# Deep Neural Network Framework Based on Backward Stochastic Differential Equations for Pricing and Hedging American Options in High Dimensions

arXiv: https://arxiv.org/abs/1909.11532 · HTML: https://mlsanity.com/paper/1909.11532/

## Key facts

- Cited by 14 papers in the map (data through 14 Oct 2026).
- First posted on arXiv 25 Sep 2019.

## Built on by (most cited first, 14 of 14)

- [Langevin dynamics based algorithm e-THvarepsilonO POULA for stochastic optimization problems with discontinuous stochastic gradient](https://mlsanity.com/paper/2210.13193/) (2022)
- [A backward differential deep learning-based algorithm for solving high-dimensional nonlinear backward stochastic differential equations](https://mlsanity.com/paper/2404.08456/) (2024)
- [Solving optimal stopping problems with Deep Q-Learning](https://mlsanity.com/paper/2101.09682/) (2021)
- [Simultaneous upper and lower bounds of American-style option prices with hedging via neural networks](https://mlsanity.com/paper/2302.12439/) (2023)
- [Efficient Pricing and Hedging of High Dimensional American Options Using Recurrent Networks](https://mlsanity.com/paper/2301.08232/) (2023)
- [Quantum-Inspired Tensor Neural Networks for Option Pricing](https://mlsanity.com/paper/2212.14076/) (2022)
- [Sampling Complexity of Deep Approximation Spaces](https://mlsanity.com/paper/2312.13379/) (2023)
- [Neural Options Pricing](https://mlsanity.com/paper/2105.13320/) (2021)
- [Deep Generative Modeling with Backward Stochastic Differential Equations](https://mlsanity.com/paper/2304.04049/) (2023)
- [A Method to Improve the Performance of Reinforcement Learning Based on the Y Operator for a Class of Stochastic Differential Equation-Based Child-Mother Systems](https://mlsanity.com/paper/2311.04014/) (2023)
- [Deep Learning vs. Black-Scholes: Option Pricing Performance on Brazilian Petrobras Stocks](https://mlsanity.com/paper/2504.20088/) (2025)
- [KANHedge: Efficient Hedging of High-Dimensional Options Using Kolmogorov-Arnold Network-Based BSDE Solver](https://mlsanity.com/paper/2601.11097/) (2026)
- [Continuous-time Optimal Stopping through Deep Reinforcement Learning](https://mlsanity.com/paper/2606.17545/) (2026)
- [Deep Learning for Reflected BSDEs: Regularization and Error Analysis](https://mlsanity.com/paper/2609.05434/) (2026)

## Abstract (arXiv)

We propose a deep neural network framework for computing prices and deltas of American options in high dimensions. The architecture of the framework is a sequence of neural networks, where each network learns the difference of the price functions between adjacent timesteps. We introduce the least squares residual of the associated backward stochastic differential equation as the loss function. Our proposed framework yields prices and deltas on the entire spacetime, not only at a given point. The computational cost of the proposed approach is quadratic in dimension, which addresses the curse of dimensionality issue that state-of-the-art approaches suffer. Our numerical simulations demonstrate these contributions, and show that the proposed neural network framework outperforms state-of-the-art approaches in high dimensions.