Fast Deep Hedging with Second-Order Optimization

TL;DR

The second-order scheme can optimize the policy in 1/4 of the number of steps that standard adaptive moment-based optimization takes and is evaluated on a challenging and practically important problem: hedging a cliquet option on a stock with stochastic volatility by trading in the spot and vanilla options.

Abstract

Hedging exotic options in presence of market frictions is an important risk management task. Deep hedging can solve such hedging problems by training neural network policies in realistic simulated markets. Training these neural networks may be delicate and suffer from slow convergence, particularly for options with long maturities and complex sensitivities to market parameters. To address this, we propose a second-order optimization scheme for deep hedging. We leverage pathwise differentiability to construct a curvature matrix, which we approximate as block-diagonal and Kronecker-factored to efficiently precondition gradients. We evaluate our method on a challenging and practically important problem: hedging a cliquet option on a stock with stochastic volatility by trading in the spot and vanilla options. We find that our second-order scheme can optimize the policy in 1/4 of the number of steps that standard adaptive moment-based optimization takes.

Figures (6)

• Figure 1: Floating grid. Implied volatility surface is stochastic; relative strike (in %) and maturity of tradable options are constant. Marked puts/calls are used in our experiments.
• Figure 2: Magnitude of largest eigenvalues differs across blocks (end of training).
• Figure 3: Validation loss during training.
• Figure 4: Gradient variance and largest eigenvalue.
• Figure 5: PnL histogram (log scale).