The CBOE Volatility Index is designed to measure the market’s expectation of stock market volatility as reflected in S&P 500 option prices. It is a complicated average of several put and call option prices with different strike prices and expiration dates. In general, call options provide the holder with the right to buy an underlying asset at a specified price (the strike price) on or sometimes before a certain date (the expiration date). Put options provide the holder with the right to sell the underlying asset at the strike price on or before the expiration date. Puts and calls both increase in value when the expected volatility of the underlying asset increases.

## Option Pay-Offs

For simplicity, consider a call option where the underlying asset is a single stock. The current stock price is $100 and the option strike price is $110. The option is “out of the money” because the strike price is less than the current strike price. Such an option has no value if it is just about to expire because no investor would pay a strike price of $110 for the stock when it is trading at $100. However, if there is enough time left before the option expires, it may move “in the money.” If, for example, at expiration the stock price is $120, the pay-off is $10 because the stock can be purchased for $110 and immediately sold for $120.

## Measures of Volatility

The most commonly used and cited measure of volatility is based on a stock return’s average squared distance away from its mean. This statistical measure is called the standard deviation, and since it is calculated from historical returns it is also called the realized volatility, which is backward-looking.

The range of a stock price is also a measure of volatility, and is far more simple to calculate. If a stock price bounces between $95 and $105, for example, the range is $105 minus $95 = $10. Implied volatility is a measure based on increases in option prices due to the market’s expectation of future volatility. The VIX is one of several types of implied volatility measures, which are intended to be forward-looking measures.

## Why More Volatility Increases Option Prices

Consider again a call option with a strike price $110 and a current stock price of $100. The price has bounced between $95 and $105 for the past month and the stock has one month left until maturity. Next, a significant event occurs and the markets have a hard time interpreting its impact on the underlying stock price, so the price starts swinging from $85 to $115. The volatility has increased from a range of $10 to a range of $30. Suppose the stock price is now $102. That's still well-below the strike price, but the option is now more valuable than before. This is because when the range was $10, there was little chance that the option would expire in the money, but when the range is $30, it is more likely that at expiration the stock will be in the money.

## The VIX as an Average of Option Prices

At any point in time, several options are trading on the S&P 500 Index. There are both calls and puts, and each has several strike prices and expirations. If there were a perfect option pricing model, only one option price would be needed to infer a forward-looking volatility measure. Although some options pricing models are very good, none have proven to be perfect, so the VIX is based on volatility information implied by several put and call option prices. It can be viewed as a complicated average of option prices. The methodology is detailed on the CBOE VIX website.

### References

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