Introduction to Signature Methods in Finance

Signatures have recently entered the area of machine learning and mathematical finance as a powerful method to extract  features from time-series data. The signature of a path or time-series is the collection of all its iterated integrals. Conceptually, those iterated integrals can be seen as the path-space analog of polynomials. Indeed, using the right path-space topology, many convenient properties of polynomial translate to signatures, for example:

• products of linear functions of the signature are again linear functions of the signature;

•  point-separation, meaning that the path is uniquely determined by its (time-extended) signature;

• linear functions of the signature are dense in continuous  path-functionals (universal approximation property), e.g.  Kalsi et al. 2020, Cuchiero et al. 2022, Bayer et al. 2023;

• sufficiently regular non-anticipative path functionals admit a  Taylor expansion in terms of signatures Dupire and Tissot-Daguette 2022, Cuchiero et al. 2024.

These properties imply that linear functions of the signature are not only universal but very tractable as path-dependent strategies in optimal control, since only a linear "read-out" needs to be trained. This implies that many optimization problems in non-Markovian situations with path-dependent strategies, for instance optimal portfolio selection, can be reduced to a convex quadratic one Cuchiero and Möller 2023.

In view of signatures of stochastic processes and further applications in finance the following properties are crucial:

• expected signature (as analog of moments) characterizes the law on path space under certain analogs of exponential moment conditions;

• the characteristic function of the signature characterizes the law on path space;

•  both quantities can be easily computed/approximated for large classes of processes, in particular for affine and polynomial ones and so-called signature SDEs, Perrez Arribas et al. 2020, Cuchiero et al. 2023.

In particular for the large class of polynomial processes, their truncated signature is again a polynomial process, resulting in an easily commutable formula (based on a matrix exponential) for conditional expectations of linear functions on the signature. More generally, the signature of  signature SDEs (including polynomial processes) are infinite dimensional affine processes Cuchiero et al. 2023, which opens the door to Fourier pricing techniques for signature-based models Cuchiero et al. 2024b, Abi Jaber et al. 2024. 

Clearly, the mentioned properties are highly relevant for financial data processing and modeling. Hence, it is no surprise that signatures have earned significant attention in various areas of finance, which can roughly be grouped into the following: 

• optimal control, e.g. Kalsi et al. 2020, Bayer et al. 2023, Futter et al. 2023 , Cuchiero and Möller 2023, Bank et al. 2024 ;

• data generation/simulation, e.g. Bühler et al. 2020, Liao et al. 2022 ; 

• financial modeling e.g. Perrez Arribas et al. 2020, Cuchiero et al. 2023b, Cuchiero et al. 2022, Cuchiero et al. 2024b, Abi Jaber et al. 2024;

• randomized signatures, signature neural processes and their applications e.g. Cuchiero et al. 2021, Compagnoni et al. 2022, Akyildirim et al. 2023, Biagini et al. 2024, Krach et al. 2022;

• signature kernel regression methods and applications to path-dependent PDEs in finance, e.g. Salvi et al. 2021, Lee and Oberhauser 2023, Pannier and Salvi 2024;

• other applications in mathematical finance and machine learning, e.g. Kiraly and Oberhauser 2019 , Chevyrev and Oberhauser 2022, Bayraktar et al. 2024, Cohen et al. 2023, Guo et al. 2023.

Additionally signatures have successfully been employed at Amazon to forecast transportation marketplace rates Guo et al. 2024.

New York University’s Tandon School of Engineering, at the Brooklyn NY campus