Some Aspects of Volatility

There are many ways to look at volatility. 

The intuitive meaning of the word is that volatility measures the level of fluctuations for a particular price. 

The way we measure it, the unit we use, the time-scale at which we are looking all have an impact and should be specified in order to transform a single volatility number into a solid understanding of how much level of fluctuation there is in that particular stock. 


By taking a more academic approach based on statistics, one can argue that the value of the stock in one year is uncertain and assign a probability distribution to it. It could be desirable to used the width or standard deviation of this distribution to link to the volatility of the stock. This point of view is exactly what has become the market standard. 


It is clear that the statistical approach is focused on the one-year horizon, whereas a trader, who wants to delta-hedge on a daily basis, is not so interested in knowing the uncertainty accumulated over the year. What he is really interested in, is to understand how the uncertainty plays a role on a much smaller scale, such that piled up over the year it leads the same distribution as the statistician has presented. 

In mathematical terms, knowing the distribution at one time (or multiple times) is not enough to complete the dynamic picture. One needs to know how the distribution changes over time. Clearly, on a very short time-scale, the uncertainty is very small and the distribution function should be sharply peaked about the current level of the stock. As the time horizon increases, the density should widen up. 


One can show that at any time t, the solution of the BS SDE, describing a model for the movement of the stock, is  a RV S(t) that behaves according to a lognormal distribution. So at any time t, we have a density that depends on the original parameters in the equation, being the drift µ and the volatility parameter σ. 

Note that the volatility is not the standard deviation of this distribution, but it does control the wideness of the distribution. 


Clearly, if we used another model for the stock price, it would lead to another family of density functions, and to other formulas for the moments. 

In a way, when people use the word volatility, they also agree on the underlying mathematical model. 



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