Christa Cuchiero: Signature methods in finance

Signature methods represent a non-parametric way for extracting characteristic features from time series data which is essential in machine learning tasks. This explains why these techniques become more and more popular in Econometrics and Mathematical Finance. Indeed, signature based approaches allow for data-driven and thus more robust model selection mechanisms, while first principles like no arbitrage can still be easily guaranteed.
In this course we shall focus on the use of signature as universal linear regression basis of continuous functionals of paths for financial applications. We first give a brief introduction to continuous rough paths and show how to embed continuous semimartingales into the rough path setting. Indeed our main focus lies on signature of semimartingales, one of the main modeling tools in finance. By relying on the Stone-Weierstrass theorem we show how to prove the universal approximation property of linear functions of the signature in appropriate topologies on path space.
In the financial applications that we have in mind one key quantity that one needs to compute is the expected signature of some underlying process. Surprisingly this can be achieved for generic classes of diffusions, called signature-SDEs (with possibly path dependent characteristics), via techniques from affine and polynomial processes. More precisely, we show how the signature process of these diffusions can be embedded in the framework of affine and polynomial processes. These classes of processes have been -- due to their tractability -- the dominating process class prior to the new era of highly over-parametrized dynamic models. Following this line we obtain that the infinite dimensional Feynman Kac PDE of the signature process can generically be reduced to an infinite dimensional ODE either of Riccati or linear type.
In terms of financial applications, we shall treat two main topics: stochastic portfolio theory and signature based asset price models.
In the context of stochastic portfolio theory we introduce a novel class of portfolios which we call linear path-functional portfolios. These are portfolios which are determined by certain transformations of linear functions of a collections of feature maps that are non-anticipative path functionals of an underlying semimartingale. As main example for such feature maps we consider signature of the (ranked) market weights. Relying on the universal approximation theorem we show that every continuous (possibly path-dependent) portfolio function of the market weights can be uniformly approximated by signature portfolios. Besides these universality features, the main numerical advantage lies in the fact that several optimization tasks like maximizing expected logarithmic utility or mean-variance optimization within the class of linear path-functional portfolios reduces to a convex quadratic optimization problem, thus making it computationally highly tractable. We apply our method to real market data, indicating out-performance on the considered out-of-sample data even under transaction costs.
In view of asset price models we consider a stochastic volatility model where the dynamics of the volatility are described by linear functions of the (time extended) signature of a primary underlying process, which is supposed to be some multidimensional continuous semimartingale. Under the additional assumption that this primary process is of polynomial type, we obtain closed form expressions for the VIX squared, exploiting the fact that the truncated signature of a polynomial process is again a polynomial process. Adding to such a primary process the Brownian motion driving the stock price, allows then to express both the log-price and the VIX squared as linear functions of the signature of the corresponding augmented process. This feature can then be efficiently used for pricing and calibration purposes. Indeed, as the signature samples can be easily precomputed, the calibration task can be split into an offline sampling and a standard optimization. For both the SPX and VIX options we obtain highly accurate calibration results, showing that this model class allows to solve the joint calibration problem without adding jumps or rough volatility. Slides: Slides 1, Slides 2, Slides 3, Slides 4
References
[1] P. Friz, Peter and N. Victoir. Multidimensional stochastic processes as rough paths: theory and applications. Vol. 120. Cambridge University Press, 2010, in particular chapters 7-9.
[2] I. Chevyrev and A. Kormilitzin. A primer on the signature method in machine learning. arXiv:1603.03788 (2016).
[3] C. Cuchiero, S. Svaluto-Ferro, and J.Teichmann. Signature SDEs from an affine and polynomial perspective. arXiv:2302.01362 (2023).
[4] C. Cuchiero, F. Primavera, and S. Svaluto-Ferro. Universal approximation theorems for continuous functions of càdlàg paths and Lévy-type signature models. arXiv:2208.02293 (2022).
[5] C. Cuchiero and J. Möller. Signature Methods in Stochastic Portfolio Theory. arXiv:2310.02322 (2023).
[6] C. Cuchiero, G. Gazzani, and S. Svaluto-Ferro. Signature-based models: Theory and calibration. SIAM journal on financial mathematics 14.3 (2023): 910-957.
[7] C. Cuchiero, G. Gazzani, J. Möller and S. Svaluto-Ferro. Joint calibration to SPX and VIX options with signature-based models. arXiv:2301.13235 (2023).