Signature Methods in Finance

An Introduction with Computational Applications

Open Access
2026
Open Access
Book

Editors: Christian Bayer; Goncalo dos Reis; Blanka Horvath; Harald Oberhauser
Book Series: Springer Finance
Publisher: Springer Nature Switzerland
Part of: Springer Professional "Business + Economics & Engineering + Technology"; Springer Professional "Business + Economics"

About this book

This Open Access volume offers an accessible entry point into the fast-growing field of signature methods in finance. It is written for early-career researchers and quantitatively minded practitioners—quant analysts and applied researchers—seeking a clear, practical introduction. It highlights recent developments and includes coding examples to help readers apply signature methods in practice.

The advantages of modeling financial markets from a path-wise perspective, rather than as a traditional series of returns, are increasingly gaining recognition. Signature methods provide a parsimonious description of paths of stochastic processes and, through the signature kernel, open a rich and compelling framework at the interface between machine learning and mathematical finance.

“I have been extraordinarily fortunate to work alongside brilliant collaborators throughout this journey, and this book beautifully reflects the richness of that shared contribution—for which I am deeply grateful.”—Prof Terry Lyons, University of Oxford, Imperial College, and PI of DataSig

“This fascinating collection, dedicated to Terry Lyons, offers invaluable insights into signature methods and their many uses.” Jim Gatheral, Presidential Professor, Baruch College, Quant of the Year 2021

"A timely and important contribution to the fast-growing field of signature methods, showcasing the theory and applications of these powerful ideas.” — Prof Ben Hambly, University of Oxford

“An impressive book on signatures with articles by the most distinguished researchers in the field. A reference from day one." – Dr Hans Buehler, co-CEO XTX Markets, Quant of the Year 2022

"This book provides a masterful exposition and development of signature methods in finance. It is concise, precise, and actionable. It will be an excellent source for anyone interested in modern financial engineering techniques." – Prof Alexander Lipton, Global Head of R&D, ADIA, and Founding Member ADIA Lab, Quant of the Year 2000 and Buy-side Quant of the Year 2021.

Frontmatter

PDF

Introduction to Signatures in Machine Learning

Frontmatter

Open Access

A Primer on the Signature Method in Machine Learning

Abstract

We provide an introduction to the signature method, focusing on its theoretical properties and machine learning applications. Our presentation is divided into two parts. In the first part, we present the definition and fundamental properties of the signature of a path. The signature is a sequence of numbers associated with a path that captures many of its important analytic and geometric properties. As a sequence of numbers, the signature serves as a compact description (dimension reduction) of a path. In presenting its theoretical properties, we assume only familiarity with classical real analysis and integration, and supplement theory with straightforward examples. We also mention several advanced topics, including the role of the signature in rough path theory. In the second part, we present practical applications of the signature to the area of machine learning. The signature method is a non-parametric way of transforming data into a set of features that can be used in machine learning tasks. In this method, data are converted into multi-dimensional paths, by means of embedding algorithms, of which the signature is then computed. We describe this pipeline in detail, making a link with the properties of the signature presented in the first part. We furthermore review some of the developments of the signature method in machine learning and, as an illustrative example, present a detailed application of the method to handwritten digit classification.

Ilya Chevyrev, Andrey Kormilitzin

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Open Access

An Introduction to Tensors for Path Signatures

Abstract

We present a fit-for-purpose introduction to tensors and their operations. It is envisaged to help the reader become acquainted with its underpinning concepts for the study of path signatures. The text includes exercises, solutions and many intuitive explanations. The material discusses direct sums and tensor products as two possible operations that make the Cartesian product of vectors spaces a vector space. The difference lies in linear vs. multilinear structures—the latter being the suitable one to deal with path signatures. The presentation is offered to understand tensors in a sense deeper than just a multidimensional array. The text concludes with the prime example of an algebra in relation to path signatures: the tensor algebra.

Jack Beda, Gonçalo dos Reis, Nikolas Tapia

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Open Access

The Signature Kernel

Abstract

The signature kernel is a positive definite kernel for sequential data. It inherits theoretical guarantees from stochastic analysis, has efficient algorithms for computation, and shows strong empirical performance. In this chapter, we provide an introduction to the signature kernel by highlighting the analytic connection between the path signature and ordinary monomials. In particular, both the classical monomials and the path signature are universal and characteristic on compact domains: they can approximate functions, and characterize probability measures. However, issues arise in practice due to unbounded domains, computational complexity, and non-robustness. We show that these issues can be avoided via kernelization and robustification. To address the computational complexity, we provide an overview of the kernel trick: algorithms to efficiently compute the signature kernel which avoid direct computations of the signature. Furthermore, the signature kernel is highly flexible, and provides a canonical way to turn a given kernel on any domain into a kernel for sequences in that domain, while retaining its theoretical and computational properties. Finally, we survey applications and recent developments of the signature kernel.

