Chaotic Hedging with Iterated Integrals and Neural Networks

TL;DR

This paper develops a novel $L^p$-chaos expansion using iterated Stratonovich integrals and neural networks, enabling universal approximation of financial derivatives and efficient $L^p$-hedging strategies.

Contribution

It introduces a new $L^p$-chaos expansion framework that combines iterated integrals with neural networks for approximation and hedging in finance.

Findings

Universal approximation of $p$-integrable derivatives achieved.

Closed-form solutions for $L^p$-hedging strategies with short runtime.

Extension of chaos expansion to non-orthogonal, $L^p$-approximate setting.

Abstract

In this paper, we derive an $L^p$-chaos expansion based on iterated Stratonovich integrals with respect to a given exponentially integrable continuous semimartingale. By omitting the orthogonality of the expansion, we show that every $p$-integrable functional, $p \in [1,\infty)$, can be approximated by a finite sum of iterated Stratonovich integrals. Using (possibly random) neural networks as integrands, we therefere obtain universal approximation results for $p$-integrable financial derivatives in the $L^p$-sense. Moreover, we can approximately solve the $L^p$-hedging problem (coinciding for $p = 2$ with the quadratic hedging problem), where the approximating hedging strategy can be computed in closed form within short runtime.