Model free implied volatility

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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:

R(t,tau) equiv ln[S(t, tau)]-ln[S(t)]

(1)   \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:

(2)   \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:

(3)   \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:

(4)   \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):

(5)   \begin{equation*} mu(t,tau) = e^{rtau}-1-frac{e^{rtau}}{2}V(t,tau)-frac{e^{rtau}}{6}W(t,tau)-frac{e^{rtau}}{24}X(t,tau) \end{equation*}

For tau-period model-free implied volatility (MFIV) is:

(6)   \begin{equation*} MFIV(t, tau) = (V(t, tau))^{1/2} \end{equation*}

For tau-period model free implied skewness(MFIS) is:

(7)   \begin{equation*} MFIS(t, tau) = frac{e^{rtau}W(t,tau)-3mu(t,tau)e^{rtau}V(t,tau)+2(mu(t,tau))^3}{(e^{rtau}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