Paper: arXiv 2510.14108

Authors: Rohan Shenoy, Peter Kempthorne

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.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 5.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper provides a rigorous mathematical framework extending characteristic functions of subordinated Brownian motion into the complex plane to extract time-change processes non-parametrically. While the theoretical derivations and mathematical proofs are strong, the empirical evaluation serves primarily as a brief illustrative example using S&P 500 log-returns.

Research Flowchart

  flowchart TD
    A[Goal: Extract Stochastic Time-Change from Financial Returns] --> B[Input: S&P 500 Log-Returns Data]
    B --> C[Review & Framework: Gaussian Variance-Mean Mixtures & VG Process]
    C --> D[Methodology: Extend Characteristic Functions to Complex Plane]
    D --> E[Computation: Non-Parametric Fourier Decomposition of Subordinated BM]
    E --> F[Outcome: Direct Characterization of Stochastic Subordinator & Empirical Proof]

Cited References (47)

  1. The Laplace Distribution and Generalizations (2001)
  2. OPTION PRICING WITH V. G. MARTINGALE COMPONENTS (1991)
  3. Lévy Processes in Finance: Theory, Numerics, and Empirical Facts (2000)
  4. Pricing European Options under Stochastic Volatility Models: Case of Five-Parameter Variance-Gamma Process (2023)
  5. The Variance Gamma Process and Option Pricing (1998)
  6. The Variance Gamma (V.G.) Model for Share Market Returns (1990)
  7. The Distribution of Share Price Changes (1972)
  8. Stochastic Volatility and Option Valuation: A Pricing-Density Approach (1995)
and 39 more
  1. Option Pricing for Log-Symmetric Distributions of Returns (2009)
  2. Stochastic Calculus: An Introduction with Applications, lecture notes (2014)
  3. Chapter 11. Volatility Smiles (2019) — link
  4. Why do we smile? On the determinants of the implied volatility function (1999)
  5. The Variance-Gamma Distribution: A Review (2023) — link
  6. On the minimal entropy martingale measure for Lévy processes (2020) — link
  7. Option pricing by Esscher transforms. (1995) — link
  8. On the probability function in the collective theory of risk (1932) — link
  9. Information Theory and Statistical Mechanics (1957) — link
  10. Characterization problems in mathematical statistics (1973)
  11. The pricing of options and corporate liabilities (1973)
  12. The Fine Structure of Asset Returns: An Empirical Investigation (2002) — link
  13. A Modern Approach to Probability Theory (1997)
  14. A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions (2019) — link
  15. The Empirical Characteristic Function and Its Applications (1977)
  16. Variance Gamma process in the option pricing model (2021)
  17. An empirical test of the variance gamma option pricing model (2002) — link
  18. Empirical Characteristic Function Estimation and Its Applications (2004) — link
  19. Parametric estimation of Lévy processes (2014) — link
  20. Lévy Processes and Stochastic Calculus (2004)
  21. On Measures of Entropy and Information (1961) — link
  22. Option Pricing in a Dynamic Variance-Gamma Model (2010)
  23. Characteristic Functions (1970)
  24. Complex Analysis (2005)
  25. A Short Note on Integral Transformations and Conversion Formulas for Sequence Generating Functions (2019) — link
  26. The exact confidence interval for the scale parameter and the mvue of the laplace distribution (1985) — link
  27. Sous-espaces denses dans $L^1$ ou $H^1$ et représentation des martingales (1978) — link
  28. A Subordinated Stochastic Process Model with Finite Variance for Speculative Prices (1973)
  29. An Analysis of Variance Test for Normality (Complete Samples) (1965) — link
  30. An Introduction to Harmonic Analysis (2004)
  31. Normal Variance-Mean Mixtures and z Distributions (1982)
  32. Probability: theory and examples (1996)
  33. A guide to Brownian motion and related stochastic processes (2018) — link
  34. Issledovaniya po teorii sluchainykh protsessov (1961)
  35. Studies in the theory of random processes. (Translated from the Russian by Scripta Technica, Inc) (1965)
  36. The Skorokhod embedding problem and its offspring (2004) — link
  37. On continuous martingales (1965) — link
  38. On Embedding Right Continuous Martingales in Brownian Motion (1972)
  39. Time change (2010) — link

0 of 47 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.