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Choosing Time-to-Expiration for Same-Day Black–Scholes Pricing

Article Quant Q&A · Author: optzh
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Summary

The document compares two ways to express the remaining time to expiration when valuing an option on its final trading day. One approach counts calendar time, converting the hours left into a fraction of a 365-day year. The other counts only trading hours, scaling the hours remaining by assumed daily trading hours and 252 trading days per year. Both answers give a worked SPXW example, but use different time conventions and therefore produce different values for T.

The choice depends on the pricing model’s time and volatility conventions. The calendar-time answer argues that option time value decays outside market hours, including weekends, while the trading-time approach reflects a convention that measures time in trading units. The discussion does not establish one universally correct method or compare the approaches empirically. Practitioners need a consistent model convention and should align the volatility input and time scaling with it; the stated exchange close and session assumptions are specific to the example.

Key ideas

  • The remaining time can be expressed as a fraction of a calendar year or a trading year.
  • The calendar-time example divides hours remaining by 24 and then by 365.
  • The trading-time example scales hours remaining by daily trading hours and 252 trading days.
  • The document argues that time value can decay outside trading hours, but acknowledges differing conventions.
  • Time scaling should be consistent with the model’s volatility and calendar assumptions.

Tags

OptionsDerivatives pricingVolatilityStatistics
Full text
# Intraday "Time to expiration" for Black-Scholes on the expiration day


# Intraday "Time to expiration" for Black-Scholes on the expiration day












In Black-Scholes, T is the % of year, how do we calculate T intraday on the expiration day?

Does the expiration happen at the exact moment of that trading session?

For example, for SPXW options that expire on Friday Jan 8th when the trading stops at 15:15 CST, at 9:15 AM CST on that day, is T=(6hours/24hours)/365=0.0006849?

## Answer by Kai (score 2)

https://quant.stackexchange.com/a/80892

In this case, at 9:15AM CST on that very day, we have 6 hours left on the trading time. However, in most models T is in fractional years (/365 most frequently) although there are some exceptions. So in this case, you should calculate it as (6/24)/365.

Note: There would be some disagreements here in the sense that traditional textbooks like Hull, which most people refer to, use 252 days to price this and only in trading hours(with compelling arguments with volatility as the main point). However, if you price a model professionally, theta decays even in non-trading hours, so you should account for this. Thinking about this logically, if you had a long option position, your option would be losing time value, even on weekends or after trading hours.

## Answer by Summer_More_More_Tea (score 0)

https://quant.stackexchange.com/a/80884

All in trading unit, i.e. if the trading hour 6 hours for Jan. 8th, the total trading hour a day is 20 hours (assume it is) and 252 trading days a year, the DTE T = (6 / 20) / 252 = 0.00119.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.