Normal distribution, standard normal distribution, and cumulative standard normal distribution

In finance, normal distribution plays a central role. This is especially true for option theory. The major reason is that it is commonly assumed that the stock prices follow a log normal distribution while the stock returns follow a normal distribution. The density of a normal distribution is defined as follows:

Here, μ is the mean and σ is the standard deviation.

By setting μ as 0 and σ as 1, the preceding general normal distribution density function collapses to the following standard normal distribution:

The following code generates a graph for the standard normal distribution. The SciPy's stats.norm.pdf() function is used for the standard normal distribution. The default setting is with a zero mean and unit standard deviation, that is, the standard normal density function:

>>>from scipy import exp,sqrt,stats
>>>stats.norm.pdf(0)
0.3989422804014327...