How Heston Correlation and Volatility of Volatility Shape the Smile
Summary
The note gives an intuition for how the Heston stochastic-volatility parameters correlation and volatility of volatility affect implied-volatility shape. It frames the smile through the risk-neutral distribution: heavier tails raise out-of-the-money option values relative to a constant-volatility Gaussian benchmark, while skewness creates different implied volatilities across puts and calls.
Negative correlation links falling spot prices with rising variance, tending to thicken the left tail and produce a downward-sloping skew. Positive correlation tends toward the opposite asymmetry. Increasing volatility of volatility generally makes the smile or skew more pronounced by widening the range of volatility outcomes; with zero volatility of volatility, the explanation approaches the flat-smile Black–Scholes case. These are qualitative intuitions, not universal quantitative predictions: the result also depends on other model parameters and maturity. The note suggests using moments from the Heston characteristic function for a more rigorous account but does not carry out that analysis.
Key ideas
- The implied-volatility smile reflects tail thickness and asymmetry in the risk-neutral return distribution.
- Negative spot–variance correlation tends to create a heavier left tail and a downward-sloping skew.
- Positive correlation tends to shift the asymmetry toward the opposite side.
- Higher volatility of volatility generally makes the smile or skew more pronounced.
- With zero volatility of volatility, the described model reduces to a flat-smile Black–Scholes case.
Tags
Full text
# parameters in Heston model and their impact on volatility smile
# parameters in Heston model and their impact on volatility smile
Consider the Heston model given by the following set of stochastic differential equations: $$\frac{dS_{t}}{S_{t}}=\mu_{t}dt+\sqrt{V_{t}}dW_{t}, S_{0}>0,$$ $$dV_{t}=\kappa(\theta-V_{t})dt+\xi\sqrt{V_{t}}dZ_{t}, V_{0}=v_{0}>0,$$ $$d<W,Z>_{t}=\rho dt$$ where $W_{t}$ and $Z_{t}$ are two brownian motion, $\kappa,\theta,\xi>0, \rho\in(-1,1).$ I don't understand what impact would $\rho$ have on the shape of volatility smile when it's negative or positive. In addition, How the volatility smile would change if $\xi$ increases? Thank you.
## Answer by Kiwiakos (score 6, accepted)
https://quant.stackexchange.com/a/17718
Intuition: You can think of the vol smile as a reflection of the risk neutral distribution (compared to the Black Scholes Gaussian density). A fat tailed distribution creates the smile: fat tail -> higher prob of exercise than Gaussian with constant stdev -> higher option price than BS with ATM vol -> higher implied vol for given strike. Skewed distributions cause skewed smile: neg skew -> left tail thicker than right -> OTM put impl vol higher than OTM call impl vol.
Also you can think of a stoch vol model as mixture of Gaussians, each with different volatility. The way that this mix is produced generates fat tailed and skewed distributions.
Non-zero Rho will produce asymmetric volatility smiles (that look more like skews). Negative correlation means that a negative spot shock is more likely accompanied by a positive vol shock -> as spot goes down we mix with higher vols -> will produce a fatter left tail than right -> negatively skewed risk neutral density -> downward sloping skew.
Large Xi produces a steeper, more pronounced smile/skew, as we increase the vol-of-vol potentially mixing with higher volatilities which generate more leptokurtic risk neutral distributions. Zero Xi would correspond to the Black Scholes case where the smile becomes flat.
To be more rigorous you can take the Heston char function, take derivatives to compute moments and show the relation of Rho/Xi with Skewness/Kurtosis.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.