Annualizing Historical Volatility for Black–Scholes Pricing
Summary
The document explains how to annualize estimated daily index volatility for use in Black–Scholes, where time to expiry is expressed as a fraction of a year. The appropriate scaling depends on how daily returns were sampled: estimates that include non-business days as zero returns use a calendar-day basis, while estimates from business days only use a business-day basis. The answer favors the business-day-only method.
It also distinguishes historical volatility, which describes past price movements, from implied volatility, which is forward-looking and is commonly quoted by markets for option pricing. Historical volatility can serve as a simple estimate for a test, but it is not equivalent to implied volatility. Consistency between the volatility annualization convention and the time convention in the pricing formula is central to applying the guidance.
Key ideas
- Annualize daily volatility according to whether the return sample includes non-business days as zero returns.
- Use a business-day scaling convention when volatility is estimated from business-day returns only.
- Keep the annualization basis consistent with the year fraction used for time to expiry.
- Implied volatility is forward-looking, while historical volatility summarizes past movements.
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Full text
# Which volatility as input in Black Scholes formula?
# 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!
## Answer by Phil-ZXX (score 2, accepted)
https://quant.stackexchange.com/a/39991
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.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.