[Paper Review] Application of Tensor Neural Networks to Pricing Bermudan Swaptions

Raj Patel, Tomas Dominguez+13 more|arXiv (Cornell University)|Apr 18, 2023

Computational Physics and Python Applications4 citations

TL;DR

This paper proposes using Tensor Neural Networks (TNNs) to price European and Bermudan swaptions under the Cheyette interest rate model, leveraging tensor decomposition to reduce parameters and improve training efficiency. TNNs outperform Dense Neural Networks and Longstaff-Schwartz regression by delivering faster convergence, lower variance, and more accurate prices, especially in high-dimensional settings.

ABSTRACT

The Cheyette model is a quasi-Gaussian volatility interest rate model widely used to price interest rate derivatives such as European and Bermudan Swaptions for which Monte Carlo simulation has become the industry standard. In low dimensions, these approaches provide accurate and robust prices for European Swaptions but, even in this computationally simple setting, they are known to underestimate the value of Bermudan Swaptions when using the state variables as regressors. This is mainly due to the use of a finite number of predetermined basis functions in the regression. Moreover, in high-dimensional settings, these approaches succumb to the Curse of Dimensionality. To address these issues, Deep-learning techniques have been used to solve the backward Stochastic Differential Equation associated with the value process for European and Bermudan Swaptions; however, these methods are constrained by training time and memory. To overcome these limitations, we propose leveraging Tensor Neural Networks as they can provide significant parameter savings while attaining the same accuracy as classical Dense Neural Networks. In this paper we rigorously benchmark the performance of Tensor Neural Networks and Dense Neural Networks for pricing European and Bermudan Swaptions, and we show that Tensor Neural Networks can be trained faster than Dense Neural Networks and provide more accurate and robust prices than their Dense counterparts.

Motivation & Objective

• Address the limitations of Monte Carlo and regression-based Longstaff-Schwartz methods in pricing Bermudan swaptions, particularly their underestimation of option values and sensitivity to dimensionality.
• Overcome the computational inefficiency and high memory demands of Dense Neural Networks (DNNs) in solving high-dimensional backward stochastic differential equations (BSDEs) for option pricing.
• Demonstrate that Tensor Neural Networks (TNNs) can achieve comparable or better accuracy than DNNs with significantly fewer parameters and faster training times.
• Investigate the scalability and robustness of TNNs in multi-factor Cheyette models, where traditional methods fail due to the curse of dimensionality.
• Provide empirical benchmarking showing that TNNs reduce price variance and improve convergence over DNNs and LS regression in both 1- and 3-factor settings.

Proposed method

• Reformulate the pricing of European and Bermudan swaptions as solving a system of forward-backward stochastic differential equations (FBSDEs) under the Cheyette model.
• Use a stack of feed-forward neural networks to model the value process at each exercise date for Bermudan swaptions, with each network corresponding to a potential exercise time.
• Implement Tensor Neural Networks (TNNs) by applying tensor decomposition (e.g., Tucker or Tensor Train) to parameterize the weight matrices of fully connected layers, drastically reducing parameter count.
• Train the TNNs via stochastic gradient descent using a loss function based on the FBSDE formulation, minimizing the mean squared error between predicted and true continuation values.
• Compare TNNs against Dense Neural Networks (DNNs) and Longstaff-Schwartz (LS) regression using identical architectures, hyperparameters, and random seeds for fair benchmarking.
• Evaluate performance across multiple metrics: convergence speed, parameter count, loss evolution, and final option price with standard error.

Experimental results

Research questions

• RQ1 Can Tensor Neural Networks provide more accurate and robust Bermudan swaption prices than Longstaff-Schwartz regression and Dense Neural Networks in low- and high-dimensional Cheyette models?
• RQ2 To what extent do TNNs reduce parameter count and training time compared to DNNs with equivalent network depth and width?
• RQ3 Does the performance gap between LS regression and DNNs diminish when increasing model complexity, and can TNNs close this gap more effectively?
• RQ4 How do TNNs compare to DNNs in terms of loss convergence and variance in option price estimation across different factor dimensions?
• RQ5 Can TNNs maintain accuracy and efficiency when scaling to higher-dimensional interest rate models where traditional Monte Carlo and finite-difference methods fail?

Key findings

• Tensor Neural Networks achieved faster training convergence than Dense Neural Networks with comparable parameter counts, as shown in loss evolution plots across 3-factor and 1-factor models.
• In the 3-factor Cheyette model with strike rate K=0.00, the TNN price was $0.120, significantly higher than the LS regression price of $0.110, suggesting TNNs avoid the underestimation common in LS methods.
• The 150 basis point price difference between LS and TNN in the 3-factor model was attributed to underfitting in LS due to low-degree polynomial regressors, which was validated by a smaller 30 basis point gap in the 1-factor model.
• In the 1-factor model, both TNN and DNN converged to the same price with negligible standard error, confirming the reliability of TNNs when model complexity is reduced.
• TNNs achieved lower variance in price estimates and smaller standard errors than DNNs, indicating greater robustness and stability in optimization.
• TNNs provided significant parameter savings—reducing the number of trainable parameters while maintaining or improving accuracy—making them more memory-efficient and scalable than DNNs.