Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

[PDS] Modeling Stock Returns: I

with one comment

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:

\displaystyle z_t = \frac{R_t - \mu}{\sigma}

and comparing their frequency distribution (probability density) with the standard Normal/Gaussian distribution (where \mu and \sigma 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.:

\displaystyle z_t = \frac{R_t - \mu}{\sigma} \sim \mathcal{N}(0,1)

Writing a draw from \mathcal{N}(0,1) as simply \epsilon then allows us to write:

\displaystyle z_t = \frac{R_t - \mu}{\sigma} = \epsilon

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

\boxed{R_t = \mu + \sigma \epsilon}

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


Written by Vineet

January 15, 2013 at 11:54 pm

One Response

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

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