Chaotic Hedging with Iterated Integrals and Neural Networks

TL;DR

This paper derives an L^p-chaos expansion based on iterated Stratonovich integrals with respect to a given exponentially integrable continuous semimartingale and obtains universal approximation results for p-integrable financial derivatives in the $L^p$-sense.

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.

Figures (9)

• Figure 1: Learning performance
• Figure 2: Payoff distribution on test set
• Figure 3: Running time and number of parameters
• Figure 4: $\theta$ and $\vartheta^{\widetilde{\varphi}_{1:N}(\widetilde{\omega})}$ for two samples of test set
• Figure 6: Learning performance

Theorems & Definitions (56)

• Remark 2.2
• Example 2.3: Polynomial diffusions
• Remark 3.1
• Lemma 3.2
• Definition 3.3
• Remark 3.4
• Lemma 3.5
• Lemma 3.6
• Definition 3.7
• Remark 3.8