Quantum-Inspired Tensor Neural Networks for Option Pricing

TL;DR

This paper introduces Tensor Neural Networks (TNN) by replacing dense neural layers with MPO-based Tensor Network layers to efficiently solve high-dimensional parabolic PDEs arising in option pricing under the Heston model. It also presents Tensor Network Initializer (TNN Init) to accelerate convergence and reduce variance. Through extensive experiments, TNNs achieve equivalent or better accuracy than dense networks with far fewer parameters, and converge significantly faster (up to ~12x) on GPU hardware, with additional benefits in stability. The approach is demonstrated on European and Bermudan options, showing potential scalability to high-dimensional problems and complex payoff structures, and it offers a path toward memory-efficient, compute-lean ML solvers for financial PDEs.

Abstract

Recent advances in deep learning have enabled us to address the curse of dimensionality (COD) by solving problems in higher dimensions. A subset of such approaches of addressing the COD has led us to solving high-dimensional PDEs. This has resulted in opening doors to solving a variety of real-world problems ranging from mathematical finance to stochastic control for industrial applications. Although feasible, these deep learning methods are still constrained by training time and memory. Tackling these shortcomings, Tensor Neural Networks (TNN) demonstrate that they can provide significant parameter savings while attaining the same accuracy as compared to the classical Dense Neural Network (DNN). In addition, we also show how TNN can be trained faster than DNN for the same accuracy. Besides TNN, we also introduce Tensor Network Initializer (TNN Init), a weight initialization scheme that leads to faster convergence with smaller variance for an equivalent parameter count as compared to a DNN. We benchmark TNN and TNN Init by applying them to solve the parabolic PDE associated with the Heston model, which is widely used in financial pricing theory.

Figures (8)

• Figure 1: The process of contracting a 2-node MPO and reshaping it into the weight matrix $\mathbf{W}$ in each forward pass.
• Figure 2: Training loss over epochs for TNN(16) with bond dimension 2 (blue), the corresponding best DNN with equivalent parameters (orange) and the DNN with equivalent neurons (green). The plots illustrates the resulting mean ± standard deviation from 100 runs with different seeds.
• Figure 3: Option Price at $t_0$ over epochs for TNN(16) with bond dimension 2 (blue), the corresponding best DNN with equivalent parameters (orange) and the DNN with equivalent neurons (green). The plots illustrate the resulting mean ± standard deviation from 100 runs. The black dotted line indicates the analytical solution obtained from solving the Heston PDE using Fourier transformation approach discussed at the end of Sec. \ref{['Problem']}.
• Figure 4: (left panel) Training loss over epochs for TNN(16) with a bond dimension 2 and that of a TNN_Init(16) with a bond dimension 2 (right panel) training loss over epochs for TNN(16) with a bond dimension 2 and that of a TNN_Init(10) with a bond dimension 2. The plots indicate resulting mean $\pm$ standard deviation from 100 runs.
• Figure 5: Training loss over epochs for DNN(16,16) (Orange) and DNN(16, 16) with TN-Initializer (Blue). The plots illustrates the resulting mean ± standard deviation from 100 runs.