Adjusting Black–Scholes Volatility Time for Weekends
Summary
The document considers how weekends should affect historical volatility estimates used in Black–Scholes pricing when option market prices are unavailable. It describes a distinction between calendar time used for discounting and effective volatility time used in the variance terms of the model. Under this approach, regular trading days contribute a full unit of volatility time, while weekends and holidays contribute a smaller amount, reflecting an assumption that price uncertainty accumulates more slowly when markets are closed.
The discussion cites a textbook example in which weekend variance is estimated at one and a half times a weekday’s variance, based on orange-juice futures and weather information. That example motivates the idea that market activity as well as incoming information affects volatility. The proposed fractional weekend weights are illustrative rather than calibrated guidance, and the document gives no empirical comparison or universal parameter. The method also changes the standard time treatment in Black–Scholes, so its suitability depends on the market and pricing purpose.
Key ideas
- The document separates calendar time for discounting from effective volatility time in Black–Scholes calculations.
- Trading days can count fully toward volatility time, while weekends and holidays can receive smaller weights.
- The cited futures example suggests that weekend variance may be lower than a simple count of calendar days implies.
- The suggested fractional weights are illustrative and require market-specific judgment.
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Full text
# Historical volatility - Black Scholes
# 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)
## Answer by nbbo2 (score 7)
https://quant.stackexchange.com/a/55357
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).Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.