Python for Finance: Analyze Big Financial Data

Normality Tests

The normal distribution can be considered the most important distribution in finance and

one of the major statistical building blocks of financial theory. Among others, the

following cornerstones of financial theory rest to a large extent on the normal distribution

of stock market returns:

Portfolio theory

When stock returns are normally distributed, optimal portfolio choice can be cast into

a setting where only the mean return and the variance of the returns (or the volatility)

as well as the covariances between different stocks are relevant for an investment

decision (i.e., an optimal portfolio composition).

Capital asset pricing model

Again, when stock returns are normally distributed, prices of single stocks can be

elegantly expressed in relationship to a broad market index; the relationship is

generally expressed by a measure for the comovement of a single stock with the

market index called beta ().

Efficient markets hypothesis

An efficient market is a market where prices reflect all available information, where

“all” can be defined more narrowly or more widely (e.g., as in “all publicly

available” information vs. including also “only privately available” information); if

this hypothesis holds true, then stock prices fluctuate randomly and returns are

normally distributed.

Option pricing theory

Brownian motion is the standard and benchmark model for the modeling of random

stock (and other security) price movements; the famous Black-Scholes-Merton option

pricing formula uses a geometric Brownian motion as the model for a stock’s random

fluctuations over time, leading to normally distributed returns.

This by far nonexhaustive list underpins the importance of the normality assumption in

finance.

Benchmark Case

To set the stage for further analyses, we start with the geometric Brownian motion as one

of the canonical stochastic processes used in financial modeling. The following can be

said about the characteristics of paths from a geometric Brownian motion S:

Normal log returns

Log returns between two times 0 < s < t are normally

distributed.

Log-normal values

At any time t > 0, the values St are log-normally distributed.