Optimal stopping with signatures

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 X <∞ 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 E X ≤N 0,T . 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.

Joint calibration to SPX and VIX options with signature-based models

“…We start by introducing basic notions related to the definition of the signature of an R dvalued continuous semimartingale. This is similar as in Cuchiero et al (2022a) or Bayer et al (2021), but to keep the paper self-contained we recall the essential definitions and properties.…”

“…A well-known and extremely useful property of the signature is that every polynomial function in the signature has a linear representation. For the precise statement we first need to introduce the following concept (see also Definition 2.4 in or Section 2.2. in Bayer et al (2021)). a I b J (e I ¡ e J ).…”

Universal approximation theorems for continuous functions of càdlàg paths and Lévy-type signature models

“…This explains why these techniques are more and more applied in econometrics and mathematical finance, see e.g. Buehler et al (2020); Kalsi et al (2020); ; ; Ni et al (2020); Bayer et al (2021); Cuchiero et al (2021); Min and Hu (2021); Akyildirim et al (2022) and the references therein. Indeed, signature-based methods allow for data-driven modeling approaches, while stylized facts or first principles from mathematical finance can still be easily guaranteed.…”