On deep portfolio optimization with stocks, bonds and options

In this paper we propose a machine learning algorithm for time-inconsistent portfolio optimization. The proposed algorithm builds on neural-network-based trading schemes, in which the asset allocation at each time point is determined by a neural network. The loss function is given by an empirical version of the objective function of the portfolio optimization problem. Moreover, various trading constraints are naturally fulfilled by choosing appropriate activation functions in the output layers of the neural networks. Besides this, our main contribution is to add options to a portfolio of risky assets and a risk-free bond and to use additional neural networks to determine the amount allocated to the options as well as their strike prices. We consider objective functions that are more in line with the rational preferences of an investor than the classical mean–variance, apply realistic trading constraints and model the assets with a correlated jump-diffusion stochastic differential equation. With an incomplete market and a more involved objective function, we show that it is beneficial to add options to the portfolio. Moreover, we show that adding options leads to a more consistent stock allocation with less demand for drastic reallocations.