Paper: arXiv 2507.17606

Authors: Jasper Rou

Abstract

In this research, we explore neural network-based methods for pricing multidimensional American put options under the BlackScholes and Heston model, extending up to five dimensions. We focus on two approaches: the Time Deep Gradient Flow (TDGF) method and the Deep Galerkin Method (DGM). We extend the TDGF method to handle the free-boundary partial differential equation inherent in American options. We carefully design the sampling strategy during training to enhance performance. Both TDGF and DGM achieve high accuracy while outperforming conventional Monte Carlo methods in terms of computational speed. In particular, TDGF tends to be faster during training than DGM.

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: The paper demonstrates high mathematical complexity through advanced PDE theory, operator splitting, and neural network optimization for free-boundary problems, while also showing strong empirical rigor with concrete backtesting results comparing TDGF and DGM against Monte Carlo methods in up to five dimensions under Black-Scholes and Heston models.

Research Flowchart

  flowchart TD
  A["Research Goal: Price Multidim American Put Options"] --> B["Methodology"]
  subgraph B ["Key Methodologies"]
      B1["Time Deep Gradient Flow TDGF"]
      B2["Deep Galerkin Method DGM"]
      B3["Monte Carlo Benchmark"]
  end
  C["Data & Inputs"] --> D["Computational Processes"]
  subgraph C ["Inputs"]
      C1["Black-Scholes &amp; Heston Models<br>1D-5D Configurations"]
      C2["Strategic Sampling Strategy"]
  end
  subgraph D ["Training &amp; Evaluation"]
      D1["Neural Network Training"]
      D2["Free-Boundary PDE Solver"]
  end
  B --> C
  D --> E["Key Findings &amp; Outcomes"]
  subgraph E ["Results"]
      E1["High Pricing Accuracy"]
      E2["TDGF Faster Training than DGM"]
      E3["Outperforms Monte Carlo Speed"]
  end

Cited References (22)

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and 14 more
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  12. Deep penalty methods: A class of deep learning algorithms for solving high dimensional optimal stopping problems — → 2405.11392
  13. DGM : A deep learning algorithm for solving partial differential equations (2018)
  14. An advanced M onte C arlo method for the multi-asset H eston model (2010)

2 of 22 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.