This paper introduces several machine learning algorithms for hedging contingent claims in incomplete markets, where sources of incompleteness include illiquidity, non-tradable risk factors, discrete hedging dates, and proportional transaction costs. The authors propose both global optimization approaches (using feedforward and recurrent neural networks to directly minimize a loss function over the entire hedging horizon) and local dynamic programming methods (solving per-step minimization problems). The global methods are particularly novel, as they allow for arbitrary loss criteria and can handle high-dimensional state spaces without the curse of dimensionality. The most effective architecture is an augmented LSTM network that combines an LSTM cell with additional feedforward layers, which captures temporal dependencies and produces superior hedging strategies compared to feedforward networks or classical LSTMs.

Numerical experiments compare the algorithms on various payoffs using a mean squared error (MSE) criterion, with dynamic programming references computed via the StOpt library. The global augmented LSTM consistently achieves lower hedging errors than local methods, which often get trapped in local minima. The paper also demonstrates flexibility by training with asymmetrical loss functions and moment-based criteria (e.g., penalizing fourth moments of losses), showing how these criteria alter the P&L distribution. For transaction costs, the authors introduce a technique to compute the entire Pareto frontier of optimal strategies balancing average cost and variance in a single training phase by randomly combining the two criteria during stochastic gradient descent. This approach is computationally efficient and provides practitioners with a range of hedging strategies to suit their risk preferences. Overall, the work shows that deep learning can effectively solve realistic hedging problems that are intractable with classical methods, particularly in high dimensions and with complex risk criteria.