# Solving high-dimensional optimal stopping problems using deep learning

@article{Becker2021SolvingHO, title={Solving high-dimensional optimal stopping problems using deep learning}, author={Sebastian Becker and Patrick Cheridito and Arnulf Jentzen and Timo Welti}, journal={European Journal of Applied Mathematics}, year={2021}, volume={32}, pages={470 - 514} }

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… Expand

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#### References

SHOWING 1-10 OF 124 REFERENCES

Deep Learning-Based BSDE Solver for Libor Market Model with Applications to Bermudan Swaption Pricing and Hedging

- Computer Science, Economics
- 2018

It is demonstrated that using backward DNN the high-dimension Bermudan swaption pricing and hedging can be solved effectively and efficiently. Expand

Implied stopping rules for American basket options from Markovian projection

- Mathematics, Economics
- 2017

This work addresses the problem of pricing American basket options in a multivariate setting, which includes among others, the Bachelier and Black–Scholes models. In high dimensions, nonlinear PDE… Expand

Beating the curse of dimensionality in options pricing and optimal stopping

- Computer Science, Mathematics
- ArXiv
- 2018

This work develops a novel pure-dual algorithm that allows one to elegantly trade-off between accuracy and runtime through a parameter epsilon controlling the associated performance guarantee, with computational and sample complexity both polynomial in T and effectively independent of the dimension, in contrast to past methods typically requiring a complexity scaling exponentially in these parameters. Expand

Machine learning for pricing American options in high-dimensional Markovian and non-Markovian models

- Computer Science, Economics
- 2019

Two efficient techniques which allow one to compute the price of American basket options, called GPR Tree and GPR Exact Integration, are proposed, which solve the backward dynamic programing problem considering a Bermudan approximation of the American option. Expand

Solving high-dimensional partial differential equations using deep learning

- Mathematics, Computer Science
- Proceedings of the National Academy of Sciences
- 2018

A deep learning-based approach that can handle general high-dimensional parabolic PDEs using backward stochastic differential equations and the gradient of the unknown solution is approximated by neural networks, very much in the spirit of deep reinforcement learning with the gradient acting as the policy function. Expand

A dynamic look-ahead Monte Carlo algorithm for pricing Bermudan options

- Mathematics
- 2007

Under the assumption of no-arbitrage, the pricing of American and Bermudan options can be casted into optimal stopping problems. We propose a new adaptive simulation based algorithm for the numerical… Expand

Neural network regression for Bermudan option pricing

- Computer Science, Mathematics
- Monte Carlo Methods Appl.
- 2021

This work proves the convergence of the well-known Longstaff and Schwartz algorithm when the standard least-square regression is replaced by a neural network approximation, assuming an efficient algorithm to compute this approximation. Expand

High Dimensional American Options

- Economics
- 2005

Pricing single asset American options is a hard problem in mathematical finance. There are no closed form solutions available (apart from in the case of the perpetual option), so many approximations… Expand

Pricing high-dimensional Bermudan options using the stochastic grid method

- Computer Science
- Int. J. Comput. Math.
- 2012

This paper proposes a stochastic grid method for estimating the optimal exercise policy and uses this policy to obtain a low-biased estimator for high-dimensional Bermudan options. Expand

Deep neural network framework based on backward stochastic differential equations for pricing and hedging American options in high dimensions

- Computer Science, Economics
- ArXiv
- 2019

This work proposes a deep neural network framework for computing prices and deltas of American options in high dimensions, and introduces the least squares residual of the associated backward stochastic differential equation as the loss function. Expand