Why Black–Scholes and Binomial Models Express Time in Years

Summary

The document explains why time to expiration in Black–Scholes and binomial option models is conventionally measured in years rather than seconds. The answers describe this as a practical convention aligned with annualized interest rates and volatility, keeping inputs at manageable scales instead of forcing users to work with very small per-second values.

The responses also emphasize consistent time units: model inputs such as rates and volatility are annualized, so using a different time scale for one variable would require rescaling the others. Annual data conventions are also more practical for obtaining some financial inputs than monthly or weekly equivalents. The document gives conceptual reasoning rather than a derivation or empirical comparison. Its point is about consistent units and convenience; another time scale could be used if all variables were converted consistently, though calendar details such as weekends and holidays complicate scaling.

Key ideas

• Years are a convention that aligns time to expiration with annualized interest rates and volatility.
• Consistent time units across model inputs are necessary for valid calculations.
• Annualized inputs keep values at convenient scales compared with per-second quantities.
• Changing the time scale requires rescaling the model variables consistently and handling calendar effects.

Tags

# Why t (time) in Black Scholes & Binomial defined as year?


# Why t (time) in Black Scholes & Binomial defined as year?












What's the logical/scientific explanation for Black Scholes & Binomial using year rather than second (SI standard for time) ?

## Answer by Bob Jansen (score 5, accepted)

https://quant.stackexchange.com/a/20669

It's just a matter of convention as it is customary to also quote the interest rate as a yearly rate. Furthermore, this time scale plays well with the size of other variables such as the interest rate and volatility. It's just not convenient to quote a tiny number for the volatility. Sure, one could scale these numbers back to something more reasonable but that only complicates matters. Especially since scaling is hard if you take into account holiday, weekends, etc.

## Answer by Robert Szóstakowski (score 1)

https://quant.stackexchange.com/a/20670

If you will take a look at the BS equation https://en.wikipedia.org/wiki/Black%E2%80%93Scholes_model particularly at the BS equation and its variables, you can see that all of them are annualized.

Moreover:

- The model would be incorrect if it had some variables annualized and some not annualized (monthly, weekly).

- It is harder to obtain data (i.e. monthly interest rates, not mentioning weekly interest rates) for some countries.

If you want you can try to scale all of the variables and see the results.

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.