Darrick Lee, Harald Oberhauser

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Applications of Signatures

Frontmatter

Open Access

Market Generators: A Paradigm Shift in Financial Modeling

Abstract

Market Generators are a rapidly evolving class of neural-network-based models to simulate financial market behavior, offering a powerful alternative to classical stochastic models. These deep learning models are trained to encode the underlying distribution of financial data and generate new synthetic market scenarios from the learned distribution. Though the expressions “Market Generator” and its related form “Market Simulator” have only entered the vocabulary of financial modeling around 2019, by today, the modeling techniques related to them have already grown into an area in their own right. This growing interest is matched by a dramatic rise in research, publications and an accelerating rate of innovation in the ambient technological arena of generative modeling. Recently, this trend has culminated in the emergence of large generative networks, particularly GPT-type language models, which are proving to be driving one of the biggest disruptions in the history of technology.

Blanka Horvath, Jonathan Plenk, Milena Vuletić

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Open Access

Signature Maximum Mean Discrepancy Two-Sample Statistical Tests

Abstract

Maximum Mean Discrepancy (MMD) is a widely used concept in machine learning research which has gained popularity in recent years as a highly effective tool for comparing (finite-dimensional) distributions. Since it is designed as a kernel-based method, the MMD can be extended to path space valued distributions using the signature kernel. The resulting signature MMD (sig-MMD) can be used to define a metric between distributions on path space. Similarly to the original use case of the MMD as a test statistic within a two-sample testing framework, the sig-MMD can be applied to determine if two sets of paths are drawn from the same stochastic process. This chapter is dedicated to understanding the possibilities and challenges associated with applying the sig-MMD as a statistical tool in practice. We introduce and explain the sig-MMD, and provide easily accessible and verifiable examples for its practical use. We present examples that can lead to Type 2 errors in the hypothesis test, falsely indicating that samples have been drawn from the same underlying process (which generally occurs in a limited data setting). We then present techniques to mitigate the occurrence of this type of error.

Andrew Alden, Blanka Horvath, Zacharia Issa

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Open Access

Signature and the Functional Taylor Expansion

Abstract

This chapter links the path signature to the functional Itô calculus (Dupire, Functional Itô calculus. SSRN (2009). Republished in Quantitative Finance 19(5), 721–729, 2019). The junction is the functional Taylor expansion (FTE), discussed in Sect. 5, which is a powerful tool to approximate functionals after an observed path.

In particular, the FTE decomposes the functional from future scenarios. In risk analysis, the FTE leads to new Greeks, one of particular importance being the Lie bracket between the space and time functional derivatives, which we call Libra. Once paired with the Lévy area, the Libra can be used to speed up the computation of risk measures such as Value at Risk and Expected Shortfall as explained in Sect. 6.1. In Sect. 6.3, we explain how the FTE can quantify the hedging error of replicating portfolios in a robust manner. In the context of financial derivatives, we demonstrate in Sect. 6.4 how the FTE generates approximations of exotic payoffs. The relevance of the FTE for cubature methods is finally outlined in Sect. 6.5.

Bruno Dupire, Valentin Tissot-Daguette

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Open Access

Signature-Based Models in Finance

Abstract

We consider two classes of asset price models where either the price or the volatility dynamics are described by a linear function of the (time extended) signature of a primary process, in general a multidimensional continuous semimartingale. These model classes are universal in the sense that classical models can be approximated arbitrarily well or are simply nested in our setup. Under the additional assumption that the primary process is polynomial, we obtain tractable option pricing formulas for so-called sig-payoffs in the first class and closed form expressions for the VIX squared and the log-price in the second one. In both cases the signature samples can be easily precomputed, hence the calibration task can be split into an offline sampling and a standard optimization. We present several applications, in particular the successfully solved joint SPX/VIX calibration problem.

