Estimating Option-Chain Implied Volatility from Across-Strike Prices

Summary

The document addresses how to summarize implied volatility across an option chain rather than relying only on the at-the-money option. It describes a model-free variance-swap approximation built from option prices over a range of strikes: calls above the underlying price and puts below it. The strike contributions are weighted by a function involving the squared strike and the log ratio of strike to underlying price.

The explanation attributes this approach to prior variance-swap literature and notes that the weighted portfolio provides variance exposure that is not dependent on the underlying price in the way a single option is. The supplied material gives the integral expression but no worked calculation from the sample implied-volatility data, implementation details for discrete strikes, or comparison with a broker’s chain-level convention. It therefore does not establish that this is the formula used by any particular platform.

Key ideas

• A chain-level volatility estimate can be based on option prices across many strikes.
• The described model-free approach combines out-of-the-money calls and puts around the underlying price.
• Strike-dependent weights are used to approximate a variance-swap rate.
• The document provides a continuous-strike expression but no discrete-data procedure or platform-specific convention.

Tags

# Option Chain Implied Volatility Calculation


# Option Chain Implied Volatility Calculation












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 by phdstudent (score 2, accepted)

https://quant.stackexchange.com/a/22581

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}

Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)

This summary was written by Stratmill's research agent from the original; it is not a copy of the source.