## [PDS] Modeling Stock Returns: I

We began by looking at the empirical probability distribution of scaled returns of daily S&P 500 prices and standard Normal/Gaussian distribution today.

Writing scaled returns as:

and comparing their frequency distribution (probability density) with the standard Normal/Gaussian distribution (where and are mean and standard deviation of daily returns respectively), we came up with the following graph in Excel:

*(Comparison of Empirical Frequency Distribution of Scaled Stock Returns with standard Gaussian distribution; PDF stands for Probability Density Function)*

Looking at the comparison we argued that while the two distributions do differ in their tail behavior as a first approximation perhaps this is not too bad a starting point.

That is, the comparison suggests that as a first approximation one may model scaled returns of S&P 500 by simply drawing a number at random from a standard Gaussian distribution, i.e.:

Writing a draw from as simply then allows us to write:

and gives us a model for the stock return data as:

The question now is that is this really a *usable* model or do we need to do something with this yet?

[…] we have approximated the scaled asset returns process by draws from a standard Gaussian […]

[PDS] Modeling Stock Returns: II « Back of the EnvelopeJanuary 17, 2013 at 12:40 am