Time Units and Annualization in the Black–Scholes Formula

Summary

The document explains how time to maturity enters the Black–Scholes option pricing formula and why calculators often request days until expiration. The input unit itself is flexible: time must be expressed consistently with the units used for the risk-free rate and volatility. Because these inputs are commonly quoted on an annual basis, a calculator typically converts the number of days into a fraction of a year before pricing.

It also notes that professional tools may accept exact start and expiration dates, sometimes including the time of day. The explanation addresses a unit-conversion question rather than comparing day-count conventions or discussing how rates and volatility should be calibrated. It does not specify a particular annualization convention, so users should check how a given calculator handles holidays, trading hours, and the fraction of a year.

Key ideas

• Time to maturity can be expressed in different units if the other model inputs use compatible units.
• Rates and volatility are commonly annualized, so calculators generally convert days to a fraction of a year.
• Entering a raw number of hours as though it were a number of years would create an inconsistent input.
• Exact dates, and sometimes times of day, can provide a more precise maturity input.

Tags

# What is the unit of $T$ in the Black-Scholes formula?


# What is the unit of $T$ in the Black-Scholes formula?












When the Black scholes formula is derived, $T$ is just some time in the future. We don't specify what it is.

So why is it that if you go to an option pricing calculator, it asks specifically for days until expiration? When these calculators then use the formula, do they just plug in these number of days remaining for $T$?

If they do this, then what if instead of saying 10 days, I said 1440 hours. That's still 10 days, yet plugging in 1440 for T obviously gives a different answer than if we plugged in 10 .....

## Answer by Alex C (score 2)

https://quant.stackexchange.com/a/32335

You describe an online calculator that takes input in days. Days is just convenient for some people, although I have seen other calculators using other units for time to maturity. Internally the calculator is translating the number of days into years before plugging it into the BSM formula. That is because mathematically the units used in the formula have to be consistent for 3 variables $r$,$\sigma$ and $T$. It just turns out $r$ and $\sigma$ are usually expressed on a yearly basis.

I would add that the calculators used professionally allow you to enter exact dates (e.g. 2017 02 08 to 2017 05 19)(maybe even the hour) rather than days or years when specifying $T$.

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