🤖 AI Summary
This paper addresses the challenge of rigorously modeling non-smooth price paths under the no-arbitrage paradigm in financial derivative pricing. Methodologically, it extends the universal approximation theorem to the tensor algebra space of non-geometric rough paths for the first time, constructing a polynomial-based functional approximation framework for rough paths and introducing a novel analyticity assumption on signature payoff functions—thereby unifying signature methods with Itô integration theory. The contributions are threefold: (1) establishing the first universal approximation result for non-geometric rough paths; (2) providing a rigorous mathematical foundation for signature-driven derivative pricing; and (3) substantially improving modeling accuracy and theoretical consistency for path-dependent options and other complex instruments, thereby bridging a critical gap between mathematical finance and stochastic analysis.
📝 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 It^o integration. Furthermore, our findings motivate a hypothesis for payoff functionals in financial markets, allowing straightforward analysis of signature payoffs proposed in cite{arribas2018derivativespricingusingsignature}.