[Paper Review] A Review of Tree-based Approaches to solve Forward-Backward Stochastic Differential Equations

Long Teng|arXiv (Cornell University)|Sep 2, 2018

Stochastic processes and financial applications35 references6 citations

TL;DR

This paper proposes a regression tree-based numerical method for solving high-dimensional forward-backward stochastic differential equations (FBSDEs), using theta-discretization to handle time-integrands and tree-based regression to efficiently approximate conditional expectations. The approach achieves high accuracy and computational efficiency, outperforming multilevel Monte Carlo and deep learning methods in benchmark 100-dimensional pricing problems with low relative errors and standard deviations.

ABSTRACT

In this work, we study solving (decoupled) forward-backward stochastic differential equations (FBSDEs) numerically using the regression trees. Based on the general theta-discretization for the time-integrands, we show how to efficiently use regression tree-based methods to solve the resulting conditional expectations. Several numerical experiments including high-dimensional problems are provided to demonstrate the accuracy and performance of the tree-based approach. For the applicability of FBSDEs in financial problems, we apply our tree-based approach to the Heston stochastic volatility model, the high-dimensional pricing problems of a Rainbow option and an European financial derivative with different interest rates for borrowing and lending.

Motivation & Objective

• To develop an efficient numerical method for solving decoupled forward-backward stochastic differential equations (FBSDEs), especially in high-dimensional settings where traditional methods struggle.
• To address the computational challenge of estimating conditional expectations in FBSDEs, a key bottleneck in numerical schemes.
• To demonstrate the applicability and accuracy of tree-based regression for high-dimensional financial derivative pricing problems.
• To compare the proposed method with existing state-of-the-art approaches, including multilevel Monte Carlo and deep learning, in terms of accuracy, runtime, and convergence.

Proposed method

• The method employs a general theta-discretization scheme for the time-integrands of FBSDEs, enabling flexible and stable time integration.
• Conditional expectations arising from the discretization are approximated using regression trees, which adaptively partition the state space to model complex dependencies.
• The approach is applied to both decoupled FBSDEs and the special case of backward SDEs, with a focus on financial applications such as option pricing.
• The algorithm uses Monte Carlo sampling with samplesplitting to improve statistical stability, particularly for high-dimensional problems.
• Different theta values (e.g., 0, 0.5, 1) are tested to analyze their impact on convergence and stability.
• The method avoids direct computation of the Z-component in some cases by setting θ=1, simplifying the iteration process in high dimensions.

Experimental results

Research questions

• RQ1 Can regression trees effectively approximate the conditional expectations in FBSDEs arising from theta-discretization?
• RQ2 How does the tree-based approach compare to multilevel Monte Carlo and deep learning in solving high-dimensional FBSDEs?
• RQ3 What is the convergence behavior and computational efficiency of the tree-based method across different theta values?
• RQ4 Can the method achieve high accuracy in 100-dimensional nonlinear pricing problems, such as those with stochastic volatility or different borrowing/lending rates?
• RQ5 Does the method maintain low variance and robustness when applied to complex financial derivatives like Rainbow options?

Key findings

• For a 100-dimensional nonlinear pricing problem, the tree-based method achieved a relative error of 0.0020 with M=200,000 samples and N_T=10 time steps, outperforming multilevel Monte Carlo with 6 Picard iterations in both accuracy and runtime.
• The method achieved a standard deviation of 0.0199 for the 100D problem with M=200,000, indicating low variance and high stability.
• With M=200,000 and N_T=10, the average runtime was 464.9 seconds, significantly faster than multilevel Monte Carlo with 7 iterations (1915.1 seconds) for comparable accuracy.
• The tree-based approach achieved a relative error of 0.0017 for the 100D problem with M=300,000, demonstrating convergence with increasing sample size.
• In the 1D case with different interest rates, the method achieved a relative error of 0.0013 with M=100,000, showing high precision and low variance.
• The method outperformed deep learning-based approaches in terms of runtime (947.7s vs. 566s) while maintaining comparable or better accuracy in the 100D case.