Deep Quadratic Hedging

Alessandro Gnoatto, Athena Picarelli, Silvia Lavagnini

Publication date: 24 December 2022

Abstract: We present a novel computational approach for quadratic hedging in a high-dimensional incomplete market. This covers both mean-variance hedging and local risk minimization. In the first case, the solution is linked to a system of BSDEs, one of which being a backward stochastic Riccati equation (BSRE); in the second case, the solution is related to the F"olmer-Schweizer decomposition and is also linked to a BSDE. We apply (recursively) a deep neural network-based BSDE solver. Thanks to this approach, we solve high-dimensional quadratic hedging problems, providing the entire hedging strategies paths, which, in alternative, would require to solve high dimensional PDEs. We test our approach with a classical Heston model and with a multi-dimensional generalization of it.

Has companion code repository: https://github.com/silvialava/deep_quadratic_hedging

Mathematics Subject Classification ID

Artificial neural networks and deep learning (68T07) Stochastic ordinary differential equations (aspects of stochastic analysis) (60H10) Optimal stochastic control (93E20) Probabilistic methods, particle methods, etc. for initial value and initial-boundary value problems involving PDEs (65M75)

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