Calendar Time and Trading Time in Black–Scholes Option Pricing
Summary
The document asks whether Black–Scholes pricing should use separate time measures for volatility and interest rates. It contrasts trading-day counting, commonly used to annualize volatility, with calendar-day counting for interest, discounting, borrowing, and forward pricing. The example considers a one-year call and tracks its remaining time after a business day or a weekend passes.
The proposed treatment reduces volatility time only as trading days elapse, while calendar time falls every day. This raises a practical modeling question about weekend option value and the passage of time. The text does not provide an answer or compare the proposed convention with market practice, alternative day-count conventions, or empirical option prices. Its weekend assumption that the underlying does not move is a simplification; actual markets can reflect price changes when they reopen, and option models may need conventions tailored to the asset and contract.
Key ideas
- Volatility and interest rates may use different time-count conventions in option pricing.
- The example uses trading days for volatility time and calendar days for interest-rate discounting.
- Under the proposed setup, a weekend reduces calendar time but leaves trading-time volatility unchanged.
- The document poses this modeling question but does not provide a resolution or market evidence.
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Full text
# Volatility Time and Interest Rate Time
# Volatility Time and Interest Rate Time
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.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.