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Fast Likelihood-Free Parameter Estimation for Lévy Processes

Nicolas Coloma, William Kleiber

TL;DR

The paper tackles parameter estimation for Lévy processes when likelihoods are intractable by introducing Neural Bayes Estimation (NBE), a simulation-based, likelihood-free method that uses a DeepSets architecture to respect permutation invariance of increments. The authors prove a risk-consistency result for the DeepSets-based Bayes estimator, demonstrate strong empirical gains in both accuracy and computational speed over classical methods, and provide uncertainty quantification via bootstrap and posterior-quantile approaches. They validate NBE through extensive simulations on standard and deep-subordination Lévy models and apply it to high-frequency cryptocurrency returns, where daily parameter estimates and credible intervals are obtained in seconds. The study highlights NBE’s scalability and practicality for nonstationary, data-rich financial applications and outlines future directions, including multivariate extensions and refined posterior inference.

Abstract

Lévy processes are widely used in financial modeling due to their ability to capture discontinuities and heavy tails, which are common in high-frequency asset return data. However, parameter estimation remains a challenge when associated likelihoods are unavailable or costly to compute. We propose a fast and accurate method for Lévy parameter estimation using the neural Bayes estimation (NBE) framework -- a simulation-based, likelihood-free approach that leverages permutation-invariant neural networks to approximate Bayes estimators. We contribute new theoretical results, showing that NBE results in consistent estimators whose risk converges to the Bayes estimator under mild conditions. Moreover, through extensive simulations across several Lévy models, we show that NBE outperforms traditional methods in both accuracy and runtime, while also enabling two complementary approaches to uncertainty quantification. We illustrate our approach on a challenging high-frequency cryptocurrency return dataset, where the method captures evolving parameter dynamics and delivers reliable and interpretable inference at a fraction of the computational cost of traditional methods. NBE provides a scalable and practical solution for inference in complex financial models, enabling parameter estimation and uncertainty quantification over an entire year of data in just seconds. We additionally investigate nearly a decade of high-frequency Bitcoin returns, requiring less than one minute to estimate parameters under the proposed approach.

Fast Likelihood-Free Parameter Estimation for Lévy Processes

TL;DR

The paper tackles parameter estimation for Lévy processes when likelihoods are intractable by introducing Neural Bayes Estimation (NBE), a simulation-based, likelihood-free method that uses a DeepSets architecture to respect permutation invariance of increments. The authors prove a risk-consistency result for the DeepSets-based Bayes estimator, demonstrate strong empirical gains in both accuracy and computational speed over classical methods, and provide uncertainty quantification via bootstrap and posterior-quantile approaches. They validate NBE through extensive simulations on standard and deep-subordination Lévy models and apply it to high-frequency cryptocurrency returns, where daily parameter estimates and credible intervals are obtained in seconds. The study highlights NBE’s scalability and practicality for nonstationary, data-rich financial applications and outlines future directions, including multivariate extensions and refined posterior inference.

Abstract

Lévy processes are widely used in financial modeling due to their ability to capture discontinuities and heavy tails, which are common in high-frequency asset return data. However, parameter estimation remains a challenge when associated likelihoods are unavailable or costly to compute. We propose a fast and accurate method for Lévy parameter estimation using the neural Bayes estimation (NBE) framework -- a simulation-based, likelihood-free approach that leverages permutation-invariant neural networks to approximate Bayes estimators. We contribute new theoretical results, showing that NBE results in consistent estimators whose risk converges to the Bayes estimator under mild conditions. Moreover, through extensive simulations across several Lévy models, we show that NBE outperforms traditional methods in both accuracy and runtime, while also enabling two complementary approaches to uncertainty quantification. We illustrate our approach on a challenging high-frequency cryptocurrency return dataset, where the method captures evolving parameter dynamics and delivers reliable and interpretable inference at a fraction of the computational cost of traditional methods. NBE provides a scalable and practical solution for inference in complex financial models, enabling parameter estimation and uncertainty quantification over an entire year of data in just seconds. We additionally investigate nearly a decade of high-frequency Bitcoin returns, requiring less than one minute to estimate parameters under the proposed approach.
Paper Structure (26 sections, 3 theorems, 29 equations, 12 figures, 7 tables)

This paper contains 26 sections, 3 theorems, 29 equations, 12 figures, 7 tables.

Table of Contents

  1. Introduction
  2. Framework
  3. Background on Lévy Processes
  4. Examples of Common Models
  5. Deep Variance Gamma Processes
  6. Estimation Methods
  7. Classical Methods
  8. Neural Bayes Estimation
  9. Loss Functions
  10. Uncertainty Quantification
  11. DeepSets Architecture for Lévy Processes
  12. Theoretical Guarantees for DeepSets-based Bayes Estimation
  13. Simulation Studies
  14. Experimental Setup
  15. Standard Lévy Models
  16. ...and 11 more sections

Key Result

Theorem 1

Let $Z(t)$ be a Lévy process on $\mathbb{R}$, then for $\omega \in \mathbb{R}$, where $\gamma \in \mathbb{R}$, $\sigma \geq 0$, and $\nu$ is a measure on $\mathbb{R}$ such that $\nu(\{0\})=0$ and $\int ( |x|^2 \wedge 1)\nu(dx) < \infty$ .

Figures (12)

  • Figure 1: Scatter plots of true versus estimated values for $\sigma^2, \alpha_1$, and $\alpha_2$, colored by their $\mathbb{L}_2^{f}$ difference. The dashed black line represents the identity.
  • Figure 2: Scatter plots of $\alpha_1$ versus $\alpha_2$, colored by their $\mathbb{L}_2^{f}$ difference. The left panel shows true parameters, while the right panel shows the corresponding estimates.
  • Figure 3: Daily parameter estimation using NBE for 2022. Each row represents bitcoin (BTC), ethereum (EHT) and ripple (XRP) respectively. The left most panel represents the log returns data. The following panels are the estimates of $\sigma^2$, $\alpha_1$, and $\alpha_2$, of a level $2$ DVG process, respectively. The dashed lines corresponds to Jan 25, May 11, Jun 14, and Nov 11.
  • Figure 4: Estimated characteristic functions (left column) and probability density functions (right column) for selected days in 2022, using estimated parameters. Each row corresponds to a different cryptocurrency: BTC (top), ETH (middle), and XRP (bottom).
  • Figure 5: Estimated daily parameters and their $90\%$ confidence intervals for of a level-2 DVG model for BTC log returns in 2022.
  • ...and 7 more figures

Theorems & Definitions (3)

  • Theorem 1: Lévy-Khintchine
  • Theorem 2
  • Corollary 3