Risk-Neutral Skewness and Kurtosis:
See this post for R code to calculate model free implied volatility.
Let stock price return for period
is given by:
(1)
The price of the volatility contract:
(2)
The price of the cubic contract:
(3)
The price of the quadratic contract:
(4)
Define :
(5)
For -period model-free implied volatility (MFIV) is:
(6)
For -period model free implied skewness(MFIS) is:
(7)
To calculate the integral in equation (ref{ref1}), (ref{ref2}), and (ref{ref3}), I require continuous option prices from 0 to . For simplicity, I set lower limit and upper limit for integrals.
In the calculation, I use lower limit of strike price as the minimum strike price for OTM put option with non zero contract size (or trade quantity).
For upper limit for strike price, , I use the maximum strike price of OTM call option with non zero trading size.
References:
Bakshi, G., Kapadia, N., & Madan, D. (2003). Stock return characteristics, skew laws, and the differential pricing of individual equity options. Review of Financial Studies, 16(1), 101-143.
Jiang, George J., and Yisong S. Tian. “The model-free implied volatility and its information content.” Review of Financial Studies 18.4 (2005): 1305-1342