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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.
