Paper: arXiv 2610.32970

Authors: Aleksandar Arandjelovic, Patrick Cheridito

Abstract

Stratification can reduce the estimator variance in Monte Carlo simulations by partitioning the sample space according to a low-dimensional factor and sampling separately within the resulting strata, making it well suited to the valuation of path-dependent claims in high-dimensional financial models. Its performance, however, depends critically on the choice of the stratification factor, and effective low-dimensional factors are generally difficult to identify in complex continuous-time models. Neural networks provide a flexible approximation class for this purpose and have proved useful in related variance-reduction problems in quantitative finance. In this paper, we study the construction of neural stratification factors for derivative pricing under proportional allocation in continuous-time models. Since the numerical construction of a stratification factor typically involves approximation, we establish continuity results for the associated conditional expectations and, more generally, conditional distributions, and provide several counterexamples showing the limits of this continuity theory. Combined with density results for neural networks, our findings yield a universal approximation theorem for admissible stratification factors. We conclude with numerical experiments involving a constant-maturity-swap range accrual in a two-factor Gaussian short-rate model and a worst-of autocallable in a Wishart stochastic covariance model, where the learned factors outperform all considered benchmarks.

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: This paper presents a highly mathematical treatment of neural stratification factors, including continuity results and a universal approximation theorem. It is complemented by solid numerical experiments on complex derivatives, demonstrating strong empirical performance. The blend of theoretical depth and practical application is commendable.

Research Flowchart

  flowchart TD
    A[Research Goal: Optimize Derivative Pricing with Stratification] --> B{Problem: Identifying Effective Stratification Factors in Complex Models};
    B --> C[Methodology: Develop Neural Network-based Stratification Factors];
    C --> D{Computational Process: Neural Network Training for Factor Identification};
    D --> E[Data/Inputs: Continuous-Time Financial Models (e.g., Gaussian Short-Rate, Wishart Stochastic Covariance)];
    E --> F[Key Findings: Universal Approximation Theorem for Admissible Stratification Factors];
    F --> G[Outcomes: Learned Factors Outperform Benchmarks for Derivative Pricing (e.g., CMS Range Accrual, Worst-of Autocallable)];