Realized Volatility

This is probably the most common volatility measure. 

One could imagine selecting a stock and a certain time period from the past, and trying to estimate the σ parameter in the BS model based on this data. 

This requires knowledge of Ito's formula (see Mathematical concepts), which allows us to transform the BS equation into a more suitable format (see BS model). The solution of this equation is following a lognormal distribution. So the logarithmic of the stock price follows a normal distribution. 

By applying Ito's lemma we can write down the dynamic this quantity follows: 


d log(St) = (µ - ½ σ2) dt + σ dWt


This equation is saying that the change in logarithmic stock price is composed of two parts. 

The drift term is proportional to the period of time over which we observe this change. 

The volatility term which is determined by noise (Brownian Motion --> See Mathematical concepts). 


If one wants to estimate the σ parameter of the stock, one can use the lognormal returns over one day and calculate the standard deviation from them. This would give the volatility over one day, which is σ√dt. This follows from the property of the BM that tells us that the variance of Wt is given by the time t. That just leaves us with the normalising effect to withdraw the value of σ. 


The first obvious question is how many days there are in a year. In your dataset, there are no quotes for weekends and holidays. So there is no volatility to observe there. Therefore a common approach is to use the number of trading days in a year. 

A fair challenge would be to ask if there is really no volatility during the weekend. In other words, is it fair to regard the return from Friday to Monday in the same manner as from Monday to Tuesday? Wouldn't that imply that the world stops turning in the weekend? For that matter, filtering techniques are sometimes applied to estimate the volatility parameter. 


 Also, it makes a difference which frequency of data you use. Typically, the longer the time period, the more normal the returns tend to be. 

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