# Optimal stopping with signatures

Source: [Optimal stopping with signatures](<https://www.sota2.com/research/paper/68fa58091e209131ce9aa9f8>)

Collection index: [llms.txt](<https://www.sota2.com/research/llms.txt>)

Data updated: 2025-10-25T15:05:34.824Z

Authors: Christian Bayer, Paul Hager, Sebastian Riedel, John Schoenmakers

Published: 2021-05-03T12:34:06.000Z

## Abstract

We propose a new method for solving optimal stopping problems (such as American option pricing in finance) under minimal assumptions on the underlying stochastic process $X$.   We consider classic and randomized stopping times represented by linear and non-linear functionals of the rough path signature $\\mathbb{X}^{\<\\infty}$ associated to $X$, and prove that maximizing over these classes of signature stopping times, in fact, solves the original optimal stopping problem. Using the algebraic properties of the signature, we can then recast the problem as a (deterministic) optimization problem depending only on the (truncated) expected signature $\\mathbb{E}\\left\[ \\mathbb{X}^{\\le N}\_{0,T} \\right\]$. By applying a deep neural network approach to approximate the non-linear signature functionals, we can efficiently solve the optimal stopping problem numerically.   The only assumption on the process $X$ is that it is a continuous (geometric) random rough path. Hence, the theory encompasses processes such as fractional Brownian motion, which fail to be either semi-martingales or Markov processes, and can be used, in particular, for American-type option pricing in fractional models, e.g. on financial or electricity markets.

## Recorded benchmark results

A research result applies to its stated dataset, split, metric and evaluation configuration. Compare like-for-like settings, preserve the metric's unit and direction, and distinguish paper-reported results from independent reproduction. The best result recorded here is not a claim of exhaustive, current global coverage.

- [Volatility Forecasting on S&P 500 Realized Volatility (test)](<https://www.sota2.com/research/sota/volatility-forecasting-on-s-and-p-500-realized-volatility-test.md>): Volatility Forecasting; S&P 500 Realized Volatility (test); R2 (Sig): 0.59.

## Original sources and implementation

[Original paper](<https://arxiv.org/abs/2105.00778>)

[PDF](<https://arxiv.org/pdf/2105.00778.pdf>)