Solving high-dimensional optimal stopping problems using deep learning

Abstract

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.

Pricing and hedging American-style options with deep learning

“…, X d n , X d+1 n ) for Our results for L, Û , V and 95% confidence intervals for different specifications of the model parameters are reported in Table 1. It can be seen that to achieve a pricing accuracy comparable to the more direct methods of [5] and [6], the networks used in the construction of the candidate optimal stopping strategy have to be trained for a longer time.…”

“…This gives a high-biased estimate and confidence intervals for the price. To achieve the same pricing accuracy as the more direct approaches of [5] and [6], we had to train the neural network approximations of the continuation values for a longer time. But computing approximate continuation values has the advantage that they can be used to break the hedging problem into a sequence of subproblems that compute the hedge only from one possible exercise date to the next.…”

“…To do that we approximate the functions h m with neural networks 7 h λ : R d → R e of the form (6) and try to find parameters λ 0 , . .…”

“…More recently, in [30] optimal stopping problems in continuous time have been solved by approximating the solutions of the corresponding free boundary PDEs with deep neural networks. In [5,6] deep learning has been used to directly learn optimal stopping strategies. The main focus of these papers is to derive optimal stopping rules and accurate price estimates.…”

Designing Universal Causal Deep Learning Models: The Geometric (Hyper)Transformer

“…This is most apparent in computational stochastic analysis and mathematical finance, where deep learning has unlocked previously intractable problems. Examples include computation of optimal hedges under market frictions and possibly rough volatility [29,32,54,74], numerical implementation of complicated local stochastic volatility models [42], numerical solutions to previously intractable principal-agent problems [31], pricing of derivatives relying on optimal stopping rules written on high-dimensional portfolios [20,21,71], data-driven prediction of price formation using ultra-high dimensional limit orderbook data [134,154].…”