# Which volatility as input in Black Scholes formula?

I am trying to price an option on an Index using Black Scholes formula. I estimated the daily volatility $\sigma_{day}$.

My question is should I use an annual volatility based on the business days of the Index ( $\sigma_{annual} = \sqrt{252}\ \sigma_{day}$) or should I choose $\sigma_{annual} = \sqrt{365}\ \sigma_{day}$ ? When using my B-S formula, $T$ is expressed in days and I use $T/365$.

Thank you for your help!

It will depend on how you estimated the daily std $\sigma_{day}$.

1) If you treated non-business days (holidays, weekends) as having a zero return, then $$\sigma_{annual} = \sqrt{365}\cdot \sigma_{day}$$

2) If you estimated the daily std using returns from actual business days only (i.e. you excluded the zero non-business day returns from your calculation), then $$\sigma_{annual} = \sqrt{252}\cdot \sigma_{day}$$

Note that method 2 is preferred.

Just to have mentioned it, the market usually quotes $\sigma_{annual}$ (= implied volatility) so you can plug it right into the BS formula (not the other way round). That is because historic volatility is backwards-looking whereas implied volatility is forward-looking. So they fundamentally describe different time horizons of the stock/index's evolution.

Of course, for a simple test using historic volatility as an estimate is absolutely fine.

• That's clear, thank you! I will use the volatility with business days then. Just another question, since I am using business days, should I use business days too when I code my B-S formula, or should I keep $T\365$ ? May 24, 2018 at 12:26
• There are different ways of computing daycount fractions, but you never have T/252 (at least I haven't seen it). So using T/365 is fine. May 24, 2018 at 13:24