🤖 AI Summary
This work addresses the high-dimensional European basket option pricing problem under jump-diffusion models. We propose a novel hybrid numerical-deep-learning method: the associated partial integro-differential equation (PIDE) is discretized via an implicit-explicit minimum-movement time-stepping scheme, and its solution is approximated layer-by-layer using a residual-type deep neural network. Crucially, we incorporate asymptotic behavior constraints into the network architecture to enforce financial consistency in the large-variable regime. To enhance high-dimensional numerical integration, we combine two complementary strategies: (i) singular value decomposition (SVD)-accelerated sparse-grid Gauss–Hermite quadrature, and (ii) a tailored artificial neural network (ANN)-based high-dimensional quadrature. Numerical experiments under the Merton model demonstrate that our method achieves superior accuracy, faster convergence, and enhanced dimensionality robustness compared to the Deep Galerkin Method and jump-adapted Deep BSDE solvers—while rigorously satisfying both the PIDE’s asymptotic properties and domain-specific financial priors.
📝 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.