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From Joshi's Quant Interviews books:

The statistics department from our tell you that the stock price has followed a mean reversion process for the last 10 years, with annual volatility 10% and daily volatility 20%. You want to sell a European option and hedge it, which volatility do you use?

Apparently the answer is daily volatiliy 20% as the option price is monotonically increasing in volatility.

I dont get this. I thought that the B-S price used annual volatility. Why should we deviate from this?

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    $\begingroup$ The BS model inputs need to be annualised (multiplied by $\sqrt{252}$ for daily returns) but annual volatility sounds like the standard deviation of annual returns. Indeed you ought to use the annualised volatility of daily returns? At least that’s how I read the question? $\endgroup$
    – Kevin
    Commented Jan 20, 2020 at 12:08
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    $\begingroup$ Yes. Note that the process is assumed to be mean-reverting... so estimates of vol from yearly returns will indeed be different from annualized estimates of vol from daily returns... Which to use ? Black Scholes assumes no mean reversion, hence does not address this. $\endgroup$
    – nbbo2
    Commented Jan 20, 2020 at 17:42

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You should always use the biggest volatility to minimise the risk and hedge the option correctly.
Don't forget to multiply daily volatility by square(252) to annualize it.

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