Paper: arXiv 2508.15237

Authors: Jingtang Ma, Xianglin Wu, Wenyuan Li

Abstract

This paper studies the pricing problem in which the underlying asset follows a non-Markovian stochastic volatility model. Classical partial differential equation methods face significant challenges in this context, as the option prices depend not only on the current state, but also on the entire historical path of the process. To overcome these difficulties, we reformulate the asset dynamics as a rough stochastic differential equation and then represent the rough paths via linear or non-linear combinations of time-extended Brownian motion signatures. This representation transforms a rough stochastic differential equation to a classical stochastic differential equation, allowing the application of standard analytical tools. We propose a deep signature approach for both linear and nonlinear representations and rigorously prove the convergence of the algorithm. Numerical examples demonstrate the effectiveness of our approach for both Markovian and non-Markovian volatility models, offering a theoretically grounded and computationally efficient framework for option pricing.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 4.0/10
  • Quadrant: Lab Rats — theoretically deep, empirically untested

Why this score: The paper provides a strong theoretical contribution by combining signature transforms with rough SDEs to solve non-Markovian option pricing problems, supported by rigorous convergence proofs. However, its empirical validation is purely numerical and lacks real-world financial data backtesting.

Research Flowchart

  flowchart TD
    A[Goal: Price Options under Non-Markovian Stochastic Volatility Models] --> B[Input: Historical Path & Rough Stochastic Differential Equations]
    B --> C[Method: Transform Rough Paths via Time-Extended Brownian Motion Signatures]
    C --> D[Model: Apply Deep Signature Framework for Linear/Nonlinear Representations]
    D --> E[Computation: Train Deep Neural Networks & Map Signatures to Classical SDEs]
    E --> F[Validation: Prove Algorithm Convergence Rigorously]
    F --> G[Outcome: Efficient, Theoretically Grounded Pricing for Markovian & Non-Markovian Models]

Cited References (27)

  1. Signature volatility models: Pricing and hedging with F ourier (2025)
  2. Path-dependent processes from signatures (2024) — link
  3. Rough PDE s for local stochastic volatility models (2025)
  4. Pricing under rough volatility (2016)
  5. Pricing A merican options under rough volatility using deep-signatures and signature-kernels — → 2501.06758
  6. Deep signature algorithm for multidimensional path-dependent options (2024)
  7. Malliavin calculus for rough stochastic differential equations (2024) — link
  8. Parameter dependent rough SDEs with applications to rough PDEs (2024) — link
and 19 more
  1. Integration of paths, geometric invariants and a generalized B aker- H ausdorff formula (1957)
  2. Signature-based models: T heory and calibration (2023)
  3. Stochastic partial differential equations: a rough paths view on weak solutions via F eynman– K ac (2017)
  4. The characteristic function of rough H eston models (2019)
  5. Rough stochastic differential equations (2021) — link
  6. Controlled rough SDEs, pathwise stochastic control and dynamic programming principles (2024) — link
  7. Approximation of dynamical systems by continuous time recurrent neural networks (1993)
  8. A Course on Rough Paths: With an Introduction to Regularity Structures (2020)
  9. Deep signature transforms (2019)
  10. Reflected backward stochastic differential equations with rough drivers (2024) — link
  11. Differential equations driven by rough signals (1998)
  12. Differential equations driven by rough paths: Ecole d’Et 'e de Probabilit 'e s de Saint-Flour XXXIV-2004 (2007)
  13. Sigformer: S ignature transformers for deep hedging (2023)
  14. On the theory of the B rownian motion (1930)
  15. Hedging with memory: shallow and deep learning with signatures — → 2508.02759
  16. Log-pde methods for rough signature kernels (2024) — link
  17. Probabilities and potential, c: potential theory for discrete and continuous semigroups (2011)
  18. Probability with martingales (1991)
  19. A continuous-time GARCH process driven by a L 'e vy process: stationarity and second-order behaviour (2004)

2 of 27 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.