Probability Distribution

Description

The probability distribution function of the price returns \(P(r_{t})\) has been extensively studied in different financial markets over different time scales [Bachelier_1900][Cont_2001][Chakraborti_2011]. It has been empirically verified that the probability distribution \(P(r_{t})\) consistently has a power-law decay in the tails

\begin{equation} P(r_{t})\propto r_{t}^{-\alpha}. \end{equation}
../_images/pos_dist.png

Fig. Averaged result for S&P500 firms daily price return

The power-law exponent satisfies \(\alpha\approx3\) for stock markets (i.e., S&P500, NewYork Stock Exchange…) at daily scale.

Code Example

import datetime as dt
import pandas_datareader.data as web
import numpy as np
import stylefact.finance as sff
import stylefact.visualize as sfv
st = dt.datetime(1990,1,1)
en = dt.datetime(2020,1,1)
data = web.get_data_yahoo('GM', start=st, end=en)
prices = data['Adj Close'].to_numpy()
log_prices = np.log(prices)
returns = np.diff(log_prices)
x,y = sff.linear_distribution(returns)
sfv.linear_distribution(x,y,'linear_distribution')
x,y = sff.log_distribution(returns,'positive')
sfv.log_distribution(x,y,'log_positive_distribution')
x,y = sff.log_distribution(returns,'negative')
sfv.log_distribution(-x,y,'log_negative_distribution')

References

[Bac00]Louis Bachelier. Théorie de la spéculation. Annales Scientifiques de l’École Normale Supérieure, 3:21–86, 1900.
[CTPA11]Anirban Chakraborti, Ioane Toke, Marco Patriarca, and Frédéric Abergel. Econophysics review: i. empirical facts. Quantitative Finance, 11(7):991–1012, 2011.
[Con01]Rama Cont. Empirical properties of asset returns: stylized facts and statistical issues. Quantitative Finance, 1(2):223–236, 2001.