Q:

Historical volatility - Black Scholes

How do you best incorporate the weekends in the calculation of the Black Scholes historical volatility? (Of course historical volatility serves as approximation, if the market price of the options is not available).
In the book "Options, Futures and other Derivatives" by John C. Hull, it is described that over the weekend (from Friday till Monday) the variance is only 1.5 times higher than during the workweek (Monday-Friday), by using as example of a future (orange-juice futures) that is dependent on information that changes just as much during the weekend as through the week namely the weather; which shows that volatility is for a big part caused by trading and not by new information coming into the market. But still it seems to me that you would need some way to adapt to the weekend.
The volatility of the daily log-normal returns: ln(P1/P0)

A:

The simplest approach is to use two different variables $T_1$ and $T_2$ instead of the single variable $T$ that denotes Time To Maturity in the classic Black Scholes Merton formula.
$T_1$, the time to maturity for interest rate  computation purposes, is the calendar time in years between now and maturity. For example the term $-Ke^{-rT}N(d_2)$ in the formula would become $-Ke^{-rT_1}N(d_2)$. The exercise is $T_1$ years away and therefore the discounting takes this into account.
$T_2$, the effective volatility time until maturity is calculated by adding up time differently depending whether the market is open or not. For regular trading days between now and maturity a "1 " is scored, but for weekends or holiday a lower number (say 0.5 or 0.6) is taken, reflecting that volatility is expected to be lower on those days. The total is then converted to yearly basis, and $T_2$ is used in the BSM formula wherever it multiplies $\sigma^2$ or wherever $\sqrt{T}$ multiplies $\sigma$.
(I am not sure which book first presented this approach, it may be Cox & Rubinstein's Option Markets).

