Paper: arXiv 2502.11706

Abstract

A deep BSDE approach is presented for the pricing and delta-gamma hedging of high-dimensional Bermudan options, with applications in portfolio risk management. Large portfolios of a mixture of multi-asset European and Bermudan derivatives are cast into the framework of discretely reflected BSDEs. This system is discretized by the One Step Malliavin scheme (Negyesi et al. [“2024, 2025”]) of discretely reflected Markovian BSDEs, which involves a $Γ$ process, corresponding to second-order sensitivities of the associated option prices. The discretized system is solved by a neural network regression Monte Carlo method, efficiently for a large number of underlyings. The resulting option Deltas and Gammas are used to discretely rebalance the corresponding replicating strategies. Numerical experiments are presented on both high-dimensional basket options and large portfolios consisting of multiple options with varying early exercise rights, moneyness and volatility. These examples demonstrate the robustness and accuracy of the method up to $100$ risk factors. The resulting hedging strategies significantly outperform benchmark methods both in the case of standard delta- and delta-gamma hedging.

Complexity vs Empirical Score

  • Math Complexity: 9.2/10
  • Empirical Rigor: 8.5/10
  • Quadrant: Holy Grail — high math complexity, high empirical rigor

Why this score: The paper employs advanced stochastic calculus including BSDEs, Malliavin calculus, and high-dimensional neural network approximations, representing a very high math complexity. The method is demonstrated with numerical experiments on portfolios up to 100 risk factors, showing backtest-ready implementation with empirical validation of hedging performance.

Research Flowchart

  flowchart TD
  A["Research Goal"] --> B["Problem Formulation"]
  B --> C["Discretization"]
  C --> D["Computational Engine"]
  D --> E["Outcomes & Validation"]

  A["Research Goal<br/>Simultaneous Pricing & Delta-Gamma Hedging<br/>High-Dim Bermudan Portfolios"]

  B["Problem Formulation<br/>Large portfolios of multi-asset options<br/>Cast as Discretely Reflected BSDEs"]

  C["Discretization<br/>One Step Malliavin Scheme<br/>(Includes Γ-process for Gamma)"]

  D["Computational Engine<br/>Neural Network Regression<br/>Monte Carlo Simulation"]

  E["Outcomes & Validation<br/>Accurate Deltas/Gammas up to 100 dimensions<br/>Superior hedging performance<br/>vs. benchmarks"]