Analysing Models for Volatility Clustering with Subordinated Processes: VGSA and Beyond

ArXiv ID: 2507.17431 “View on arXiv”

Authors: Sourojyoti Barick, Sudip Ratan Chandra

Abstract

This paper explores a comprehensive class of time-changed stochastic processes constructed by subordinating Brownian motion with Levy processes, where the subordination is further governed by stochastic arrival mechanisms such as the Cox Ingersoll Ross (CIR) and Chan Karolyi Longstaff Sanders (CKLS) processes. These models extend classical jump frameworks like the Variance Gamma (VG) and CGMY processes, allowing for more flexible modeling of market features such as jump clustering, heavy tails, and volatility persistence. We first revisit the theory of Levy subordinators and establish strong consistency results for the VG process under Gamma subordination. Building on this, we prove asymptotic normality for both the VG and VGSA (VG with stochastic arrival) processes when the arrival process follows CIR or CKLS dynamics. The analysis is then extended to the more general CGMY process under stochastic arrival, for which we derive analogous consistency and limit theorems under positivity and regularity conditions on the arrival process. A simulation study accompanies the theoretical work, confirming our results through Monte Carlo experiments, with visualizations and normality testing (via Shapiro-Wilk statistics) that show approximate Gaussian behavior even for processes driven by heavy-tailed jumps. This work provides a rigorous and unified probabilistic framework for analyzing subordinated models with stochastic time changes, with applications to financial modeling and inference under uncertainty.

Keywords: Stochastic Time Change, Lévy Processes, Cox-Ingersoll-Ross (CIR), Variance Gamma (VG), Jump Diffusion, Equity (General Derivatives)

Complexity vs Empirical Score

  • Math Complexity: 9.5/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats
  • Why: The paper is highly mathematically dense, featuring proofs of strong consistency and asymptotic normality for subordinated processes using advanced probability theory and stochastic calculus. However, empirical rigor is low as it lacks real-world backtesting, implementation details, or dataset usage, relying solely on Monte Carlo simulations to validate theoretical results.
  flowchart TD
    A["Research Goal: Analyze volatility clustering in<br/>subordinated processes VGSA & beyond"] --> B["Methodology: &quot;Prove consistency &amp; asymptotic normality<br/>for VG/CGMY under CIR/CKLS time change&quot;"]
    B --> C["Data/Input: &quot;Stochastic arrivals (CIR/CKLS) +<br/>Heavy-tailed Levy drivers (VG/CGMY)&quot;"]
    C --> D["Computational Process: &quot;Monte Carlo Simulation &amp;<br/>Shapiro-Wilk Normality Testing&quot;"]
    D --> E["Outcome 1: &quot;Strong Consistency<br/>(Gamma subordination)&quot;"]
    D --> F["Outcome 2: &quot;Asymptotic Normality<br/>(VGSA &amp; CGMYSA)&quot;"]
    D --> G["Outcome 3: &quot;Validated Heavy-Tail &amp;<br/>Volatility Persistence Models&quot;"]