How the Heston Parameters Control the Implied Volatility Surface

In the Heston model, both the dynamic of the asset price and its instantaneous variance nu are stochastic. The model assumes that the variance follows a mean-reverting Cox-Ingersoll Ross process, and it is correlated with the asset price.

The Heston model has five unknown parameters:
– ν0: the initial variance
– κ: the speed of reversion
– θ: the long-term mean of the variance
– ξ: the volatility
– ⍴: the correlation of the two Wiener processes or spot / vol correlation

The implied volatility surface is not flat under the Heston model, and the five parameters have different impacts on the shape of the volatility surface.

The initial variance ν0 and the long-term mean of the variance θ control the level of the implied volatility curve. They control the second moment, the variance, of the underlying asset return distribution implied by option prices.

The spot / vol correlation ⍴ controls the slope, the skew of the implied volatility curve. It controls the third moment, the skewness, of the underlying asset return distribution implied from option prices

The vol of vol ξ controls the smile of the implied volatility curve.
It controls the fourth moment, the kurtosis, of the return distribution implied from option prices.

The term structure of the expected annualised variance is a function of the initial variance ν0, the long-term mean of the variance θ, and the speed of mean-reversion κ.
When the long-term variance is lower (resp. higher) than the initial one, the curve is downward (resp. upward) sloping.
When we increase the speed of reversion, it increases the slope on the front-end of the curve as the volatility will converge to its long-term average in a faster way.

Presentation

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