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Forward volatility

From Wikipedia, the free encyclopedia

Forward volatility is a measure of theimplied 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

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The variance is thesquare of differences of measurements from themean divided by the number of samples. Thestandard deviation is thesquare root of thevariance. The standard deviation of the continuously compounded returns of afinancial instrument is calledvolatility.

The (yearly) volatility in a given asset price or rate over a term that starts from $t_{0}=0$ corresponds to the spot volatility for that underlying, for the specific term. A collection of such volatilities forms a volatility term structure, similar to theyield curve. Just asforward rates can be derived from a yield curve, forward volatilities can be derived from a given term structure of volatility.

Derivation

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Given that the underlyingrandom variables for non overlapping time intervals areindependent, the variance is additive (seevariance). So for yearly time slices we have the annualized volatility as

${\begin{aligned}\sigma _{0,j}^{2}&={\frac {1}{j}}(\sigma _{0,1}^{2}+\sigma _{1,2}^{2}+\cdots +\sigma _{j-2,j-1}^{2}+\sigma _{j-1,j}^{2})\\\Rightarrow \sigma _{j-1,j}&={\sqrt {j\sigma _{0,j}^{2}-\sum _{k=1}^{j-1}\sigma _{k-1,k}^{2}}},\end{aligned}}$

where

j=1,2,\ldots

is the number of years and the factor

{\frac {1}{j}}

scales the variance so it is a yearly one

\sigma _{i,\,j}

is the current (at time 0) forward volatility for the period

[i,\,j]

\sigma _{0,\,j}

the spot volatility for maturity

j {\displaystyle j}

To ease computation and get a non-recursive representation, we can also express the forward volatility directly in terms of spot volatilities:^[1]

${\begin{aligned}\sigma _{0,j}^{2}&={\frac {1}{j}}(\sigma _{0,1}^{2}+\sigma _{1,2}^{2}+\cdots +\sigma _{j-1,j}^{2})\\&={\frac {j-1}{j}}\cdot {\frac {1}{j-1}}(\sigma _{0,1}^{2}+\sigma _{1,2}^{2}+\cdots +\sigma _{j-2,j-1}^{2})+{\frac {1}{j}}\sigma _{j-1,j}^{2}\\&={\frac {j-1}{j}}\,\sigma _{0,j-1}^{2}+{\frac {1}{j}}\sigma _{j-1,j}^{2}\\\Rightarrow {\frac {1}{j}}\sigma _{j-1,j}^{2}&=\sigma _{0,j}^{2}-{\frac {(j-1)}{j}}\sigma _{0,j-1}^{2}\\\sigma _{j-1,j}^{2}&=j\sigma _{0,j}^{2}-(j-1)\sigma _{0,j-1}^{2}\\\sigma _{j-1,j}&={\sqrt {j\sigma _{0,j}^{2}-(j-1)\sigma _{0,j-1}^{2}}}\end{aligned}}$

Following the same line of argumentation we get in the general case with $t_{0}<t<T$ for the forward volatility seen at time $t_{0}$ :

$\sigma _{t,T}={\sqrt {\frac {(T-t_{0})\sigma _{t_{0},T}^{2}-(t-t_{0})\sigma _{t_{0},t}^{2}}{T-t}}}$ ,

which simplifies in the case of $t_{0}=0$ to

$\sigma _{t,T}={\sqrt {\frac {T\sigma _{0,T}^{2}-t\sigma _{0,t}^{2}}{T-t}}}$ .

Example

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The volatilities in the market for 90 days are 18% and for 180 days 16.6%. In our notation we have $\sigma _{0,\,0.25}$ = 18% and $\sigma _{0,\,0.5}$ = 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 $t_{0}=0$ we get

$\sigma _{0.25,\,0.5}={\sqrt {\frac {0.5\cdot 0.166^{2}-0.25\cdot 0.18^{2}}{0.25}}}=0.1507\approx 15.1\%$ .

References

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^Taleb, Nassim Nicholas (1997).Dynamic Hedging: Managing Vanilla and Exotic Options. New York: John Wiley & Sons.ISBN 0-471-15280-3, pg 154

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Category:

Mathematical finance

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