Volatility (finance)
In finance, volatility (symbol σ) is the degree of variation of a trading price series over time as measured by the standard deviation of logarithmic returns.
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:
Now turning to implied volatility, we have:
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.
For any fund that evolves randomly with time, the square of volatility is the variance of the sum of infinitely many instantaneous rates of return, each taken over the nonoverlapping, infinitesimal periods that make up a single unit of time.
Thus, "annualized" volatility σ is the standard deviation of an instrument's yearly logarithmic returns.
The generalized volatility σT for time horizon T in years is expressed as:
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