Forward volatility is a measure of the
implied volatility of a financial instrument over a period in the future, extracted from the term structure of volatility (which refers to how implied volatility differs for related financial instruments with different maturities).
Underlying principle
The variance is the
square
In Euclidean geometry, a square is a regular quadrilateral, which means that it has four equal sides and four equal angles (90- degree angles, π/2 radian angles, or right angles). It can also be defined as a rectangle with two equal-length a ...
of differences of measurements from the
mean
There are several kinds of mean in mathematics, especially in statistics. Each mean serves to summarize a given group of data, often to better understand the overall value ( magnitude and sign) of a given data set.
For a data set, the '' ar ...
divided by the number of samples. The
standard deviation
In statistics, the standard deviation is a measure of the amount of variation or dispersion of a set of values. A low standard deviation indicates that the values tend to be close to the mean (also called the expected value) of the set, whil ...
is the
square root
In mathematics, a square root of a number is a number such that ; in other words, a number whose '' square'' (the result of multiplying the number by itself, or ⋅ ) is . For example, 4 and −4 are square roots of 16, because .
...
of the
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
.
The standard deviation of the continuously compounded returns of a
financial instrument
Financial instruments are monetary contracts between parties. They can be created, traded, modified and settled. They can be cash (currency), evidence of an ownership interest in an entity or a contractual right to receive or deliver in the form ...
is called
volatility.
The (yearly) volatility in a given asset price or rate over a term that starts from
corresponds to the spot volatility for that underlying, for the specific term. A collection of such volatilities forms a volatility term structure, similar to the
yield curve. Just as
forward rate
The forward rate is the future yield on a bond. It is calculated using the yield curve. For example, the yield on a three-month Treasury bill six months from now is a ''forward rate''..
Forward rate calculation
To extract the forward rate, we n ...
s can be derived from a yield curve, forward volatilities can be derived from a given term structure of volatility.
Derivation
Given that the underlying
random variable
A random variable (also called random quantity, aleatory variable, or stochastic variable) is a mathematical formalization of a quantity or object which depends on random events. It is a mapping or a function from possible outcomes (e.g., the po ...
s for non overlapping time intervals are
independent
Independent or Independents may refer to:
Arts, entertainment, and media Artist groups
* Independents (artist group), a group of modernist painters based in the New Hope, Pennsylvania, area of the United States during the early 1930s
* Independe ...
, the variance is additive (see
variance
In probability theory and statistics, variance is the expectation of the squared deviation of a random variable from its population mean or sample mean. Variance is a measure of dispersion, meaning it is a measure of how far a set of numbe ...
). So for yearly time slices we have the annualized volatility as
where
:
is the number of years and the factor
scales the variance so it is a yearly one
:
is the current (at time 0) forward volatility for the period