A Deep BSDE Approach for the Simultaneous Pricing and Delta-Gamma Hedging of Large Portfolios Consisting of High-Dimensional Multi-asset Bermudan Options

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, Andersson, and Oosterlee, IMA J. Numer. Anal., 44 (2024), pp. 3595–3647] of discretely reflected Markovian BSDEs, which involves a [Formula: see text] 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 in the case of both standard delta and delta-gamma hedging.