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.

中文翻译：

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使用深度学习解决高维最优停止问题

如今，许多金融衍生品，例如美国或百慕大期权，都属于提前行使类型。通常，早期行使期权的定价会引起高维最优止损问题，因为该维对应于标的资产的数量。然而，由于众所周知的维数诅咒，高维最优停止问题是出了名的难以解决。在这项工作中，我们提出了一种解决此类问题的算法，该算法基于深度学习，并在早期行使期权定价的背景下计算最佳行使策略的近似值和所考虑期权的价格。所提出的算法也可以应用于其他领域出现的最佳停止问题，在这些领域可以有效地模拟潜在的随机过程。我们提供了大量示例问题的数值结果，其中包括许多高维美国和百慕大期权的定价，例如高达 5000 维的百慕大最大看涨期权。大多数获得的结果与通过利用特定问题设计计算的参考值进行比较，或者在可用的情况下与文献中的参考值进行比较。这些数值结果表明，就准确性和速度而言，所提出的算法在许多底层的情况下都非常有效。参考文献中的值。这些数值结果表明，就准确性和速度而言，所提出的算法在许多底层的情况下都非常有效。参考文献中的值。这些数值结果表明，就准确性和速度而言，所提出的算法在许多底层的情况下都非常有效。