Deep nets are becoming larger and larger in practice, with no respect for (non)-factors that ought to limit growth including the so-called curse of dimensionality (CoD). Donoho suggested that dimensionality can be a blessing as well as a curse. Current practice in industry is well ahead of theory, but there are some recent theoretical results from Weinan E’s group suggesting that errors may be independent of dimensions $d$. Current practice suggests an even stronger conjecture: deep nets are not merely immune to CoD, but actually, deep nets thrive on scale.

Nowadays many financial derivatives, such as American or Bermudan options, are of early exercise type. Often the pricing of early exercise options gives rise to high-dimensional optimal stopping problems, since the dimension corresponds to the number of underlying assets. High-dimensional optimal stopping problems are, however, notoriously difficult to solve due to the well-known curse of dimensionality. In this work, we propose an algorithm for solving such problems, which is based on deep learning and computes, in the context of early exercise option pricing, both approximations of an optimal exercise strategy and the price of the considered option. The proposed algorithm can also be applied to optimal stopping problems that arise in other areas where the underlying stochastic process can be efficiently simulated. We present numerical results for a large number of example problems, which include the pricing of many high-dimensional American and Bermudan options, such as Bermudan max-call options in up to 5000 dimensions. Most of the obtained results are compared to reference values computed by exploiting the specific problem design or, where available, to reference values from the literature. These numerical results suggest that the proposed algorithm is highly effective in the case of many underlyings, in terms of both accuracy and speed.