Realized volatility is a standard deviation of daily price fluctuations of an underlying asset, expressed as a percentage. Historical volatility measures the size of price fluctuations over a certain period of time. As a rule, the value of volatility is translated into an annual expression. For example, on July 31, 2014 the volatility is translated into an annual expression. The S&P 500 index has fallen by 1.75%. Thus, in one day the historical volatility was:

1,75% x sqrt(252) = 27,8%

**Historical volatility**

When determining the price of an option, traders try to predict the future realized volatility of the asset price. After an option is exercised, we can look back and calculate what the historical volatility was during the lifetime of the option.

**Implied volatility**

Implied volatility is the volatility that, when substituted with the Black-Scholes formula, will give the market price of the option. Implied volatility is the expectation of the market regarding future realized volatility.

If the spot price of the underlying is 50, the interest rate is 5%, the option is a year away from expiration, and the call with strike $50 is traded at $4.85, then implied volatility is the kind of volatility that would give a call option a price of $4.85 if the option were subscribed to the Black-Scholes formula. In this case, the implied volatility is 18%.

**Volatility term structure**

Volatility term structure - a graph displaying the ratio between implied volatility and expiry date of options with a certain strike. Usually, the time structure takes the form of a curve because the implied volatility in a short end of the structure is subject to more significant fluctuations than the implied volatility of long-term options.

Term structure of S&P 500 volatility for ATM options (May 1, 2015)

In periods of stable stock growth, the volatility of short-term options is lower than the volatility of options with a later expiry date. During rapid index declines (e.g. in 2009) the term structure is inverted, i.e. short volatility is more important than the volatility of long-term options.

**Volatility surface**

Volatility surface – is a three-dimensional image reflecting the implied volatility of options relative to two variables:

1) strike price and

2) time to expiration.

You can view the smile (or skew) of the volatility and term structure graphs in two dimensions. However, the volatility surface combines the volatility skew graphs for all the expirations and the term structure graphs for all strikes into one three-dimensional image.