jump-diffusion-estimation 0.2.2

Simulation and maximum-likelihood estimation of jump-diffusion processes with pluggable asymmetric, heavy-tailed jump distributions, plus likelihood-based inference and a test for the presence of jumps.

Jump-Diffusion Parameter Estimation

A comprehensive Python library for simulating and estimating parameters of jump-diffusion processes with asymmetric jump distributions.

🚀 Features

• Flexible Simulation: Generate jump-diffusion paths with customizable parameters
• Maximum Likelihood Estimation: Robust parameter estimation using mixture distributions
• Pluggable Jump Distributions: Skew-normal, Normal (Merton), and the Skewed Generalized Error Distribution (SGED) built in, with a simple interface (jump_distribution=) to add more
• Goodness-of-Fit Comparison: Rank candidate jump distributions on the same data via AIC/BIC and a simulation-based Kolmogorov-Smirnov test
• Validation Tools: Monte Carlo experiments for method validation
• Extensible Architecture: Easy to add new models, jump distributions, and estimation methods
• Educational Focus: Comprehensive documentation and tutorials

📊 Model

Our implementation focuses on jump-diffusion processes of the form:

dX_t = μ dt + σ dW_t + J_t dN_t

Where:

• μ: drift parameter
• σ: diffusion volatility
• W_t: Brownian motion
• J_t: jump sizes (asymmetrically distributed)
• N_t: jump arrival times (Bernoulli approximation)

🛠️ Installation

# Install from PyPI
pip install jump-diffusion-estimation

Or install the latest development version from source:

git clone https://github.com/jdospina/jump-diffusion-estimation.git
cd jump-diffusion-estimation
pip install -e .

🎯 Quick Start

import numpy as np
from jump_diffusion import JumpDiffusionSimulator, JumpDiffusionEstimator

# Create simulator
simulator = JumpDiffusionSimulator(
    mu=0.05,           # 5% annual drift
    sigma=0.2,         # 20% annual volatility
    jump_prob=0.1,     # 10% jump probability per period
    jump_scale=0.15,   # jump magnitude scale
    jump_skew=2.0      # positive skewness
)

# Simulate a path
times, path, jumps = simulator.simulate_path(T=1.0, n_steps=252)

# Estimate parameters
increments = np.diff(path)
dt = times[1] - times[0]
estimator = JumpDiffusionEstimator(increments, dt)
results = estimator.estimate()

print(f"Estimated drift: {results['parameters']['mu']:.4f}")
print(f"Estimated volatility: {results['parameters']['sigma']:.4f}")

Using a different jump distribution

Jumps follow a skew-normal distribution by default. Other distributions can be plugged in via jump_distribution, both when simulating and when estimating:

from jump_diffusion.distributions import SGEDJump

simulator = JumpDiffusionSimulator(
    mu=0.05, sigma=0.2, jump_prob=0.1,
    jump_distribution=SGEDJump(),
    jump_loc=0.0, jump_scale=0.15, jump_nu=1.5, jump_xi=2.0,
)
times, path, jumps = simulator.simulate_path(T=1.0, n_steps=252)

increments = np.diff(path)
dt = times[1] - times[0]
estimator = JumpDiffusionEstimator(increments, dt, jump_distribution=SGEDJump())
results = estimator.estimate()

Distributions without a known closed-form likelihood (like SGED) fall back to a generic FFT-based convolution to approximate the mixture density, so adding a new distribution only requires implementing its pdf.

Robust estimation with differential evolution

The default L-BFGS-B optimizer needs a reasonable initial guess and can stall on harder mixture likelihoods (SGED in particular). Differential evolution searches globally instead, needing no initial guess — the applied finding of the thesis this library is based on:

results = estimator.estimate(method="differential_evolution", seed=42)

It costs thousands of likelihood evaluations (seconds instead of milliseconds), with defaults ported from the thesis (rand/1 strategy, DEoptim-style population sizing, early stopping on convergence).

