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

TL;DR

The paper tackles fast, arbitrage-free pricing of European basket options under jump-diffusion dynamics by solving a partial integro-differential equation (PIDE). It introduces a deep implicit-explicit minimizing movement (IMEX) framework that evolves the solution with a residual-type neural network at each time step, using two integral-operator discretizations: a sparse-grid Gauss–Hermite approach and an ANN-based quadrature, along with a decomposition $u=w+v$ into time value and intrinsic value. A domain truncation strategy and warm-starting across steps enhance accuracy and efficiency, and the method is validated on the Merton model with $d=5$ and $d=15$ assets, showing competitive accuracy and favorable scalability relative to deep Galerkin and deep BSDE solvers. The work yields a practical, space-time solution framework that facilitates Greeks computation and offers a competitive alternative for high-dimensional derivative pricing under jump risks.

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.

Figures (11)

• Figure 2.1: A two-dimensional example for the domain truncation. The green boundary defines the projection surface for the option price in extreme moneynesses.
• Figure 3.1: Flowchart of the deep neural network modeling the solution of the PIDE for a time instance $t=t_k$.
• Figure 3.2: The architecture of a DGM layer.
• Figure 3.3: Sparse sampling in $\mathbb R^2$ using the Gauss--Hermite quadrature for normally distributed random variables with correlation equal to $\frac{1}{2}$.
• Figure 4.1: Basket option prices (top right), differences between price and payoff (bottom right) and differences between the proposed method and quasi Monte Carlo (left), using the implicit-explicit Euler scheme and the Gauss--Hermite quadrature for the integral, for $d=5$ .

Theorems & Definitions (1)

• Remark 3.1