# 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) ?

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.

• I've done some experiment using very tiny number for volatility (with r=0 & q=0), but still I can't get the correct value (when compared using t as year versus t as day or month) – Tidy Star Sep 8 '15 at 12:08
• It's hard for me to comment without knowing your experiment. Are you scaling correctly (square root rule) and did you check that your tiny values do not result in rounding errors? – Bob Jansen Sep 8 '15 at 12:13
• My bad, the software I used set r automatically to 0.05. You're right. – Tidy Star Sep 8 '15 at 12:17

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:

1. The model would be incorrect if it had some variables annualized and some not annualized (monthly, weekly).
2. 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.

• Lets say r=0, q=0. The part I don't understand is why the d1 equation involve t defined as year? Is it possible to convert BSM formula to using second rather than year? – Tidy Star Sep 8 '15 at 11:58