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On Time-subordinated Brownian Motion Processes for Financial Markets

Rohan Shenoy, Peter Kempthorne

TL;DR

This work develops a Fourier-based framework for time-subordinated Brownian motion in financial markets by leveraging Gaussian variance-mean mixtures to model log-returns and introducing a time-change transform that recovers the subordinating process from price data. It shows that the Variance-Gamma family arises from a Gamma subordinator and provides a semi-parametric method to infer the mixing distribution V via the generalized variance-mixing transform, enabling direct empirical characterization of stochastic time-change. An explicit transform S^θ[X] is proposed to extract the subordinator density from observed price paths, with existence established through analytic extension of characteristic functions. Empirical analysis on SPX daily returns supports a Gamma-subordinator interpretation while highlighting limitations of simple Lévy-subordinator assumptions and pointing to richer future models.

Abstract

In the context of time-subordinated Brownian motion models, Fourier theory and methodology are proposed to modelling the stochastic distribution of time increments. Gaussian Variance-Mean mixtures and time-subordinated models are reviewed with a key example being the Variance-Gamma process. A non-parametric characteristic function decomposition of subordinated Brownian motion is presented. The theory requires an extension of the real domain of certain characteristic functions to the complex plane, the validity of which is proven here. This allows one to characterise and study the stochastic time-change directly from the full process. An empirical decomposition of S\&P log-returns is provided to illustrate the methodology.

On Time-subordinated Brownian Motion Processes for Financial Markets

TL;DR

This work develops a Fourier-based framework for time-subordinated Brownian motion in financial markets by leveraging Gaussian variance-mean mixtures to model log-returns and introducing a time-change transform that recovers the subordinating process from price data. It shows that the Variance-Gamma family arises from a Gamma subordinator and provides a semi-parametric method to infer the mixing distribution V via the generalized variance-mixing transform, enabling direct empirical characterization of stochastic time-change. An explicit transform S^θ[X] is proposed to extract the subordinator density from observed price paths, with existence established through analytic extension of characteristic functions. Empirical analysis on SPX daily returns supports a Gamma-subordinator interpretation while highlighting limitations of simple Lévy-subordinator assumptions and pointing to richer future models.

Abstract

In the context of time-subordinated Brownian motion models, Fourier theory and methodology are proposed to modelling the stochastic distribution of time increments. Gaussian Variance-Mean mixtures and time-subordinated models are reviewed with a key example being the Variance-Gamma process. A non-parametric characteristic function decomposition of subordinated Brownian motion is presented. The theory requires an extension of the real domain of certain characteristic functions to the complex plane, the validity of which is proven here. This allows one to characterise and study the stochastic time-change directly from the full process. An empirical decomposition of S\&P log-returns is provided to illustrate the methodology.
Paper Structure (6 sections, 6 theorems, 40 equations, 6 figures)

This paper contains 6 sections, 6 theorems, 40 equations, 6 figures.

Table of Contents

  1. Introduction
  2. Gaussian Variance-Mean Mixtures
  3. Time-Subordinated Brownian Motion Processes
  4. Time-Change Decomposition
  5. Example Empirical Analysis of a Decomposed Time-Change Process
  6. Discussion

Key Result

Proposition 1

Let $X = \theta V + \sigma\sqrt{V}Z$ be a Gaussian variance-mean mixture as in Definition defvariancemeanmix. Then the characteristic function of the $X$, $\psi_X$ is given by

Figures (6)

  • Figure 1: Gamma distribution fit for the stochastic variance $V = \mathcal{V}^0[X]$ for S$\&$P500 daily log returns data $X$ (January 2022 to January 2024)
  • Figure 2: Realised components of a Time-subordinated Brownian motion process: $VG(t;\theta = 0.1, \sigma = 0.05, \nu = 0.05)$
  • Figure 3: Components of $\mathcal{S}^{\theta}[\cdot]$
  • Figure 4: Fit of Gamma process as the subordinator process for S$\&$P500 daily log returns (January 2022 to January 2024) under the time-subordinator transform
  • Figure 5: Evolution of (a) the empirical time-change process and (b) the fitted Gamma process for S$\&$P500 daily log returns (January 2022 to January 2024) from the time-change transform
  • ...and 1 more figures

Theorems & Definitions (24)

  • Definition 1: Gaussian variance-mean mixture
  • Remark 1
  • Proposition 1
  • Remark 2
  • proof
  • Corollary 1
  • Remark 3
  • Definition 2: Variance-mixing transform
  • Remark 4
  • Definition 3: Brownian motion
  • ...and 14 more