Standard errors via Likelihood Profiling

In complex jump-diffusion mixture models, the numerical Hessian is often unstable or ill-conditioned. Standard errors and 95% confidence intervals can be robustly calculated using Profile Likelihood. After estimating the parameters (preferably with global optimization), you can run:

# Compute standard errors and confidence intervals using a Wilks' theorem threshold
se_results = estimator.estimate_standard_errors(n_points=5, confidence_level=0.95)

# The results table now includes standard errors and CI bounds
estimator.diagnostics()

# Visualize the profile log-likelihood curves
estimator.plot_profiles()

Comparing jump distributions

JumpDistributionComparison fits several candidate jump distributions to the same data and ranks them by AIC/BIC plus a simulation-based Kolmogorov-Smirnov test:

from jump_diffusion.distributions import NormalJump, SGEDJump, SkewNormalJump
from jump_diffusion.validation import JumpDistributionComparison

comparison = JumpDistributionComparison(increments, dt)
comparison.fit("Normal", NormalJump())
comparison.fit("SkewNormal", SkewNormalJump())
comparison.fit("SGED", SGEDJump())

print(comparison.compare())  # ranked by AIC, includes KS statistic/p-value
comparison.plot_comparison()

📚 Examples

Ready-to-run scripts live in the examples/ directory:

• basic_usage.py – demonstrates basic library usage
• validation_experiment.py – runs Monte Carlo validation experiments

Interactive notebooks live in the notebooks/ directory. Every notebook is available in English and in Spanish (suffix _spanish):

• getting_started.ipynb – The canonical end-to-end tutorial: simulate, estimate (L-BFGS-B and Differential Evolution), quantify uncertainty via the three inference routes (profile / Wald / bootstrap), test for jumps, and compare jump distributions. Start here.
  • 🇪🇸 getting_started_spanish.ipynb – Tutorial completo de principio a fin: simulación, estimación (L-BFGS-B y Evolución Diferencial), inferencia por tres rutas (perfil / Wald / bootstrap), contraste de saltos y comparación de distribuciones.
• jump_diffusion_playground.ipynb – Interactive playground: pick a jump distribution (Normal, Skew-Normal, SGED, Kou, Student-t), wiggle the sliders, simulate, and play the "guess the parameters" game.
  • 🇪🇸 jump_diffusion_playground_spanish.ipynb – Laboratorio interactivo: elige una distribución de saltos, mueve los controles, simula y juega a adivinar los parámetros.
• differential_evolution_showcase.ipynb – Differential Evolution vs. L-BFGS-B on the multimodal mixture likelihood of the SGED jump-diffusion model.
  • 🇪🇸 differential_evolution_showcase_spanish.ipynb – Evolución Diferencial frente a L-BFGS-B sobre la verosimilitud multimodal del modelo de difusión con saltos SGED.
• sp500_case_study.ipynb – Real S&P 500 data: parameter estimation, simulated-vs-real comparison, and a ranking of all five jump distributions by AIC/BIC and bootstrap KS.
  • 🇪🇸 sp500_case_study_spanish.ipynb – Datos reales del S&P 500: estimación de parámetros, comparación simulado-vs-real y ranking de las cinco distribuciones de salto por AIC/BIC y KS bootstrap.

Notebook setup

Install optional dependencies and launch Jupyter to explore the notebooks:

pip install notebook ipywidgets matplotlib
jupyter notebook

📖 Academic References

The numerical methods and statistical models implemented in this library build on the following literature:

1. Calibration with Differential Evolution and the SGED:

   • Ospina Arango, J. D. (2009). MSc thesis. Universidad Nacional de Colombia. (Foundations of applying the SGED and Differential Evolution to jump-diffusion processes.)
   • Ardia, D., Ospina, J. D., & Giraldo, N. D. (2011). Jump-diffusion calibration using differential evolution. Wilmott, 2011(55), 76-79.
   • Storn, R., & Price, K. (1997). Differential evolution–a simple and efficient heuristic for global optimization over continuous spaces. Journal of global optimization, 11(4), 341-359.

2. Jump-diffusion models and distributions:

   • Merton, R. C. (1976). Option pricing when underlying stock returns are discontinuous. Journal of financial economics, 3(1-2), 125-144.
   • Theodossiou, P. (2015). Skewed Generalized Error Distribution of Financial Assets and Option Pricing. Multinational Finance Journal, 19(4), 223-266.

3. Statistical inference (likelihood profiling):

   • Wilks, S. S. (1938). The large-sample distribution of the likelihood ratio for testing composite hypotheses. The Annals of Mathematical Statistics, 9(1), 60-62.

🤝 Contributing

We welcome contributions! Please see our Contributing Guidelines for details.

📄 License

This project is licensed under the MIT License - see the LICENSE file for details.

🙏 Acknowledgments

• Inspired by classical jump-diffusion literature
• Built with love for the quantitative finance community
• Special thanks to contributors and users