Discrete Models of Equity Returns
The basic standard model of equity returns assumes that the distribution of returns is normal. Returns were initially measure as aritmetic returns :
{% r_t = \frac{p_t}{p_{t-1}} %}
or more recently as the difference of log prices.
{% r_t = log(p_{t} - log(p_{t-1}) %}
Departures from Normality and Stylized Facts
It has been known for some time that stock returns do not follow a normal distribution. In particular, the returns display "fat tails", which makes the risk of extreme moves larger than what would be expected. In addition, it has been noted that exteme moves to the downside occur with a higher frequency than extreme moves higher.These two phenomena are entailed by the return distribution have both nonzero skew and kurtosis.