A deep BSDE approach for the simultaneous pricing and delta-gamma hedging of large portfolios consisting of high-dimensional multi-asset Bermudan options

TL;DR

This paper introduces a deep BSDE-based method for pricing and hedging large, high-dimensional portfolios of Bermudan options, demonstrating robustness and superior performance over benchmarks in complex scenarios.

Contribution

It develops a neural network Monte Carlo approach for discretely reflected BSDEs with second-order sensitivities, enabling efficient high-dimensional portfolio hedging.

Findings

Method accurately prices and hedges high-dimensional Bermudan options.

Outperforms benchmark methods in delta and delta-gamma hedging.

Effective for portfolios with up to 100 risk factors.

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 $\Gamma$ 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.