If the log$-$prices follow a random walk log$($St$)=$log$($St$1)+$Wt

$,$ then the returns are described as

rt$=\backslash$mu $+\backslash$sigma Zt$,$Zt$$N$(0,1).$

For this exercise, it is useful to isolate the random term, also called the normalized return,

Zt$=$rt$\backslash$mu $\backslash$sigma $.$

Note how we simply have to rewrite the equation for rt

to get to Zt

$.$

In our model, we assume $\backslash$mu $,\backslash$sigma

to be the parameters driving the process. In reality, we do not know their exact values; therefore, we estimate them from the data, using np$.$mean and np$.$std$.$

Plot a histrogram of the daily normalized returns of Apple, Microsoft, and Tesla shares. Use the following bins: np$.$linspace$(-5,5,50).$ In the same figure, plot the normal distribution with zero mean and unit variance, that is N$(0,1)$

$.$

How well do the figures agree? What conclusions do you draw?