dist_fit_spy.py

import numpy as np
import yfinance as yf
import scipy.stats as stats
from scipy.integrate import quad
from scipy.optimize import minimize

# Get the monthly price data for the SPY ticker
df = yf.download(
    'SPY', 
    interval='1mo',
)

# Convert the closing prices to percentage changes
data = 100*df['Close'].pct_change().dropna().values

# Define a list of candidate distributions to fit
dist_list = [
    stats.norm, 
    stats.lognorm,
    stats.genlogistic,
]

# Estimate the parameters of each candidate distribution, by default, the 
# fitting is performed with maximum likelihood estimation
param_estimates = [dist.fit(data) for dist in dist_list]

# Compute the log-likelihood of the data for each distribution, this
# serves as a method of finding the "best fit" distribution, i.e., which
# distribution maximises the likelihood function
loglikes = [
    dist.logpdf(data, *params).sum() 
    for dist, params in zip(dist_list, param_estimates)
]

# Select the best-fit distribution based on the minimum negative log-likelihood
best_dist = dist_list[np.argmax(loglikes)]
params = param_estimates[np.argmax(loglikes)]

# Print the name of the best-fit distribution
print(f'The best-fit distribution is {best_dist.name}')

# Let's get the x position of the peak of the distribution, and work out the 
# volume to the right of the peak
x_pos_peak = minimize(lambda x: -best_dist.pdf(x, *params), x0=0).x[0]
vol_right = quad(lambda x: best_dist.pdf(x, *params), x_pos_peak, np.inf)[0]

# Print out the statistics
print('\nStatistics:')
print('- Mean of the distribution: ', np.mean(data))
print('- Standard deviation of the distribution: ', np.std(data))
print('- Median of the distribution: ', np.median(data))
print('- Interquartile range: ', stats.iqr(data))
print('- Peak of the fitted distribution: ', x_pos_peak)
print('- Volume to the right of the peak: ', 100*vol_right)