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In Sheldon Natenberg's book "Option Volatiliy & Pricing (2nd)", he mentioned that (on page 65), only trading days (roughly 252 in a year) are counted when computing vol time and all calendar days (roughly 365 in a year) are used when computing interest rate (hence discount factor, borrow, forward, and etc.)

My question is: When we use the Black-Scholes formula, do we use two time-to-maturity numbers, one for vol and the other for others?

For example, for a one-year (365 calendar days and 252 trading days) expiry call, do we use the following?

\begin{align} C = e^{-r\times (365/365)}\left[Se^{r\times365/365}N(d_1) - KN(d_2)\right] \end{align} where \begin{align} d_{1,2} = \frac{\log \frac{S}{K} \pm \frac12\sigma^2 \times (252/252)}{\sigma\sqrt{(252/252)}} \end{align} and for the greeks, we change the two time-to-expiry numbers accordingly. For example, with one business day passing by, the price of the call becomes: \begin{align} C^* = e^{-r\times (364/365)}\left[Se^{r\times364/365}N(d_1) - KN(d_2)\right] \end{align} where \begin{align} d_{1,2}^* = \frac{\log \frac{S}{K} \pm \frac12\sigma^2 \times (251/252)}{\sigma\sqrt{(251/252)}} \end{align} But with two weekend days passing by, the price of the call is instead: \begin{align} C^{**} = e^{-r\times (363/365)}\left[Se^{r\times363/365}N(d_1) - KN(d_2)\right] \end{align} where \begin{align} d_{1,2}^{**} = \frac{\log \frac{S}{K} \pm \frac12\sigma^2 \times (252/252)}{\sigma\sqrt{(252/252)}} \end{align} because the underlier's price does not move during weekends.

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  • $\begingroup$ I am not sure what you are asking. But yes, Natenberg is suggesting calculating the Black Scholes value in this manner. And many ppl do it this way. $\endgroup$
    – nbbo2
    Aug 26 at 18:22
  • $\begingroup$ Agreed. Banks pay interest on every day including weekends. As @nbbo2 said many people use business day time to measure expiration of options, but many also go one step further and attribute more ‘volatility time’ to event days such as Fed meetings and less time to business days with no scheduled event. $\endgroup$
    – dm63
    Aug 26 at 19:45
  • $\begingroup$ @nbbo2 Thank you so much for your reply. I'm trying to ask, in the BS formula, do we use the same number in $rt$ and $\sigma \sqrt{t}$? Say, if from the pricing day to the expiry, there are 7 calendars left and 5 business days left, do I use 7/365 for $rt$ but $5/252$ for $\sigma \sqrt{t}$? Or do I use, for example, $5/252$ for both? $\endgroup$
    – Michael
    Aug 27 at 17:07
  • $\begingroup$ The question about interest rates boils down to this: if you and I agree on the PV of a fixed cashflow that is paid at time $t$ (which we hopefully do) we agree on a discount factor and not how we express this with some interest rate $r$ and with some $\tau$ calculated with some daycount method. What applies to your $r$ and its "interest rate time" is a matter of pure convention. $\endgroup$
    – Kurt G.
    Aug 27 at 18:03
  • $\begingroup$ That's right. We could even write a Python function where you give it a date (maybe in the form 20240827 or maybe the Excel Julian Date Number for that date) and it returns to us the value $exp(−rt)$ all ready to use in the option formula. And a different function that returns a numeric value for $\sigma^2 t$ along the lines dm63 brought up. In the main program we would not even have a variable $t$. $\endgroup$
    – nbbo2
    Aug 27 at 20:25

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