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Universal approximation on non-geometric rough paths and applications to financial derivatives pricing

Fabian A. Harang, Fred Espen Benth, Fride Straum

TL;DR

The paper addresses pricing of complex, path-dependent financial derivatives by extending universal signature-based approximation to non-geometric rough paths, thereby incorporating Itô (Itô) integration into a rough-path framework. It introduces a polynomial-enriched feature class over signatures and, under a separability hypothesis, proves a generalized universal approximation with explicit error bounds, applicable even when time-extended, non-geometric rough paths are used. The approach yields practical representations of payoff functionals as polynomials in a finite set of signature coordinates and enables pricing through low-dimensional correlators, with concrete illustrations for Asian, spread, and quanto-style derivatives and energy-market applications. This framework promises computational efficiency and theoretical robustness for Itô-driven derivatives pricing in finance, linking rough-path theory with practical, model-free pricing techniques.

Abstract

We present a novel perspective on the universal approximation theorem for rough path functionals, introducing a polynomial-based approximation class. We extend universal approximation to non-geometric rough paths within the tensor algebra. This development addresses critical needs in finance, where no-arbitrage conditions necessitate Itô integration. Furthermore, our findings motivate a hypothesis for payoff functionals in financial markets, allowing straightforward analysis of signature payoffs proposed in \cite{arribas2018derivativespricingusingsignature}.

Universal approximation on non-geometric rough paths and applications to financial derivatives pricing

TL;DR

The paper addresses pricing of complex, path-dependent financial derivatives by extending universal signature-based approximation to non-geometric rough paths, thereby incorporating Itô (Itô) integration into a rough-path framework. It introduces a polynomial-enriched feature class over signatures and, under a separability hypothesis, proves a generalized universal approximation with explicit error bounds, applicable even when time-extended, non-geometric rough paths are used. The approach yields practical representations of payoff functionals as polynomials in a finite set of signature coordinates and enables pricing through low-dimensional correlators, with concrete illustrations for Asian, spread, and quanto-style derivatives and energy-market applications. This framework promises computational efficiency and theoretical robustness for Itô-driven derivatives pricing in finance, linking rough-path theory with practical, model-free pricing techniques.

Abstract

We present a novel perspective on the universal approximation theorem for rough path functionals, introducing a polynomial-based approximation class. We extend universal approximation to non-geometric rough paths within the tensor algebra. This development addresses critical needs in finance, where no-arbitrage conditions necessitate Itô integration. Furthermore, our findings motivate a hypothesis for payoff functionals in financial markets, allowing straightforward analysis of signature payoffs proposed in \cite{arribas2018derivativespricingusingsignature}.

Paper Structure

This paper contains 19 sections, 12 theorems, 128 equations, 1 figure.

Table of Contents

  1. Introduction
  2. Main ideas and contribution
  3. Organization of the paper
  4. Notation
  5. Basics of words and signatures
  6. Words
  7. Signatures
  8. Non-geometric rough paths
  9. The regime $p<3$
  10. The regime $p>3$
  11. Functional approximation with signatures
  12. Universal approximation for geometric rough paths
  13. General universal approximation on rough paths
  14. Signature associated to price paths
  15. Examples of functions
  16. ...and 4 more sections

Key Result

Theorem 2.11

[Lyons' extension theorem Lyons2007] Let $\mathbf{X}$ be a p-rough path. Then for any $n\geq \lfloor p\rfloor +1$ there exists a unique continuous map such that is a multiplicative functional with finite $p$-variation. We call $\mathbf{X}^{\leq \infty}$ the signature of $\mathbf{X}$. Moreover, we have that where $|\cdot|_k$ is the canonical Hilbert space norm on $H^{\otimes k}$ and $\beta(p)$ i

Figures (1)

  • Figure 5.1: Plot of $\bar{f}(x)=\frac{x}{1+e^{-Nx}}$

Theorems & Definitions (51)

  • Definition 2.1
  • Definition 2.2
  • Example 2.3
  • Remark 2.4
  • Definition 2.5
  • Definition 2.6
  • Remark 2.7
  • Example 2.8
  • Definition 2.9
  • Definition 2.10
  • ...and 41 more