# Forward volatility

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{{One source|date=June 2025}}

'''Forward volatility''' is a measure of the [implied volatility](/source/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](/source/Square_(algebra)) of differences of measurements from the [mean](/source/mean) divided by the number of samples. The [standard deviation](/source/standard_deviation) is the [square root](/source/square_root) of the [variance](/source/variance). 
The standard deviation of the continuously compounded returns of a [financial instrument](/source/financial_instrument)  is called [volatility](/source/Volatility_(finance)).

The (yearly) volatility in a given asset price or rate over a term that starts from <math>t_0=0</math> 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](/source/yield_curve). Just as [forward rate](/source/forward_rate)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](/source/random_variable)s for non overlapping time intervals are [independent](/source/Independence_(probability_theory)), the variance is additive (see [variance](/source/variance)). So for yearly time slices we have the annualized volatility as

<math>\begin{align}
\sigma_{0,j}^2
&= \frac{1}{j}(\sigma_{0,1}^2 + \sigma_{1,2}^2 + \cdots  + \sigma_{j-2,j-1}^2 + \sigma_{j-1,j}^2)\\
\Rightarrow \sigma_{j-1,j}
&=\sqrt{j\sigma_{0,j}^2-\sum_{k=1}^{j-1}\sigma_{k-1,k}^2},
\end{align}
</math>

where

:<math>j=1,2,\ldots</math> is the number of years and the factor <math>\frac{1}{j}</math> scales the variance so it is a yearly one

:<math>\sigma_{i,\,j}</math> is the current (at time 0) forward volatility for the period <math>[i,\,j]</math>

:<math>\sigma_{0,\,j}</math> the spot volatility for maturity <math>j</math>.

To ease computation and get a non-recursive representation, we can also express the forward volatility directly in terms of spot volatilities:<ref>Taleb, Nassim Nicholas (1997). ''Dynamic Hedging: Managing Vanilla and Exotic Options''. New York: John Wiley & Sons. {{ISBN|0-471-15280-3}}, pg 154</ref>

<math>\begin{align}
\sigma_{0,j}^2
&= \frac{1}{j}(\sigma_{0,1}^2 + \sigma_{1,2}^2 + \cdots  + \sigma_{j-1,j}^2)\\
&= \frac{j-1}{j}\cdot\frac{1}{j-1}(\sigma_{0,1}^2 + \sigma_{1,2}^2 + \cdots  + \sigma_{j-2,j-1}^2) + \frac{1}{j}\sigma_{j-1,j}^2\\
&= \frac{j-1}{j}\,\sigma_{0,j-1}^2 + \frac{1}{j}\sigma_{j-1,j}^2 \\
\Rightarrow \frac{1}{j} \sigma_{j-1,j}^{2} &= \sigma_{0,j}^2-\frac{(j-1)}{j}\sigma_{0,j-1}^{2}\\
\sigma_{j-1,j}^{2} &= j \sigma_{0,j}^2-(j-1)\sigma_{0,j-1}^{2}\\
\sigma_{j-1,j} &= \sqrt{j \sigma_{0,j}^2-(j-1)\sigma_{0,j-1}^{2}}
\end{align}
</math>

Following the same line of argumentation we get in the general case with <math>t_0<t<T</math> for the forward volatility seen at time <math>t_0</math>:

<math>\sigma_{t,T}=\sqrt{\frac{(T-t_0)\sigma_{t_0,T}^2-(t-t_0)\sigma_{t_0,t}^2}{T-t}}</math>,

which simplifies in the case of <math>t_0=0</math> to

<math>\sigma_{t,T}=\sqrt{\frac{T\sigma_{0,T}^2-t\sigma_{0,t}^2}{T-t}}</math>.

==Example==

The volatilities in the market for 90 days are 18% and for 180 days 16.6%. In our notation we have <math>\sigma_{0,\,0.25}</math> = 18% and <math>\sigma_{0,\,0.5}</math> = 16.6% (treating a year as 360 days). 
We want to find the forward volatility for the period starting with day 91 and ending with day 180. Using the above formula and setting <math>t_0=0</math> we get

<math>\sigma_{0.25,\,0.5}=\sqrt{\frac{0.5\cdot 0.166^2-0.25\cdot 0.18^2}{0.25}}=0.1507\approx 15.1\%</math>.

==References==

<references/>

Category:Mathematical finance

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Adapted from the Wikipedia article [Forward volatility](https://en.wikipedia.org/wiki/Forward_volatility) by Wikipedia contributors ([contributor history](https://en.wikipedia.org/wiki/Forward_volatility?action=history)). Available under [Creative Commons Attribution-ShareAlike 4.0 International](https://creativecommons.org/licenses/by-sa/4.0/). Changes may have been made.