Back of the Envelope

Observations on the Theory and Empirics of Mathematical Finance

[PDS] Modeling Stock Returns: I

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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?