Paper: arXiv 2501.06758

Authors: Christian Bayer, Luca Pelizzari, Jia-Jie Zhu

Abstract

We extend the signature-based primal and dual solutions to the optimal stopping problem recently introduced in [Bayer et al.: Primal and dual optimal stopping with signatures, to appear in Finance & Stochastics 2025], by integrating deep-signature and signature-kernel learning methodologies. These approaches are designed for non-Markovian frameworks, in particular enabling the pricing of American options under rough volatility. We demonstrate and compare the performance within the popular rough Heston and rough Bergomi models.

Complexity vs Empirical Score

  • Math Complexity: 8.0/10
  • Empirical Rigor: 7.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper extends signature-based optimal stopping methods for non-Markovian rough volatility models using deep signatures and signature kernels. It combines sophisticated rough path theory and deep learning with thorough numerical evaluation on realistic financial models (rough Heston and rough Bergomi), achieving remarkably tight bid-ask bounds.

Research Flowchart

  flowchart TD
    A[Research Goal: Price American Options<br/>under Rough Volatility] --> B[Inputs: Non-Markovian Models<br/>Rough Heston & Rough Bergomi]
    B --> C[Methodology: Extend Signature-Based<br/>Primal & Dual Optimal Stopping]
    C --> D[Computational Engine: Deep-Signatures<br/>& Signature-Kernel Learning]
    D --> E[Outcomes: Efficient Pricing & Performance<br/>Comparison in Rough Volatility Frameworks]

Cited References (72)

  1. Optimal stopping with signatures (2023)
  2. Signature moments to characterize laws of stochastic processes (2022)
  3. Brownian motion and stochastic calculus (1991)
  4. Global universal approximation of functional input maps on weighted spaces — → 2306.03303
  5. Functional I t ^o calculus (2019)
  6. Functional I t ^o calculus and stochastic integral representation of martingales (2013)
  7. Pricing A merican options by exercise rate optimization (2020)
  8. Machine learning for pricing A merican options in high-dimensional M arkovian and non- M arkovian models (2020)
and 64 more
  1. Solving optimal stopping problems via empirical dual optimization (2013) — link
  2. Pathwise optimization for optimal stopping problems (2012)
  3. Integration of paths, geometric invariants and a generalized B aker- H ausdorff formula (1957)
  4. Rough PDE s for local stochastic volatility models (2025)
  5. Stochastic programming models (2003)
  6. Deep optimal stopping (2019)
  7. From optimal martingales to randomized dual optimal stopping (2021) — link
  8. Unified signature cumulants and generalized M agnus expansions (2022)
  9. Multidimensional stochastic processes as rough paths: theory and applications (2010)
  10. A generalization of the strict topology (1971)
  11. A course on rough paths (2020)
  12. Rough stochastic differential equations (2021) — link
  13. Rough semimartingales and $ p $-variation estimates for martingale transforms (2020) — link
  14. Stochastic partial differential equations: a rough paths view on weak solutions via F eynman– K ac (2017)
  15. Contingent claims and market completeness in a stochastic volatility model (1997)
  16. Pricing under rough volatility (2016)
  17. The characteristic function of rough H eston models (2019)
  18. Solving optimal stopping problems via randomization and empirical dual optimization (2023)
  19. Dual pricing of A merican options by W iener chaos expansion (2018)
  20. Valuing A merican options by simulation: a simple least-squares approach (2001)
  21. Primal-dual simulation algorithm for pricing multidimensional A merican options (2004)
  22. Regression methods for pricing complex A merican-style options (2001)
  23. Optimal execution with rough path signatures (2020)
  24. Methods of mathematical finance (1998)
  25. Monte Carlo valuation of A merican options (2002)
  26. Optimal stopping and free-boundary problems (2006)
  27. An analysis of a least squares regression method for A merican option pricing (2002)
  28. Measure theory (2007)
  29. Monte C arlo sampling methods (2003)
  30. Universal approximation on path spaces and applications in finance (2022)
  31. Differential equations driven by rough signals (1998)
  32. Support Vector Machines (2008)
  33. Volatility Is Rough (2018)
  34. Volatility is (mostly) path-dependent (2023)
  35. Trades, quotes and prices: financial markets under the microscope (2018)
  36. Statistical models for earthquake occurrences and residual analysis for point processes (1988)
  37. Fractional B rownian motion and the M arkov property. (1998)
  38. Generalized F eller processes and M arkovian lifts of stochastic V olterra processes: the affine case (2020)
  39. Efficient option pricing in the rough H eston model using weak simulation schemes — → 2310.04146
  40. An extension theorem to rough paths (2007)
  41. A note on the notion of geometric rough paths (2006)
  42. True upper bounds for B ermudan products via non-nested M onte C arlo (2009)
  43. $\mathfrak X $PDE for $\mathfrak X \in \ \mathrm BS ,\mathrm FBS , \mathrm P \ $: a rough volatility context (2023) — link
  44. Conditioning on Stochastic Processes via Signatures (2023)
  45. The Signature Kernel is the solution of a G oursat PDE (2021)
  46. Primal and dual optimal stopping with signatures — → 2312.03444
  47. Efficient option pricing in the rough H eston model using weak simulation schemes (2024)
  48. Turbocharging M onte C arlo pricing for the rough B ergomi model (2018)
  49. The iisignature library: efficient calculation of iterated-integral signatures and log signatures (2018) — link
  50. Neural network regression for B ermudan option pricing (2021)
  51. Random F ourier signature features (2023) — link
  52. Multilayer feedforward networks with a nonpolynomial activation function can approximate any function (1993)
  53. Calculation of iterated-integral signatures and log signatures (2017) — link
  54. Deep signature transforms (2019)
  55. A generalized representer theorem (2001)
  56. On the N ystr "o m Method for Approximating a Gram Matrix for Improved Kernel-Based Learning. (2005)
  57. Using the N ystr "o m method to speed up kernel machines (2000)
  58. The signature of a rough path: uniqueness (2016)
  59. A high order solver for signature kernels (2024) — link
  60. Kernel mean embedding of distributions: A review and beyond (2017)
  61. Lecture notes on rough paths and applications to machine learning (2024) — link
  62. Seq2Tens: An efficient representation of sequences by low-rank tensor projections (2021)
  63. Rough volatility (2023)
  64. Optimal stopping via distribution regression: a higher rank signature approach (2023) — link

3 of 72 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.