# Application of Tensor Neural Networks to Pricing Bermudan Swaptions

arXiv: https://arxiv.org/abs/2304.09750 · HTML: https://mlsanity.com/paper/2304.09750/

## Key facts

- No citing papers in the map yet (data through 14 Oct 2026).
- Builds on 20 of its 44 references that are in the map.
- First posted on arXiv 18 Apr 2023.

## Builds on (most cited first, 20 of 20)

- [Very Deep Convolutional Networks for Large-Scale Image Recognition](https://mlsanity.com/paper/1409.1556/) (2014)
- [Solving high-dimensional partial differential equations using deep learning](https://mlsanity.com/paper/1707.02568/) (2017)
- [Physics Informed Deep Learning (Part I): Data-driven Solutions of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10561/) (2017)
- [Tensorizing Neural Networks](https://mlsanity.com/paper/1509.06569/) (2015)
- [Deep learning-based numerical methods for high-dimensional parabolic partial differential equations and backward stochastic differential equations](https://mlsanity.com/paper/1706.04702/) (2017)
- [Physics Informed Deep Learning (Part II): Data-driven Discovery of Nonlinear Partial Differential Equations](https://mlsanity.com/paper/1711.10566/) (2017)
- [Unsupervised Generative Modeling Using Matrix Product States](https://mlsanity.com/paper/1709.01662/) (2017)
- [Machine learning approximation algorithms for high-dimensional fully nonlinear partial differential equations and second-order backward stochastic differential equations](https://mlsanity.com/paper/1709.05963/) (2017)
- [Tree Tensor Networks for Generative Modeling](https://mlsanity.com/paper/1901.02217/) (2019)
- [Forward-Backward Stochastic Neural Networks: Deep Learning of High-dimensional Partial Differential Equations](https://mlsanity.com/paper/1804.07010/) (2018)
- [Learning Relevant Features of Data with Multi-scale Tensor Networks](https://mlsanity.com/paper/1801.00315/) (2017)
- [Machine Learning by Unitary Tensor Network of Hierarchical Tree Structure](https://mlsanity.com/paper/1710.04833/) (2017)
- [From probabilistic graphical models to generalized tensor networks for supervised learning](https://mlsanity.com/paper/1806.05964/) (2018)
- [Partial Differential Equations Meet Deep Neural Networks: A Survey](https://mlsanity.com/paper/2211.05567/) (2022)
- [Generative Tensor Network Classification Model for Supervised Machine Learning](https://mlsanity.com/paper/1903.10742/) (2019)
- [Modeling Sequences with Quantum States: A Look Under the Hood](https://mlsanity.com/paper/1910.07425/) (2019)
- [Extension of PCA to Higher Order Data Structures: An Introduction to Tensors, Tensor Decompositions, and Tensor PCA](https://mlsanity.com/paper/1803.00704/) (2018)
- [Matrix Product State Based Quantum Classifier](https://mlsanity.com/paper/1905.01426/) (2019)
- [Quantum-Inspired Tensor Neural Networks for Partial Differential Equations](https://mlsanity.com/paper/2208.02235/) (2022)
- [Quantum-Inspired Tensor Neural Networks for Option Pricing](https://mlsanity.com/paper/2212.14076/) (2022)

## Abstract (arXiv)

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.