Time Deep Gradient Flow Method for pricing American options

ArXiv ID: 2507.17606 “View on arXiv”

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.

Keywords: Deep Galerkin Method (DGM), Time Deep Gradient Flow (TDGF), American Put Options, Multidimensional Pricing, Heston Model, Equity Derivatives

Complexity vs Empirical Score

  • Math Complexity: 8.5/10
  • Empirical Rigor: 7.0/10
  • Quadrant: Holy Grail
  • Why: 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.
  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