Paper: arXiv 2312.03444

Authors: Christian Bayer, Luca Pelizzari, John Schoenmakers

Abstract

We propose two signature-based methods to solve the optimal stopping problem - that is, to price American options - in non-Markovian frameworks. Both methods rely on a global approximation result for $L^p-$functionals on rough path-spaces, using linear functionals of robust, rough path signatures. In the primal formulation, we present a non-Markovian generalization of the famous Longstaff-Schwartz algorithm, using linear functionals of the signature as regression basis. For the dual formulation, we parametrize the space of square-integrable martingales using linear functionals of the signature, and apply a sample average approximation. We prove convergence for both methods and present first numerical examples in non-Markovian and non-semimartingale regimes.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 5.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents a rigorous theoretical framework extending the Longstaff-Schwartz algorithm and dual formulations to non-Markovian settings using rough path signatures, supported by convergence proofs. While mathematically sophisticated and highly novel, the empirical section focuses on basic numerical examples rather than extensive backtests on historical market data.

Research Flowchart

  flowchart TD
    A[Research Goal: Price American Options in Non-Markovian Frameworks] --> B[Core Theory: Global Approximation of Lp-functionals via Path Signatures]
    
    B --> C[Primal Method: Non-Markovian Longstaff-Schwartz via Signature Regression]
    B --> D[Dual Method: Martingale Parameterization & Sample Average Approximation]
    
    E[Inputs: Rough Path Spaces & Asset Price Paths] --> C
    E --> D
    
    C --> F[Empirical Evaluation & Convergence Proofs]
    D --> F
    
    F --> G[Outcomes: Proven Convergence & Validated Pricing in Non-Semimartingale Regimes]

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2 of 51 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.