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 X0 only, while we present a version with random X0 and a version that learns portfolio values at intermediate times as well.
Summary
This paper extends backward deep BSDE methods to handle nonlinear generators, specifically the differential rates problem where borrowing and lending rates differ. Unlike the forward deep BSDE method of Han, Jentzen, E (2018) which time-steps the BSDE forward and minimises terminal mismatch E|Y_T - g(X_T)|^2, the backward method starts from the known terminal condition Y_T = g(X_T) and time-steps backward, minimising the variance of Y_0 (for fixed X_0) or the distance to a learned initial value function Y_init(X_0) (for random X_0). The authors derive both exact analytical backward step solutions and first-order Taylor approximations for the differential rates generator, and demonstrate that the two give nearly identical results. The method is validated against the HJB PDE solver of Forsyth and Labahn (2007) on call combination and straddle pricing problems.
Key Contributions
- First application of backward deep BSDE to nonlinear generators (previous backward methods only treated zero or linear generators)
- Exact analytical backward step solution for the differential rates problem, derived by case-splitting on the sign of the cash position
- First-order Taylor expansion backward step as an alternative that avoids problem-specific derivation
- Novel loss function variants: batch-variance method, learned Y_0 parameter, and learned Y_init(X_0) network for random initial conditions
- Extension to learn portfolio values at intermediate times via additional loss terms E|Y_{t_i} - Y_learned_i(X_{t_i})|
- Demonstration that exact and Taylor backward steps give differences on the order of 10^{-5}, validating the Taylor approach for practical use
Methodology
The paper formulates the FBSDE for a nonlinear pricing problem. The forward SDE for asset dynamics X_t is time-stepped forward, while the BSDE for the portfolio value Y_t is time-stepped backward from Y_T = g(X_T). At each backward step, one must solve Y_{t_i} - f(t_i, X_{t_i}, Y_{t_i}, Pi_{t_i}) Delta t = Y_{t_{i+1}} - Pi_{t_i}^T sigma(t_i, X_{t_i}) Delta W_i for Y_{t_i}, which is nonlinear when f depends on Y. For the differential rates generator f(t,X,Y,Pi) = -r^l Y + (r^b - r^l)(sum pi_i - Y)^+, there are two cases depending on whether sum pi_i > Y (borrowing) or sum pi_i ⇐ Y (lending), each admitting an explicit solution. The hedging strategy Pi_{t_i} is approximated by neural networks at each time step. Three loss functions are presented: (1) batch variance var(Y_0) for fixed X_0, (2) learned scalar parameter Y_0 with loss E|Y_0^B - Y_0|^2 for fixed X_0, (3) learned DNN Y_init(X_0) with loss E|Y_0^B - Y_init(X_0)|^2 for random X_0. The architecture uses two hidden layers with dim+10 neurons, ELU activation, and batch size 512.
Key Findings
- Backward deep BSDE produces results within the confidence intervals of the Forsyth-Labahn PDE solver for both upper and lower prices of the straddle (upper: ~24.07 vs PDE 24.02-24.07; lower: ~23.10-23.13 vs PDE 23.06-23.09)
- The learned Y_0 parameter method converges more slowly initially but more smoothly at convergence than batch-variance methods
- Computing Y_0 as a mean over 100 mini-batches (rather than 1) gives faster and smoother convergence for the batch-variance method
- Exact and Taylor backward steps give differences of order 10^{-5} across all methods, making the generic Taylor approach attractive for problems where exact solutions are unavailable
- The method works with batch sizes as small as 512, and accuracy improves with larger batch sizes (512, 1024)
- Trading strategies (delta hedges) and borrowing/lending regions are visualised and match financial intuition
Important References
- Han, Jentzen, E (2018) — original forward deep BSDE method (1940 citations)
- E, Han, Jentzen (2017) — Deep learning-based numerical methods for high-dimensional parabolic PDEs and BSDEs (881 citations)
- Forsyth, Labahn (2007) — numerical methods for controlled HJB PDEs in finance, benchmark comparison (201 citations)
- Hure, Pham, Warin (2019) — machine learning schemes for high-dimensional nonlinear PDEs (97 citations)
- Wang et al. (2018) — first backward deep BSDE method (zero drift only), LIBOR market model application (25 citations)
- Hientzsch (2019) — introduction to solving quant finance problems with time-stepped FBSDE and deep learning (10 citations)
Atomic Notes
- forward deep BSDE vs backward deep BSDE
- differential rates problem
- backward time-stepping for BSDEs
- nonlinear Feynman-Kac formula