# Beating the curse of dimensionality in options pricing and optimal stopping

@article{Goldberg2018BeatingTC, title={Beating the curse of dimensionality in options pricing and optimal stopping}, author={David Alan Goldberg and Yilun Chen}, journal={ArXiv}, year={2018}, volume={abs/1807.02227} }

The fundamental problems of pricing high-dimensional path-dependent options and optimal stopping are central to applied probability and financial engineering. Modern approaches, often relying on ADP, simulation, and/or duality, have limited rigorous guarantees, which may scale poorly and/or require previous knowledge of basis functions. A key difficulty with many approaches is that to yield stronger guarantees, they would necessitate the computation of deeply nested conditional expectations…

## 17 Citations

Solving high-dimensional optimal stopping problems using deep learning

- Computer ScienceEuropean Journal of Applied Mathematics
- 2021

An algorithm is proposed for solving high-dimensional optimal stopping 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.

Pricing and Exercising American Options: an Asymptotic Expansion Approach

- EconomicsJournal of Economic Dynamics and Control
- 2019

Deep Reinforcement Learning for Optimal Stopping with Application in Financial Engineering

- Computer ScienceArXiv
- 2021

This paper presents for the first time a comprehensive empirical evaluation of the quality of optimal stopping policies identified by three state of the art deep RL algorithms: double deep Q-learning (DDQN), categorical distributional RL (C51), and Implicit Quantile Networks (IQN).

Interpretable Optimal Stopping

- Computer ScienceManag. Sci.
- 2022

This paper proposes a new approach to optimal stopping wherein the policy is represented as a binary tree, in the spirit of naturally interpretable tree models commonly used in machine learning, and shows that the class of tree policies is rich enough to approximate the optimal policy.

Nonlinear Monte Carlo methods with polynomial runtime for high-dimensional iterated nested expectations

- MathematicsArXiv
- 2020

This article proves under suitable assumptions that these MLP approximation schemes can approximately calculate multiply iterated nested expectations with a computational effort growing at most polynomially in the number of nestings.

Bridging optimal stopping and max-flow min-cut : a theory of duality

- Mathematics
- 2020

In this work, we draw a novel connection between martingale duality for optimal stopping and the celebrated max-flow min-cut duality in the theory of network flows. First, we prove a new dual…

Toward breaking the curse of dimensionality: an FPTAS for stochastic dynamic programs with multidimensional actions and scalar states

- Computer ScienceSIAM J. Optim.
- 2019

This paper enlarges the class of dynamic programs that admit an FPTAS by showing, under suitable conditions, how to deal with multidimensional action, scalar state, convex costs, and linear state transition function.

A nonparametric algorithm for optimal stopping based on robust optimization

- Computer Science, MathematicsSSRN Electronic Journal
- 2021

A new method for solving stochastic optimal stopping problems with known probability distributions is introduced and it is shown that this combination of robust optimization and simulation can find policies that match, and in some cases significantly outperform, those from state-of-the-art algorithms on low-dimensional, non-Markovian optimal stopped problems from options pricing.

Blind Network Revenue Management and Bandits with Knapsacks under Limited Switches

- Computer Science
- 2019

This work reveals a surprising result: the optimal regret rate is completely characterized by a piecewise-constant function of the switching budget, which further depends on the number of resource constraints --- notably, this is the first time theNumber of resources constraints is shown to play a fundamental role in determining the statistical complexity of online learning problems.

Semitractability of optimal stopping problems via a weighted stochastic mesh algorithm

- Computer Science, MathematicsMathematical Finance
- 2020

It is shown that in the discrete‐time case the WSM algorithm leads to semitractability of the corresponding optimal stopping problem in the sense that its complexity is bounded in order by ε−4logd+2(1/ε) with d being the dimension of the underlying Markov chain.

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