A deep implicit-explicit minimizing movement method for option pricing in jump-diffusion models

Paper: arXiv 2401.06740

Abstract

We develop a novel deep learning approach for pricing European basket options written on assets that follow jump-diffusion dynamics. The option pricing problem is formulated as a partial integro-differential equation, which is approximated via a new implicit-explicit minimizing movement time-stepping approach, involving approximation by deep, residual-type Artificial Neural Networks (ANNs) for each time step. The integral operator is discretized via two different approaches: (a) a sparse-grid Gauss-Hermite approximation following localised coordinate axes arising from singular value decompositions, and (b) an ANN-based high-dimensional special-purpose quadrature rule. Crucially, the proposed ANN is constructed to ensure the appropriate asymptotic behavior of the solution for large values of the underlyings and also leads to consistent outputs with respect to a priori known qualitative properties of the solution. The performance and robustness with respect to the dimension of these methods are assessed in a series of numerical experiments involving the Merton jump-diffusion model, while a comparison with the deep Galerkin method and the deep BSDE solver with jumps further supports the merits of the proposed approach.

Complexity vs Empirical Score

• Math Complexity: 9.5/10
• Empirical Rigor: 7.0/10
• Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper presents advanced mathematical theory involving minimizing movement methods, implicit-explicit schemes, and PIDEs, indicating high math complexity, while the numerical experiments and comparison to benchmarks provide significant empirical validation but lack full backtest-ready details.

Research Flowchart

  flowchart TD
  A["Research Goal<br>Pricing European Basket Options<br>in Jump-Diffusion Models"] --> B["Formulate Problem<br>Partial Integro-Differential Equation"]
  B --> C["Deep Implicit-Explicit<br>Minimizing Movement Method"]
  C --> D{"Integral Operator<br>Discretization"}
  D --> E["Sparse-Grid Gauss-Hermite<br>SVD Approximation"]
  D --> F["ANN-based<br>High-Dim Quadrature Rule"]
  E & F --> G["Deep Residual ANN Training<br>with Asymptotic Constraints"]
  G --> H{"Performance Evaluation"}
  H --> I["Experiments on<br>Merton Jump-Diffusion Model"]
  H --> J["Comparison with<br>Deep Galerkin & Deep BSDE"]
  I & J --> K["Key Findings<br>Effective High-Dim Pricing<br>Robustness & Consistency Verified"]

Cited References (16)

1. Jumps and Stochastic Volatility: Exchange Rate Processes Implicit in D eutsche Mark Options (1996)
2. Option Valuation Using the Fast F ourier Transform (1999)
3. Optimal Damping with Hierarchical Adaptive Quadrature for Efficient F ourier Pricing of Multi-Asset Options in L évy Models (2023)
4. Quasi- M onte C arlo for Efficient F ourier Pricing of Multi-Asset Options (2024)
5. A Jump-Diffusion Model for Option Pricing (2002)
6. Option Pricing When Underlying Stock Returns are Discontinuous (1976)
7. Jump-Diffusion Models (2003)
8. DGM : A deep learning algorithm for solving partial differential equations (2018)

0 of 16 cited works are in the corpus; resolved via arXiv source bundles, Semantic Scholar and OpenAlex. Corpus-wide: 2518 linked papers.