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In the Black (1976) model:

  1. We should use the settlement prices of the underlying futures contract in order to estimate the volatility, right? Or can we also use the spot prices? Because the behaviors of these series of prices are quite different.

  2. Should the volatility always be annualized?

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In standard Black model

1) We have to use the volatility of underlying. It is similar for futures and stocks, but it is not the same! Futures prices differ from stock prices not only by discounting, but also by dividends, for example.

2) Volatility should be annualized if you want to use formula "as is" without amendments.

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You should use the volatility of your hedge instrument. If you do hedging with the underlying you use the underlyings vol. If you do hedging with futures then derive the vol form there...

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In theory, since the futures price is $F(t,T)=S(t)e^{r(T-t)}$, the only risk source is coming from S(t), so the volatilities of the two series should be the same.

The thing is that the theory doesn't consider the fact that the interest rates could be stochastic, or that the future market is more liquid and so incorporates more easily information. Also if the underlying of the future contract is a commodity then you also have the volatility coming from storage cost and convenience yield. So indeed the two series could represent different volatilities.

And, the one you should use is of course the volatility obtained from the futures prices.

In what regards weather you should use annualised or monthly volatilities, it shouldn't matter, in theory, as long as you fix the time unit and you are consistent with it throughout the calculations. It does come handy though to use the year as a unit time.

The reason why it doesn't matter what time unit you chose resides in the scaling properties of the BM (in case you are interested).

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yes, in theory the unit of time shouldn't matter. However, if we ignore that fact usually we use the annualized volatility right? can we get big differences if we simply use the standard deviation of the daily returns (without adjusting the time scale)? – Joao Serafim Aug 11 '13 at 2:55
the unit of time doesn't matter because there supposed to be a relation between daily volatility and annual volatility given by the $\sigma_{year}=\sigma_{day}*DaysInAYear$. – KAT Aug 12 '13 at 7:40
Moreover, if you chose the daily volatility and put the daily interest rates as well as expressing time in days in your formula then you should get the same result. – KAT Aug 12 '13 at 7:43

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