Log Periodic Power Law Singularity (LPPLS) Model

lppls is a Python module for fitting the LPPLS model to data.

Overview

The LPPLS model provides a flexible framework to detect bubbles and predict regime changes of a financial asset. A bubble is defined as a faster-than-exponential increase in asset price, that reflects positive feedback loop of higher return anticipations competing with negative feedback spirals of crash expectations. It models a bubble price as a power law with a finite-time singularity decorated by oscillations with a frequency increasing with time.

Here is the model:

where:

• expected log price at the date of the termination of the bubble
• critical time (date of termination of the bubble and transition in a new regime)
• expected log price at the peak when the end of the bubble is reached at
• amplitude of the power law acceleration
• amplitude of the log-periodic oscillations
• degree of the super exponential growth
• scaling ratio of the temporal hierarchy of oscillations
• time scale of the oscillations

The model has three components representing a bubble. The first, , handles the hyperbolic power law. For < 1 when the price growth becomes unsustainable, and at the growth rate becomes infinite. The second term, , controls the amplitude of the oscillations. It drops to zero at the critical time . The third term, , models the frequency of the osciallations. They become infinite at .

Important links

• Official source code repo: https://github.com/Boulder-Investment-Technologies/lppls
• Download releases: https://pypi.org/project/lppls/
• Issue tracker: https://github.com/Boulder-Investment-Technologies/lppls/issues

Installation

Dependencies

lppls requires:

• Python (>= 3.7)
• Matplotlib (>= 3.1.1)
• Numba (>= 0.51.2)
• NumPy (>= 1.17.0)
• Pandas (>= 0.25.0)
• SciPy (>= 1.3.0)
• Pytest (>= 6.2.1)

User installation

pip install -U lppls

Example Use

from lppls import lppls, data_loader
import numpy as np
import pandas as pd
%matplotlib inline

# read example dataset into df 
data = data_loader.sp500()

# convert index col to evenly spaced numbers over a specified interval
time = np.linspace(0, len(data)-1, len(data))

# create list of observation data, in this case, 
# daily adjusted close prices of the S&P 500
# use log price
price = np.log(data['Adj Close'].values)

# create Mx2 matrix (expected format for LPPLS observations)
observations = np.array([time, price])

# set the max number for searches to perform before giving-up
# the literature suggests 25
MAX_SEARCHES = 25

# instantiate a new LPPLS model with the S&P 500 dataset
lppls_model = lppls.LPPLS(observations=observations)

# fit the model to the data and get back the params
tc, m, w, a, b, c, c1, c2 = lppls_model.fit(observations, MAX_SEARCHES, minimizer='Nelder-Mead')

# visualize the fit
lppls_model.plot_fit()

# should give a plot like the following...

# define custom filter condition
filter_conditions_config = [
  {'condition_1':[
      (0.0, 0.1), # tc_range
      (0,1), # m_range
      (4,25), # w_range
      2.5, # O_min
      0.5, # D_min
  ]},
]

# compute the confidence indicator
res = lppls_model.mp_compute_indicator(
    workers=4, 
    window_size=120, 
    smallest_window_size=30, 
    increment=5, 
    max_searches=25,
    filter_conditions_config=filter_conditions_config
)
res_df = res_to_df(res, 'condition_1')
lppls_model.plot_confidence_indicators(res_df, title='Short Term Indicator 120-30')

# should give a plot like the following...

If you wish to store res as a pd.DataFrame, use res_to_df.

Example

res_df = lppls_model.res_to_df(res, condition_name='condition_1')
res_df.tail()
# gives the following...

Other Search Algorithms

Shu and Zhu (2019) proposed CMA-ES for identifying the best estimation of the three non-linear parameters (, , ).

The CMA-ES rates among the most successful evolutionary algorithms for real-valued single-objective optimization and is typically applied to difficult nonlinear non-convex black-box optimization problems in continuous domain and search space dimensions between three and a hundred. Parallel computing is adopted to expedite the fitting process drastically.

This approach has been implemented in a subclass and can be used as follows... Thanks to @paulogonc for the code.

from lppls import lppls_cmaes
lppls_model = lppls_cmaes.LPPLSCMAES(observations=observations)
tc, m, w, a, b, c, c1, c2 = lppls_model.fit(max_iteration=2500, pop_size=4)

Performance Note: this works well for single fits but can take a long time for computing the confidence indicators. More work needs to be done to speed it up.

References

• Filimonov, V. and Sornette, D. A Stable and Robust Calibration Scheme of the Log-Periodic Power Law Model. Physica A: Statistical Mechanics and its Applications. 2013
• Shu, M. and Zhu, W. Real-time Prediction of Bitcoin Bubble Crashes. 2019.
• Sornette, D. Why Stock Markets Crash: Critical Events in Complex Financial Systems. Princeton University Press. 2002.
• Sornette, D. and Demos, G. and Zhang, Q. and Cauwels, P. and Filimonov, V. and Zhang, Q., Real-Time Prediction and Post-Mortem Analysis of the Shanghai 2015 Stock Market Bubble and Crash (August 6, 2015). Swiss Finance Institute Research Paper No. 15-31.