Rough kernel hedging
Nicola Muca Cirone, Cristopher Salvi
TL;DR
This work addresses path-dependent hedging in incomplete markets by embedding the problem in a kernelized rough-path framework that is largely model-free. It replaces the hedging functional with a vector-valued RKHS $\mathcal{H}_K$ built from an operator-valued kernel $K$ and a feature map $\Phi$, so the stochastic integral is expressed as an inner product in the Hilbert space, leading to a standard kernel regression problem. Key contributions include a representer theorem guaranteeing a unique global minimizer and an analytic solution for broad loss functions, plus a practical, data-driven scheme that leverages a Gram matrix $\mathcal{K}_{\Phi}(\mathbb{X},\mathbb{X})$ for finite-sample computation. The framework supports incorporating auxiliary data (e.g., trading signals, news analytics) via the kernel, enabling scalable, high-dimensional hedging with rough-path dynamics, and a GBM-based demonstration shows convergence of the kernel hedge to the delta hedge as data increases.
Abstract
Building on the functional-analytic framework of operator-valued kernels and un-truncated signature kernels, we propose a scalable, provably convergent signature-based algorithm for a broad class of high-dimensional, path-dependent hedging problems. We make minimal assumptions about market dynamics by modelling them as general geometric rough paths, yielding a fully model-free approach. Furthermore, through a representer theorem, we provide theoretical guarantees on the existence and uniqueness of a global minimum for the resulting optimization problem and derive an analytic solution under highly general loss functions. Similar to the popular deep hedging approach, but in a more rigorous fashion, our method can also incorporate additional features via the underlying operator-valued kernel, such as trading signals, news analytics, and past hedging decisions, closely aligning with true machine-learning practice.