Deep Quadratic Hedging

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 Foellmer-Schweizer decomposition and is also linked to a BSDE. We apply (recursively) a deep neural network-based BSDE solver. Thanks to these 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. Due to the unboundedness of the variance process, existence and uniqueness results for the BSRE must be considered.