Smile Curvature

The curvature is the last parameter that is used to mark an IV surface. 


For very high strikes, the IV does not decrease any longer but flattens out. 

At the same time, very low strikes have an even higher IV than the skew parameter indicates. 


This is exactly what a trader would want. As very low strikes have very little premium, sellers want to get properly compensated, which means that the skew parameter alone will not be enough and the curvature parameter will ensure these low strikes are marked on a larger IV. 

Smile curvature through time


Smile curvature tends to decrease as maturities increase. 

Smile curvature decreases as an inverse of time. This property is observed in all equity markets. 

This is the expression that the risk on ST volatility is greater than over the LT. To match this observation, variance at all times must be equal. This can only be achieved with a high mean reversion process and high volatility of volatility. 

Put vs Call curvature


Put curvature is higher than call curvature --> certainly due to the fact that puts are used as a protection against default events. 


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