Chaotic Hedging with Iterated Integrals and Neural Networks

In this paper, we extend the Wiener-Ito chaos decomposition to the class of diffusion processes, whose drift and diffusion coefficient are of linear growth. By omitting the orthogonality in the chaos expansion, we are able to show that every $p$-integrable functional, for $p \in [1,\infty)$, can be represented as sum of iterated integrals of the underlying process. Using a truncated sum of this expansion and (possibly random) neural networks for the integrands, whose parameters are learned in a machine learning setting, we show that every financial derivative can be approximated arbitrarily well in the $L^p$-sense. Since the hedging strategy of the approximating option can be computed in closed form, we obtain an efficient algorithm that can replicate any integrable financial derivative with short runtime.