Option Chain Implied Volatility Calculation

Question

I have the following EOD options data for the SPY containing IV data for each strike.

Date        Symbol  Exp         Strike  P/C  ImpVol
2015-07-01, SPY,    2015-07-10, 185.5,  C,   0.272986
2015-07-01, SPY,    2015-07-10, 186,    C,   0.267097
2015-07-01, SPY,    2015-07-10, 186.5,  C,   0.261214
2015-07-01, SPY,    2015-07-10, 187,    C,   0.255573
.
.

I'd like to calculate the IV for the SPY Option Chain using this data.

I believe the Option Chain IV is related to the ATM strike IV, but I'm not 100% sure how ThinkOrSwim calculates it.

Is there a formula I can use to calculate the Option Chain IV?

Answer 1

Using that data the best way to compute implied volatility is tho use the methodology to approximate the variance swap rate closely following the model-free estimate proposed by Demeter et al. (1999) and Carr and Madan (1998) who show that if one owns a portfolio of options across all strikes inversely weighted by the squared strike then one gets a variance exposure that does not depend on the price. The variance swap rate or implied volatility is approximated by: \begin{equation} \sigma_{i,t,\tau}^2=\int_{S_i(t)}^{\infty}\frac{2\Big(1-\log[\frac{K}{S_i(t)}]\Big)}{K^2}C_i(t,\tau,K)dK+\int_{0}^{S_i(t)}\frac{2\Big(1-\log[\frac{K}{S_i(t)}]\Big)}{K^2}P_i(t,\tau,K)dK \end{equation}