)

- mean revertion and correlation between brownian motions both control ATM skew.

- long term variance controls overall level of skew (moves whole skew graph higher)

In practice these parameters are calibrated to market quotes of vanilla options using Heston's semi analytic formula and numerical optimizer.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.