Forward volatility

Given that the underlying random variables for non overlapping time intervals are independent, the variance is additive (see variance). So for yearly time slices we have the annualized volatility as

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

where

${\displaystyle j=1,2,\ldots }$ is the number of years and the factor ${\displaystyle {\frac {1}{j}}}$ scales the variance so it is a yearly one

${\displaystyle \sigma _{i,\,j}}$ is the current (at time 0) forward volatility for the period ${\displaystyle [i,\,j]}$

${\displaystyle \sigma _{0,\,j}}$ the spot volatility for maturity ${\displaystyle j}$.

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

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

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

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

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

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

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

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