Paper: arXiv 2412.15971

Authors: Martin Friesen, Stefan Gerhold, Kristof Wiedermann

Abstract

We study small-time central limit theorems for stochastic Volterra integral equations with Hölder continuous coefficients and general locally square integrable Volterra kernels. We prove the convergence of the finite-dimensional distributions, a functional CLT, and limit theorems for smooth transformations of the process, which covers a large class of Volterra kernels that includes rough models based on Riemann-Liouville kernels with short- and long-range dependencies. To illustrate our results, we derive asymptotic pricing formulae for digital calls on the realized variance in three different regimes. The latter provides a robust and model-independent pricing method for small maturities in rough volatility models. Finally, for the case of completely monotone kernels, we introduce a flexible framework of Hilbert space-valued Markovian lifts and derive analogous limit theorems for such lifts.

Complexity vs Empirical Score

  • Math Complexity: 9.0/10
  • Empirical Rigor: 2.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides deep mathematical proofs for small-time central limit theorems for stochastic Volterra integral equations and their Markovian lifts, particularly in the context of rough volatility. However, it is purely theoretical with mathematical derivations and lacks empirical backtesting or real market data analysis.

Research Flowchart

  flowchart TD
    A[Goal: Small-time CLTs for Stochastic Volterra Integral Equations & Markovian Lifts] --> B[Inputs: Hölder Coefficients, Locally L2 Volterra Kernels & Completely Monotone Kernels]
    B --> C[Derive Convergence of Finite-Dimensional Distributions & Functional CLT]
    C --> D[Lift Process to Hilbert Space Framework for Completely Monotone Kernels]
    C --> E[Apply CLTs to Smooth Transformations for Option Pricing]
    D --> F[Outcomes: Analogous Limit Theorems for Markovian Lifts]
    E --> F[Outcomes: Robust Small-Maturity Asymptotic Pricing Formulae for Digital Calls]

Cited References (28)

  1. Comparison principles for stochastic V olterra equations (2024)
  2. Limits of stochastic V olterra equations driven by G aussian noise (2024)
  3. Regular occupation measures of V olterra processes (2024)
  4. Self-similar G aussian M arkov processes (2020)
  5. Affine V olterra processes (2019) — link
  6. On the existence of weak solutions to stochastic V olterra equations (2023)
  7. Weak well-posedness of stochastic V olterra equations with completely monotone kernels and non-degenerate noise (2023)
  8. The central limit theorem for stochastic V olterra equations with singular kernels (2023)
and 20 more
  1. Pathwise uniqueness for singular stochastic V olterra equations with H "o lder coefficients (2022)
  2. Uniqueness for V olterra-type stochastic integral equations (2015)
  3. Volterra Equations Driven by Semimartingales (1985)
  4. A weak solution theory for stochastic Volterra equations of convolution type (2021)
  5. Non- M arkovianity of $2 K - B $ and a degeneration (2024)
  6. Local times of self-intersection and sample path properties of V olterra G aussian processes (2024)
  7. Markovian structure of the V olterra H eston model (2019)
  8. Markovian Lifts of Stochastic V olterra Equations in S obolev Spaces: Solution theory, an I to Formula and Invariant Measures (2024)
  9. Multifactor Approximation of Rough Volatility Models (2019) — link
  10. Introduction to Analytic and Probabilistic Number Theory (1995) — link
  11. Completely Monotone Functions: A Digest (2014) — link
  12. Special Functions (1999)
  13. Asymptotic analysis of a double integral occurring in the rough Bergomi model (2020)
  14. Stochastic V olterra equations with H "o lder diffusion coefficients (2023) — link
  15. Continuous Martingales and Brownian Motion (1999)
  16. The characteristic function of rough H eston models (2019) — link
  17. Perfect hedging in rough H eston models (2018)
  18. Volatility is rough (2018) — link
  19. Pricing under rough volatility (2016) — link
  20. Stochastic V olterra equations in B anach spaces and stochastic partial differential equation (2010) — link

0 of 28 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.