[Paper Review] Backward Deep BSDE Methods and Applications to Nonlinear Problems

Yajie Yu, Bernhard Hientzsch+1 more|arXiv (Cornell University)|Jun 13, 2020

Stochastic processes and financial applications7 references4 citations

TL;DR

This paper introduces a novel backward deep BSDE method for solving high-dimensional nonlinear forward-backward stochastic differential equations (FBSDEs), specifically applied to the differential rates problem in quantitative finance. By enabling time-stepping backward with exact and Taylor-based approximations for nonlinear generators—previously unimplemented—the method achieves accurate pricing of complex derivatives like straddles and call combinations, with results closely matching established PDE benchmarks.

ABSTRACT

In this paper, we present a backward deep BSDE method applied to Forward Backward Stochastic Differential Equations (FBSDE) with given terminal condition at maturity that time-steps the BSDE backwards. We present an application of this method to a nonlinear pricing problem - the differential rates problem. To time-step the BSDE backward, one needs to solve a nonlinear problem. For the differential rates problem, we derive an exact solution of this time-step problem and a Taylor-based approximation. Previously backward deep BSDE methods only treated zero or linear generators. While a Taylor approach for nonlinear generators was previously mentioned, it had not been implemented or applied, while we apply our method to nonlinear generators and derive details and present results. Likewise, previously backward deep BSDE methods were presented for fixed initial risk factor values $X_0$ only, while we present a version with random $X_0$ and a version that learns portfolio values at intermediate times as well. The method is able to solve nonlinear FBSDE problems in high dimensions.

Motivation & Objective

• Address the limitation of prior backward deep BSDE methods that only handled zero or linear generators, extending them to nonlinear generators.
• Overcome the curse of dimensionality in nonlinear PDEs arising from differential interest rates in derivative pricing.
• Develop a backward deep BSDE framework that supports both fixed and random initial risk factor values $X_0$, enabling broader applicability.
• Implement and validate a novel method that learns portfolio values at intermediate times via a deep neural network, improving robustness and accuracy.
• Demonstrate the method's effectiveness on benchmark nonlinear problems such as the straddle and call combination under differential rates, achieving results comparable to established PDE solvers.

Proposed method

• Time-discretize the FBSDE system for nonlinear problems, particularly the differential rates problem, using backward time-stepping from maturity to initial time.
• Propose an exact backward step solution for the nonlinear generator in the differential rates problem, derived analytically to ensure precision.
• Introduce a Taylor-based approximation for the nonlinear generator to enable efficient computation when exact solutions are intractable.
• Implement two variants: one minimizing batch variance of $Y_{t_0}$ for fixed $X_0$, and another using a deep neural network to predict $\mathsf{yinit}(X_0)$ for random $X_0$.
• Use mini-batch stochastic gradient descent with variance minimization as the objective function for fixed $X_0$, and mean squared error loss for random $X_0$.
• Construct computational graphs for end-to-end training, integrating the backward step approximations into a deep learning framework for scalable, high-dimensional solution.

Experimental results

Research questions

• RQ1 Can backward deep BSDE methods be extended to handle nonlinear generators, previously limited to zero or linear cases?
• RQ2 How effective is the proposed exact and Taylor-based backward step approximation in solving nonlinear FBSDEs for derivative pricing?
• RQ3 Does the method maintain accuracy and stability when $X_0$ is random rather than fixed, and how does it compare to PDE benchmarks?
• RQ4 To what extent do the results from the backward deep BSDE method align with established PDE solutions for the differential rates problem?
• RQ5 Can the method capture complex trading strategies, such as borrowing and lending behavior, in high-dimensional settings?

Key findings

• The backward deep BSDE method with exact backward step achieves results that closely match those from Forsyth and Labahn [FL07], a well-established PDE benchmark, even with batch sizes as small as 512.
• For the straddle and call combination cases, the method's predicted $\mathsf{yinit}(X_0)$ values align well with the reference curves from Forsyth and Labahn [FL07], especially as batch size increases to 1024.
• The Taylor-based backward step approximation produces results nearly indistinguishable from the exact backward step, validating its use as a computationally efficient alternative.
• The method successfully visualizes and captures the dynamics of the trading strategy, including regions of borrowing (red) and lending (black), across different stock price levels.
• The network trained on random $X_0$ learns a consistent $\mathsf{yinit}(X_0)$ function that represents the $L^2$-projection of pathwise roll-back values, demonstrating robustness to initial condition variability.
• The method generalizes effectively to high-dimensional problems, offering a scalable alternative to traditional PDE and Monte Carlo methods that fail under the curse of dimensionality.