Rolling At-The-Money Implied Volatility is inferred from options quoted on the market, and is calculated for the following tenors: 1 Week (1W), 1 Month (1M), 3 Months (3M) and 6 Months (6M).

Given a 'hypothetical' option starting today with expiry T (equal 1W, 1M, 3M or 6M), we select two market options such as \(T_{1} \leq T \leq T_{2}\).

We start by calculating the variance of these two contracts:

\begin{equation*}

Var_{i} = \sigma ^{2}_{i} \times \tau_{i}

\end{equation*}

where \(\sigma_{i}\) is the ATM volatility and \( \tau_{i}\) is the time-to-maturity (in years) for contract i.

The theoretical option variance is:

\begin{equation*}

Var^{th} = Var_{1} + (Var_{2} - Var_{1}) \times \frac{\tau^{th} - \tau_{1}}{\tau_{2} - \tau_{1}}

\end{equation*}

The rolling ATM implied volatility is then:

\begin{equation*}

\sigma_{th} = \sqrt{\frac{Var^{th}}{\tau^{th}}}

\end{equation*}