## How is VIX Calculated? Step-by-Step

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.

where var = (365/30) x Expected 30-day variance.

The 30-day variance is the sum of squared standard deviations s_{t} ("volatilities") of the S&P 500 rate of return at every point in time t during the 30 days:

For any N, the expected value of the N-day variance of the S&P 500 return can be estimated by the forward price P of a strip of options expiring in N days. This is because the forward price of the strip represents the market's risk-neutral expectation of that variance. The forward price is equal to e^{rt} times the spot price of the option strip, where t expresses the number of days N as a fraction of a year, i.e. t = N/365. The calculation of VIX is actually based on the number of minutes to expiration, and t therefore expresses the number of minutes to expiration as a fraction of a year.

**Expected N-day variance = Forward price of strip of options = P = e ^{rt} Spot price of option strip**

Since 30-day options are usually not available, a 30-day expected variance is inter- or extrapolated as a weighted average of the forward prices P_{1} and P_{2} of two options strips with the two closest nearby expirations T_{1} and T_{2}, but no closer than eight days from their expiration dates. The option with the near-term expiration is dropped from the calculation on the Monday preceding its expiration (3rd Saturday of the month) and a new far-term option is added. The weights used to average the forward prices P_{1} and P_{2} are w = (T_{2} -30)/(T_{2}-T_{1}) and 1- w:

**Expected 30-day variance = wP _{1} + (1-w)P_{2}**

The option strips whose prices are used to calculate VIX are portfolios of out-of-the-money SPX puts and calls, with moneyness referenced to the first strike K_{0} below the forward price F_{0} of the S&P 500. For example if the date-T_{1} forward price of the S&P 500 is 1011, and the date-T_{2} forward price is 1016, then the T_{1} (T_{2}) strip contains puts at strikes at and below 1010 (1015) and calls at strikes at and above 1010 (1015). Each strip includes 2△K/K^{2} calls or puts, where △K is the average of the strike intervals adjacent to the strike K. The price of the option strip is adjusted to compensate for the difference between the forward price of the S&P 500 and a listed strike.

In summary:

or, if 30-day options are available

### The Theory behind the VIX 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^{2} 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^{3} .

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^{4} 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^{5}. 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^{6} 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^{7} 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 below in greater detail.

### Replication of Total Variance

The construction of the portfolio which replicates total variance is based on two observations:

(1) (1) The total variance of the instantaneous rate of return of the S&P 500 over a period T is equal to:

(2) A Taylor expansion of the logarithmic term ln(F_{T}/F_{0}) shows that it can be replicated by trading stock index futures and a continuum of out-of-the-money options

Combining (1) and (2), the total variance is

The first term of the variance isthe return to a position in index futures dynamically rebalanced to maintain a constant dollar exposure to the stock indexthe second term is the return to a static position of F0 index futures held to the date of maturity, and the last term is the return to a static portfolio of out-of-the-money puts and calls, with moneyness defined relative to the forward price F_{0}. This portfolio contains dK/K^{2} puts with strike price K where K is smaller than or equal to F_{0}, and dK/K^{2} calls with strike price K, where K is greater than or equal to F_{0}. At expiration, the return to a put at strike K is ( K- F_{T})^{+} and the return to a call at strike K is (F_{T} - K)^{+}.

The replicating portfolio for total variance assumes continuous trading, a continuum of listed strike prices, and a listed strike at-the-money. Two discrete approximations are needed to assemble and trade this portfolio. First, the dynamic component of the futures position is rebalanced at discrete intervals △_{ti}, second the strip of options consists of all listed puts with strikes at or below K_{0} , and all listed calls with strikes at or above K_{0}, where K_{0} is the closest listed strike below F_{0}. This leads to the discrete approximation:

_{i}is the change in the futures position over △t

_{i}, △K is half the distance between the two strikes adjacent to K, or the distance to the adjacent strike for the initial and final strikes in the put and call series. The last term of the approximation is an adjustment compensating for the fact that the strip is not centered around a strike exactly at-the-money. This term drops out if there is a listed strike at-the-money, i.e. K

_{0}= F

_{0}.

### Forward Pricing of Total Variance

The forward price P of total variance is determined from the discrete approximation to this variance. Specifically, the forward price of total variance is the expected value of this approximation, with the expectation taken under the risk-neutral distribution. Next taking into account the fact that the expected value of the futures positions is zero^{8}, the forward price of total variance simplifies to the forward price of the strip of options plus the adjustment term

where r is the money market rate, and e^{rt} Put_{K} and e^{rt} Call_{K} are the forward prices of out-of-the-money put and calls.

The forward price of a 30-calendar day variance is interpolated from the forward prices P_{1} and P_{2} of variances over the terms T_{1} and T_{2} of nearby and second-nearby listed options, with the nearby option at least a week from expiration. This price is annualized and VIX is the square root.

* The methodology of the CBOE Volatility Index is owned by CBOE and may be covered by one or more patents or pending patent applications.