\begin{equation}ABV \approx ( \delta_i - \delta_f ) ( k * \delta_e )^{-1} \end{equation}
Where \delta_e is the density of ethanol
\begin{equation} \delta_e = 0.7892 \text{ g/cm}^3 \end{equation}
And k is the proportionality constant between the weights of CO_2 and \text{ethanol} that are produced during the fermentation process.
W_{CO_2} = k*W_e
We can derive the value of k by having a look at the fermentation reaction:
C_6H_{12}O_{6} \rightarrow 2 C_2H_5OH + 2 CO_2
In other words, for each molecule of carbon dioxide released in fermentation, one ethanol molecule is produced. Since the molar mass of ethanol is 46.07 g/mol, compared to 44.01 g/mol for carbon dioxide; we have
\begin{equation}k = \frac{W_{CO_2}}{W_e}= \frac{44.01 \text{g/mol}}{ 46.07 \text{g/mol}} \approx 0.9553 \end{equation}
And now it is time to plug the numbers...Using the values of (2) & (3) into (1), we finally get
ABV \approx ( \delta_i - \delta_f ) ( 0.9553*0.7892\text{ g/cm}^3)^{-1}
\Rightarrow ABV \approx ( \delta_i - \delta_f ) *1.326\text{ cm}^3/ \text{g}
In case the hydrometer readings for \delta_i and \delta_f are given in kg/m^3 instead of g/cm ^3, we need to divide by 1000; and if we want the Alcohol by Volume to be expressed as a percentage, we have to multiply by 100. In such case,
\boxed{ \%ABV \approx ( \delta_i - \delta_f ) *0.1326 \text{ [m}^3/ \text{kg]} }