[Paper Review] Deep-learning based numerical BSDE method for barrier options

Bing Yu, Xiaojing Xing+1 more|arXiv (Cornell University)|Apr 11, 2019

Stochastic processes and financial applications10 references3 citations

TL;DR

This paper proposes a deep learning-based numerical method for pricing barrier options by reformulating the PDE with terminal and boundary conditions as a standard backward stochastic differential equation (BSDE) problem. By embedding barrier breach probabilities into the terminal condition via Brownian motion bridges, the method leverages neural networks to approximate the Z-process in the BSDE, achieving high accuracy across diverse market conditions, especially with batch normalization for convergence stability.

ABSTRACT

As is known, an option price is a solution to a certain partial differential equation (PDE) with terminal conditions (payoff functions). There is a close association between the solution of PDE and the solution of a backward stochastic differential equation (BSDE). We can either solve the PDE to obtain option prices or solve its associated BSDE. Recently a deep learning technique has been applied to solve option prices using the BSDE approach. In this approach, deep learning is used to learn some deterministic functions, which are used in solving the BSDE with terminal conditions. In this paper, we extend the deep-learning technique to solve a PDE with both terminal and boundary conditions. In particular, we will employ the technique to solve barrier options using Brownian motion bridges.

Motivation & Objective

• To address the challenge of pricing barrier options, which involve both terminal and boundary conditions, using machine learning.
• To extend the deep learning-based BSDE framework—previously limited to terminal-condition PDEs—to handle Dirichlet-type boundary conditions.
• To improve convergence and stability in training neural networks for option pricing under complex boundary constraints.
• To validate the method against analytical solutions across a wide range of market parameters.

Proposed method

• Reformulate the PDE for barrier options as a BSDE with terminal conditions by incorporating the probability of barrier breach through Brownian motion bridges.
• Use a neural network to approximate the Z-process in the BSDE, parameterized by θ, with time-dependent batch normalization parameters β_{t_i} to stabilize training.
• Apply the Euler scheme to discretize both forward (X_t) and backward (Y_t, Z_t) processes in discrete time steps.
• Define a loss function based on the terminal condition error: E[(Y_T - h(X_T))²], minimized via stochastic gradient descent.
• Introduce batch normalization at each layer or only at input to reduce covariate shift and accelerate convergence.
• Use a probabilistic identity (Lemma 2) to transform conditional expectations involving barrier events into expectations over the full probability space, enabling tractable computation.

Experimental results

Research questions

• RQ1 Can the deep learning-based BSDE framework be extended to solve PDEs with boundary conditions, such as those for barrier options?
• RQ2 How can boundary conditions in barrier options be embedded into the terminal condition of a BSDE without using reflected BSDEs or penalty functions?
• RQ3 What role does batch normalization play in improving convergence and stability when training neural networks for option pricing with complex boundary constraints?
• RQ4 How does the performance of the proposed method compare to analytical solutions across diverse market conditions?

Key findings

• The method achieves relative errors below 1% on average across 100 test cases, with median relative error of 0.39% and 25th percentile error of 0.21%.
• In isolated cases with high volatility or extreme moneyness, the relative error dropped from over 18% (without batch normalization) to 0.65% (with batch normalization).
• Applying batch normalization at the input layer only reduced total training time by nearly 3x (75s vs. 200s) compared to layer-wise batch normalization, despite requiring more iterations.
• The use of Brownian motion bridges successfully embeds boundary conditions into the terminal condition, enabling the use of the standard deep BSDE framework without modifying the loss function.
• The method maintains robustness across a wide range of maturities, strike prices, and volatilities, with consistent performance even in high-dimensional or path-dependent settings.
• Theoretical justification via Lemma 2 ensures that the transformation from conditional expectations to unconditional expectations is mathematically sound and computationally feasible.