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The Normal-Generalised Gamma-Pareto process: A novel pure-jump Lévy process with flexible tail and jump-activity properties

Fadhel Ayed, Juho Lee, François Caron

TL;DR

The paper introduces the Normal Generalised Gamma-Pareto (NGGP) process, a four-parameter pure-jump Lévy framework that decouples tail behavior and jump activity and supports exact sampling of increments for likelihood-free inference. By subordinating Brownian motion to a Generalised Gamma-Pareto subordinator, the NGGP yields a flexible class that captures heavy-tailed, power-law jumps with a tunable BG index, finite or infinite activity, and finite variance when appropriate. The authors develop both exponentiated Lévy and Ornstein-Uhlenbeck based Lévy-driven stochastic volatility models, and provide a pseudo-marginal MCMC approach for posterior inference, demonstrated on simulated and real stock data. Empirical results show improved tail fitting and predictive performance over classical Lévy models, with NGGP-based SV models offering robust risk assessment and competitive forecasting in financial time series.

Abstract

Pure-jump Lévy processes are popular classes of stochastic processes which have found many applications in finance, statistics or machine learning. In this paper, we propose a novel family of self-decomposable Lévy processes where one can control separately the tail behavior and the jump activity of the process, via two different parameters. Crucially, we show that one can sample exactly increments of this process, at any time scale; this allows the implementation of likelihood-free Markov chain Monte Carlo algorithms for (asymptotically) exact posterior inference. We use this novel process in Lévy-based stochastic volatility models to predict the returns of stock market data, and show that the proposed class of models leads to superior predictive performances compared to classical alternatives.

The Normal-Generalised Gamma-Pareto process: A novel pure-jump Lévy process with flexible tail and jump-activity properties

TL;DR

The paper introduces the Normal Generalised Gamma-Pareto (NGGP) process, a four-parameter pure-jump Lévy framework that decouples tail behavior and jump activity and supports exact sampling of increments for likelihood-free inference. By subordinating Brownian motion to a Generalised Gamma-Pareto subordinator, the NGGP yields a flexible class that captures heavy-tailed, power-law jumps with a tunable BG index, finite or infinite activity, and finite variance when appropriate. The authors develop both exponentiated Lévy and Ornstein-Uhlenbeck based Lévy-driven stochastic volatility models, and provide a pseudo-marginal MCMC approach for posterior inference, demonstrated on simulated and real stock data. Empirical results show improved tail fitting and predictive performance over classical Lévy models, with NGGP-based SV models offering robust risk assessment and competitive forecasting in financial time series.

Abstract

Pure-jump Lévy processes are popular classes of stochastic processes which have found many applications in finance, statistics or machine learning. In this paper, we propose a novel family of self-decomposable Lévy processes where one can control separately the tail behavior and the jump activity of the process, via two different parameters. Crucially, we show that one can sample exactly increments of this process, at any time scale; this allows the implementation of likelihood-free Markov chain Monte Carlo algorithms for (asymptotically) exact posterior inference. We use this novel process in Lévy-based stochastic volatility models to predict the returns of stock market data, and show that the proposed class of models leads to superior predictive performances compared to classical alternatives.

Paper Structure

This paper contains 60 sections, 2 theorems, 104 equations, 11 figures, 11 tables.

Table of Contents

  1. Introduction
  2. Notations.
  3. The NGGP process
  4. Generalised gamma subordinator
  5. Generalised gamma-Pareto subordinator
  6. Definition
  7. Properties
  8. Positive stable process.
  9. Scaled GG process.
  10. Moments and cumulants.
  11. Inverse scale parameter.
  12. Activity of the jumps and BG index.
  13. Heavy tails and power-law behaviour.
  14. Simulation of the increments.
  15. Self-decomposability.
  16. ...and 45 more sections

Key Result

Proposition 1

The random variable $Z_t\mathop{\mathrm{GGP}}\nolimits(t\eta, \sigma, \tau, c)$ is self-decomposable if $\sigma\geq 0$. That is, for any $a\in(0,1)$, there is $Z_t^{(a)}$ independent of $Z_t$ such that

Figures (11)

  • Figure 1: Histograms (left) and trace plots (right) of the posterior distributions of the parameters on the simulated data experiment. The blue line represents the value of the parameter used to generate the data. We also report the Gelman-Rubin scores to assess convergence of the chains (the lower the better, the empirical threshold for convergence is $1.1$)
  • Figure 2: Ranked squared increments on the tech companies dataset. From top to bottom row: Apple, Amazon, Facebook. The line represents the ranked $y^2$ in the test dataset; the shaded area represent the 95% credible region. Results are given for the NGGP, GH, NS and Student models in this order.
  • Figure 3: Posterior samples of the parameters on simulated data from OU-basd stochastic volatility model with NGGP marginal.
  • Figure 4: Posterior mean (solid red line) and 95% credible intervals (shaded area) of the integrated volatility. True volatility is in dashed green line.
  • Figure 5: Posterior estimates (solid line) and 95% credible intervals of the integrated volatility under the NG (left) and NGGP (right) models for the AORD index. The true integrated volatility is represented by a green dashed line.
  • ...and 6 more figures

Theorems & Definitions (2)

  • Proposition 1
  • Proposition 2