### Model free implied volatility

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*