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'''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).
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The variance is the [[Square (algebra)|square]] of differences of measurements from the [[mean]] divided by the number of samples. The [[standard deviation]] is the [[square root]] of the [[variance]].  
The standard deviation of the continuously compounded returns of a [[financial instrument]]  is called [[Volatility (finance)|volatility]].
 
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]]. Just as [[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]]s for non overlapping time intervals are [[Independence (probability theory)|independent]], the variance is additive (see [[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 \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 formular 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]]

Latest revision as of 15:23, 7 January 2015

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