Application of Tensor Neural Networks to Pricing Bermudan Swaptions

TL;DR

Tensor Neural Networks can be trained faster than Dense Neural Networks and provide more accurate and robust prices than their Dense counterparts and are proposed to be leveraged for solving the backward Stochastic Differential Equation associated with the value process for European and Bermudan Swaptions.

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.

Figures (9)

• Figure 1: Learning pipeline for the European Swaption Neural Network.
• Figure 2: Learning pipeline for the Bermudan Swaption stacked Neural Network.
• Figure 3: The process of contracting a 2-node MPO and reshaping it into the weight matrix $\mathbf{W}$ in each forward pass.
• Figure 4: (top panel) Initial option price evolution for fixed rate $K = 0.00$ (left) and fixed rate $K=0.01$ (right) for TNN(64, 64) with a bond dimension 2 (red), the corresponding DNN(64,64) with similar neuron count (green) and the best DNN(24,27) with equivalent parameter count (blue). The plots display the mean and $95\%$ confidence interval for the runs. To benchmark the results, the dotted line (black) indicates the MC price from $10^5$ runs while the grey shaded region indicates its associated $95\%$ confidence interval. (bottom panel) Training loss evolution for fixed rate $K=0.00$ (left) and fixed rate $K=0.01$ (right) for TNN(64, 64) with a bond dimension 2 (red), the corresponding DNN(64,64) with similar neuron count (green) and the best DNN(24,27) with equivalent parameter count (blue). The plots display the mean and $95\%$ confidence interval for the runs.
• Figure 5: European Swaption Price for different architectures in 3-factor Cheyette Model with fixed rate $K=0.00$.

Theorems & Definitions (1)

• Proposition 1