VIX is a measure of expected volatility calculated as 100 times the square root of the expected 30-day variance (var) of the S&P 500 rate of return. The variance is annualized and VIX expresses volatility in percentage points. This manner of calculating the VIX emerged in September of 2003 and is documented with an example by the CBOE. In this article, the calculation of the VIX is reproduced in a Microsoft Excel template to automate the calculation. Further, one can also apply other option series to calculate a VIX-type analysis for the underlying security, which is of great benefit because the calculation is independent of option-pricing model biases.
The Theory behind the VIX (Volatility) calculation
VIX is obtained as the square root of the price of variance, and this price is derived as the forward price of a particular strip of SPX options. The justification for this derivation is that variance is replicated by delta-hedging the options in the strip. An intuitive explanation of the mechanics of this replication based on Demeterfi, Derman, Kamal and Zou is:
- The price of a stock index option varies with the index level and with its total variance to expiration. This suggests using S&P 500 options to design a portfolio that isolates the variance.
The portfolio which isolates variance is centered around two strips of out-of the money S&P 500 calls and puts. Its exposure to the risk of stock index variations is eliminated by delta hedging with a forward position in the S&P 500.
A clean exposure to volatility risk independent of the value of the stock index is obtained by calibrating the options to yield a constant sensitivity to variance. If each option is weighed by the inverse of the square of its strike price times a small strike interval centered around its strike price, the sensitivity of the portfolio to total variance is equal to one. Holding the portfolio to expiration therefore replicates the total variance.
Arbitrage implies that the forward price of variance must be equal to the forward price of the portfolio which replicates it. Observing that the S&P 500 forward positions in the portfolio contribute nothing to its value, the forward price of variance reduces to the forward price of the strips of options.
VIX Volatility versus Black-Scholes Volatility
The formula for VIX is very different from the Black-Scholes implied volatility familiar to option traders, as is its derivation. VIX is based on a weighted sum of option prices, while a Black-Scholes implied volatility is backed out of an option price. This raises two questions: First, why use VIX rather than a Black Scholes volatility to measure expected volatility. Second, what is the relationship between VIX and a Black Scholes implied volatility?
Why Use VIX?
The Black-Scholes derivation is justified when the instantaneous return of the index is normally distributed with a constant volatility until expiration. Given these assumptions, implied volatilities should be the same at every strike price. However, a notorious pattern called the skew emerged after the stock market crash of October 1987. Since then, Black-Scholes implied volatilities of stock index options have decreased with the strike price. One prominent explanation of the skew in option prices is that the market marks up the prices of out-of-the-money puts to reflect the impact of stochastic variations in volatility on the distribution of equity returns, such as its skew and fatter than normal tails.
The skew of stock index implied volatilities signals that the Black-Scholes model mis-specifies the underlying return, and that random volatility matters a great deal. This suggests using a more consistent and robust measure of expected volatility, one which will not depend on the strike and will not be model-dependent. VIX is such a measure as it does not constrain volatility to be constant. VIX pools the information from options with different strike prices and extracts the full information conveyed by the skew to reconstitute expected volatility.
There is a secondary benefit to using VIX: its construction clearly lays out the portfolio of stock index options and futures that replicates total variance. The possibility of replication facilitates hedging and arbitrage of VIX contracts, and this ensures that the VIX futures price can converge to the special opening quotation of VBI at expiration.
Relationship between VIX and Black-Scholes Implied Volatilities
Dupire and Hagan characterize Black-Scholes implied volatilities when the underlying volatility is not constant. The Black-Scholes implied volatility of an option with strike price K is approximately equal to the expected volatility over the most probable price path whose ending value at expiration is K. This is in contrast to VIX which is the square root of expected variance over all possible volatility paths. Carr and Lee find that a Black-Scholes implied volatility comes closest to expected volatility when the strike is at-the-money.
Mathematical Derivation of VIX
VIX is the square root of the annualized forward price of the 30-day variance of the S&P 500 return. This forward price is based on the replication of total variance by a portfolio of options delta-hedged with stock index futures. The construction of the replicating portfolio and the determination of its forward price from listed S&P 500 option prices is described in the VIX White Paper.
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VIX Calculation in Excel
For a better understanding of the VIX calculation we have made an excel document with two example calculations. The first calculation is an exact copy of the calculation in the VIX White Paper (also using the same data set). The second calculation is an example using a DJX option chain data set. This calculation of the VIX is the same calculation we use in our mathematical models.
This file requires Microsoft Excel running under Microsoft Windows or Mac OS X.
Note: If you intend to distribute a modified version of our calculation, you must ask us for permission first, please contact us for further details. If you intend to distribute an unmodified version of our calculation please provide an appropriate link to this post.