Volatility is a measure of the statistical dispersion of returns for a given asset. Volatility refers to the amount of uncertainty about the scale of changes in the price. A higher volatility means that the price can be spread out over a larger range of values, and that it could change drastically over a short time period in either direction. Conversely, lower volatility means that the price does not fluctuate drastically, but changes in value at a steady pace over a period of time.
Historical Volatility can be measured by using either the standard deviation or variance between returns on the underlying.
Volatility is a principal input into option pricing formulas, where it estimates the extent to which the price of the underlying asset will fluctuate between now and the option's expiry date. Volatility is expressed as a percentage coefficient within option pricing models.
Implied Volatility refers to the volatility inherent in an options price, and is calculated by reversing the Black-Scholes model to determine what volatility value was used to produce the option price. Implied volatility is often compared with Historical volatility to establish whether option prices are under or over-valued.
At-the-money options tend to have lower implied volatilities than In-the-Money or Out-the-money options. When implied volatility is plotted against strike price, the resulting graph is typically downward sloping for equity derivatives, giving rise to the term "Volatility Skew". For other markets, such as FX and index options, the typical graph turns up at either end, hence the term "volatility smile" is used.