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 $\tau$ is given by:

$$\begin{equation} R(t,\tau) \equiv ln[S(t, \tau)]-ln[S(t)] \end{equation}$$

$$\begin{equation} H[S]\left\{\begin{matrix} V(t,\tau) \equiv R(t, \tau)^{2}    \text{              Volatility contract }  & \\W(t,\tau) \equiv R(t, \tau)^{3} \text{            Cubic contract }  & \\ X(t,\tau) \equiv R(t, \tau)^{4}\text{       Quadrartic contract }  & \end{matrix}\right. \end{equation}$$

The price of the volatility contract:
$$\begin{equation} V(t,\tau)=\int_{S(t)}^{\infty }\frac{2(1-ln(K/S(t)))}{K^{2}}C(t,\tau;K)dK+\int_{0}^{S(t)}\frac{2(1-ln(K/S(t)))}{K^{2}}P(t,\tau;K)dK \label{ref1} \end{equation}$$

The price of the cubic contract:
$$\begin{equation} W(t,\tau)=\int_{S(t)}^{\infty }\frac{6ln[\frac{K}{S(t)}]-3([\frac{K}{S(t)}])^2}{K^{2}}C(t,\tau;K)dK+\int_{0}^{S(t)}\frac{6ln[\frac{K}{S(t)}]+3([\frac{K}{S(t)}])^2}{K^{2}}P(t,\tau;K)dK \label{ref2} \end{equation}$$

The price of the quadratic contract:
$$\begin{equation} X(t,\tau)=\int_{S(t)}^{\infty }\frac{12(ln[\frac{K}{S(t)}])^2-4([\frac{K}{S(t)}])^3}{K^{2}}C(t,\tau;K)dK+\int_{0}^{S(t)}\frac{12(ln[\frac{K}{S(t)}])^2-4([\frac{K}{S(t)}])^3}{K^{2}}P(t,\tau;K)dK \label{ref3} \end{equation}$$

Define $\mu(t, \tau)$:
$$\begin{equation} \mu(t,\tau) = e^{r\tau}-1-\frac{e^{r\tau}}{2}V(t,\tau)-\frac{e^{r\tau}}{6}W(t,\tau)-\frac{e^{r\tau}}{24}X(t,\tau) \end{equation}$$

For $\tau$-period model-free implied volatility (*MFIV*) is:
$$\begin{equation} MFIV(t, \tau) = (V(t, \tau))^{1/2} \end{equation}$$


For $\tau$-period model free implied skewness(*MFIS*) is:
$$\begin{equation} MFIS(t, \tau) = \frac{e^{r\tau}W(t,\tau)-3\mu(t,\tau)e^{r\tau}V(t,\tau)+2(\mu(t,\tau))^3}{(e^{r\tau}V(t,\tau)-(\mu(t,tau))^2)^{3/2}} \end{equation}$$

To calculate  the integral in equation ($\ref{ref1}$), ($\ref{ref2}$), and ($\ref{ref3}$), I require continuous option prices from 0 to $\infty$. For simplicity, I set lower limit and upper limit for integrals.

In the calculation, I use lower limit of strike price $K_{L}$ as the minimum strike price for OTM put option with non zero contract size (or trade quantity).

For upper limit for strike price, $K_{U}$, 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