[Paper Review] Quantum-Inspired Tensor Neural Networks for Option Pricing

Raj Patel, Chia-Wei Hsing+13 more|arXiv (Cornell University)|Dec 28, 2022

Tensor decomposition and applications4 citations

TL;DR

This paper proposes Quantum-Inspired Tensor Neural Networks (TNN) and a novel Tensor Network Initializer (TNN Init) to accelerate and reduce memory usage in deep learning for high-dimensional option pricing. By replacing dense layer weights with tensor train decompositions, TNN achieves comparable or better accuracy than DNNs with significantly fewer parameters and faster convergence, validated on the Heston model and Bermudan option pricing in 5–10 dimensions.

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.

Motivation & Objective

• Address the curse of dimensionality in solving high-dimensional parabolic PDEs for financial derivatives pricing.
• Reduce memory consumption and training time in deep learning models used for option pricing.
• Develop a parameter-efficient neural network architecture that maintains or improves accuracy compared to standard DNNs.
• Introduce a new weight initialization scheme (TNN Init) to accelerate convergence and reduce variance in training.
• Benchmark TNN and TNN Init against DNNs with equivalent parameter counts on realistic financial models like the Heston model and Bermudan options.

Proposed method

• Replace dense layer weight matrices in DNNs with tensor train (TT) decompositions, forming a Tensor Neural Network (TNN) to reduce parameter count.
• Use Matrix Product Operator (MPO) representation to parameterize the weight tensors, enabling efficient storage and computation.
• Implement a new weight initialization scheme, TNN Init, tailored for TNNs to improve training stability and convergence speed.
• Map the parabolic PDE of the Heston model into a forward-backward stochastic differential equation (FBSDE), solvable via neural networks.
• Train the TNN and DNN models using the Adam optimizer with mean squared error (MSE) loss on simulated paths of the underlying assets.
• Perform recursive backward induction using feedforward networks to estimate continuation values at each exercise date in Bermudan option pricing.

Experimental results

Research questions

• RQ1 Can TNNs achieve comparable or better accuracy than DNNs while using significantly fewer parameters in high-dimensional option pricing?
• RQ2 Does the TNN Init initialization scheme lead to faster convergence and lower variance in training compared to standard DNN initialization?
• RQ3 How does the training speed and memory consumption of TNNs compare to DNNs with the same number of parameters on the Heston model?
• RQ4 To what extent can TNNs reduce the computational cost of pricing early-exercise options compared to the Longstaff-Schwartz method?
• RQ5 Can TNNs effectively solve high-dimensional PDEs arising in quantitative finance without relying on Monte Carlo sampling or mesh-based methods?

Key findings

• TNNs achieve the same accuracy as DNNs with significantly fewer parameters, demonstrating substantial parameter savings.
• For the Heston model, TNNs converge faster and with lower variance than DNNs with equivalent parameter counts.
• On 5-asset Bermudan options, TNNs achieve a price of 26.13 ± 0.18, outperforming DNNs at 26.01 ± 0.18 with lower loss across all exercise dates.
• For 10-asset Bermudan options, TNNs achieve a price of 38.14 ± 0.25, compared to 37.92 ± 0.27 for DNNs, with consistently lower loss during training.
• TNNs reduce training time and memory usage while maintaining or improving performance, especially in high-dimensional settings.
• The TNN Init initialization scheme leads to faster convergence and reduced variance in training, outperforming standard DNN initialization empirically.