Historic volatility is derived from time series of past market prices. An implied volatility is derived from the market price of a market traded derivative (in particular an option).
Volatility as described here refers to the actual volatility, more specifically:
- actual current volatility of a financial instrument for a specified period (for example 30 days or 90 days), based on historical prices over the specified period with the last observation the most recent price.
- actual historical volatility which refers to the volatility of a financial instrument over a specified period but with the last observation on a date in the past
- actual future volatility which refers to the volatility of a financial instrument over a specified period starting at the current time and ending at a future date (normally the expiry date of an option)
Now turning to implied volatility, we have:
- historical implied volatility which refers to the implied volatility observed from historical prices of the financial instrument (normally options)
- current implied volatility which refers to the implied volatility observed from current prices of the financial instrument
- future implied volatility which refers to the implied volatility observed from future prices of the financial instrument
For a financial instrument whose price follows a Gaussian random walk, or Wiener process, the width of the distribution increases as time increases. This is because there is an increasing probability that the instrument’s price will be farther away from the initial price as time increases. However, rather than increase linearly, the volatility increases with the square-root of time as time increases, because some fluctuations are expected to cancel each other out, so the most likely deviation after twice the time will not be twice the distance from zero.
Since observed price changes do not follow Gaussian distributions, others such as the Lévy distribution are often used. These can capture attributes such as “fat tails“. Volatility is a statistical measure of dispersion around the average of any random variable such as market parameters etc.