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Deep Learning for Energy Market Contracts: Dynkin Game with Doubly RBSDEs

Nacira Agram, Ihsan Arharas, Giulia Pucci, Jan Rems

TL;DR

This work addresses pricing a Contract for Difference (CfD) with early exit options in energy markets by formulating it as a two-player zero-sum Dynkin game and representing the value via a Doubly Reflected BSDE (DRBSDE). It introduces a backward, trajectory-based neural solver (Deep DRBSDE) that learns both the value process $Y$ and the optimal stopping regions in high dimensions, with convergence guarantees. The method is validated on a 20-dimensional symmetric benchmark and a 24-dimensional European-zone CfD model calibrated to weekly price data, demonstrating scalability and robustness. The approach enables data-driven, high-dimensional contract design and risk sharing considerations, with potential extensions to jump dynamics and richer penalty structures.

Abstract

We formulate a Contract for Difference (CfD) with early exit options as a two-player zero-sum Dynkin game, reflecting the strategic interaction between an electricity producer and a regulatory entity. The game incorporates penalties for early termination and mean-reverting price dynamics, with the value characterized through a doubly reflected backward stochastic differential equation (DRBSDE). To compute the contract value and optimal stopping strategies, we develop a neural solver that approximates the DRBSDE solution using a sequence of neural networks trained on simulated trajectories. The method avoids discretizing the state space, supports time-dependent barriers, and scales to high-dimensional settings. We establish a convergence result and test the method on two scenarios: a benchmark symmetric game in 20 dimensions, and a CfD model with 24-dimensional electricity prices representing multiple European zones. The results demonstrate that the proposed solver accurately captures the contract's value and optimal stopping regions, with consistent performance across dimensional settings.

Deep Learning for Energy Market Contracts: Dynkin Game with Doubly RBSDEs

TL;DR

This work addresses pricing a Contract for Difference (CfD) with early exit options in energy markets by formulating it as a two-player zero-sum Dynkin game and representing the value via a Doubly Reflected BSDE (DRBSDE). It introduces a backward, trajectory-based neural solver (Deep DRBSDE) that learns both the value process and the optimal stopping regions in high dimensions, with convergence guarantees. The method is validated on a 20-dimensional symmetric benchmark and a 24-dimensional European-zone CfD model calibrated to weekly price data, demonstrating scalability and robustness. The approach enables data-driven, high-dimensional contract design and risk sharing considerations, with potential extensions to jump dynamics and richer penalty structures.

Abstract

We formulate a Contract for Difference (CfD) with early exit options as a two-player zero-sum Dynkin game, reflecting the strategic interaction between an electricity producer and a regulatory entity. The game incorporates penalties for early termination and mean-reverting price dynamics, with the value characterized through a doubly reflected backward stochastic differential equation (DRBSDE). To compute the contract value and optimal stopping strategies, we develop a neural solver that approximates the DRBSDE solution using a sequence of neural networks trained on simulated trajectories. The method avoids discretizing the state space, supports time-dependent barriers, and scales to high-dimensional settings. We establish a convergence result and test the method on two scenarios: a benchmark symmetric game in 20 dimensions, and a CfD model with 24-dimensional electricity prices representing multiple European zones. The results demonstrate that the proposed solver accurately captures the contract's value and optimal stopping regions, with consistent performance across dimensional settings.

Paper Structure

This paper contains 21 sections, 9 theorems, 73 equations, 7 figures, 3 tables, 1 algorithm.

Table of Contents

  1. Introduction
  2. Paper Organization.
  3. Contract for Differences with Exit Options in En- ergy Markets
  4. Market Price Modeling
  5. Exit Options and Strategic Decision-Making
  6. Solution via DRBSDEs
  7. Application to Exponentially Decaying Penalty Structures
  8. Deep Learning for DRBSDEs
  9. Neural Network Architecture
  10. Time Discretization of the Forward and Backward Components
  11. Neural Network Approximation of the DRBSDE Solution
  12. Convergence Analysis
  13. Numerical Implementation
  14. Benchmark Problem: Symmetric Fair Game
  15. Parameter Calibration for the Underlying Price Process
  16. ...and 6 more sections

Key Result

Theorem 4.1

Let $(\widehat{Y}_{n}^N,\widehat{Z}^N_n)$ for $n=0, \ldots, N$ be neural network approximations of the DRBSDE solution $(Y_t, Z_t)$ for $t \in [0,T]$. Then the error converges towards 0 as we increase number of timesteps $N$ and the number of neural networks' hidden parameters $\theta_n$ for each $n = 0, \ldots, N$.

Figures (7)

  • Figure 1: Convergence of loss function at different timesteps $n$ in the benchmark problem
  • Figure 2: Three realisations of the $Y_t$ dynamics and the distribution of estimated $Y_0$ over 100 independent training processes in the benchmark problem
  • Figure 3: Estimated distribution of the first exit times in the benchmark problem
  • Figure 4: Calibration performance visual tests
  • Figure 5: Convergence of loss function at different timesteps $n$ in the CfD example
  • ...and 2 more figures

Theorems & Definitions (18)

  • Theorem 4.1
  • proof
  • Lemma 4.2
  • proof
  • Theorem 4.3
  • proof
  • Corollary 4.4
  • proof
  • Definition A.2
  • Theorem A.3
  • ...and 8 more