Christa Cuchiero, Guido Gazzani, Janka Möller, Sara Svaluto-Ferro

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Open Access

Signature Trading Strategies

Abstract

In this chapter, the primary objective is to recall the concept of a trading strategy and to extend this concept to the setting of signatures. We discuss how one can extend classical methods of optimal trading by tackling stochastic control problems through the lens of signatures, in order to incorporate path-dependencies. First introduced in (Signatures in machine learning and finance. PhD thesis, University of Oxford, 2020), the notion of a signature trading strategy allows to solve path-dependent optimization problems in a simple and explicit way. We first outline the key concepts of signature trading and highlight its associated benefits. Additionally, we present a solution to the mean-variance criterion and illustrate the advantages of signature-based methods using intuitive examples. Finally, we explore how signature trading strategies can be applied to obtain approximate solutions to optimal execution problems.

Owen Futter, Magnus Wiese

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Open Access

Optimal Stopping for Non-Markovian Asset Price Processes

Abstract

Some of the most liquidly traded options in equity markets are American and Bermudan options, whose owner may choose the option’s exercise date—within a certain range. Hence, these options are optimal stopping problems from a mathematical perspective. There is a huge literature on solving optimal stopping problems, and most of the prevalent methods (e.g., solving the Hamilton-Jacobi-Bellman PDE, least squares Monte Carlo, dual martingale methods) strongly rely on the Markov property for the underlying dynamics, to avoid the curse of dimensionality. In this chapter, we will show how the signature can be used to adopt classical, Markovian numerical methods for the non-Markovian case.

Christian Bayer, Paul P. Hager, Sebastian Riedel

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Open Access

Adapted Topologies and Higher-Rank Signatures

Abstract

Two stochastic processes can have similar laws but yield a vastly different outcome in applications such as optimal stopping or stochastic programming. The reason is that the usual weak topology does not account for the different available information in time that is stored in the filtration of the underlying process. To address the resulting discontinuities, Aldous introduced the extended weak topology, and subsequently, Hoover and Keisler showed that both, weak topology as well as extended weak topology, are just the first two topologies in a sequence of the so-called adapted weak topologies that get increasingly finer. In this short survey, we give a brief introduction to the recent advances of the applications of signature theory, in particular the higher-rank (expected) signatures, in adapted weak topologies and related fields, and highlight theoretical and computational properties.

Chong Liu, Gudmund Pammer

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Open Access

On Expected Signatures and Signature Cumulants in Semimartingale Models

Abstract

The signature transform, a Cartan type development, translates paths into high-dimensional feature vectors, capturing their intrinsic characteristics. Under natural conditions, the expectation of the signature determines the law of the signature, providing a statistical summary of the data distribution. This property facilitates robust modeling and inference in machine learning and stochastic processes. Building on previous work by the present authors (Friz et al., Unified signature cumulants and generalized Magnus expansions. In Forum of Mathematics, Sigma, vol. 10, p. e42, 2022) we here revisit the actual computation of expected signatures, in a general semimartingale setting. Several new formulae are given. A log-transform of (expected) signatures leads to log-signatures (signature cumulants), offering a significant reduction in complexity.

Peter K. Friz, Paul P. Hager, Nikolas Tapia

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Accessibility

Title: Signature Methods in Finance
Editors: Christian Bayer
Goncalo dos Reis
Blanka Horvath
Harald Oberhauser
Publisher: Springer Nature Switzerland
Electronic ISBN: 978-3-031-97239-3
Print ISBN: 978-3-031-97238-6
DOI: https://doi.org/10.1007/978-3-031-97239-3

PDF files of this book have been created in accordance with the PDF/UA-1 standard to enhance accessibility, including screen reader support, described non-text content (images, graphs), bookmarks for easy navigation, keyboard-friendly links and forms and searchable, selectable text. We recognize the importance of accessibility, and we welcome queries about accessibility for any of our products. If you have a question or an access need, please get in touch with us at accessibilitysupport@springernature.com.

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Signature Methods in Finance

An Introduction with Computational Applications

About this book

Table of Contents

Frontmatter

Introduction to Signatures in Machine Learning

Frontmatter

A Primer on the Signature Method in Machine Learning

An Introduction to Tensors for Path Signatures

The Signature Kernel

Applications of Signatures

Frontmatter

Market Generators: A Paradigm Shift in Financial Modeling

Signature Maximum Mean Discrepancy Two-Sample Statistical Tests

Signature and the Functional Taylor Expansion

Signature-Based Models in Finance

Signature Trading Strategies

Optimal Stopping for Non-Markovian Asset Price Processes

Adapted Topologies and Higher-Rank Signatures

On Expected Signatures and Signature Cumulants in Semimartingale Models

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