Chaotic Hedging with Iterated Integrals and Neural Networks
This provides a method for financial hedging that is incremental, improving approximation and computational efficiency for practitioners in quantitative finance.
The paper tackles the problem of approximating p-integrable financial derivatives and solving the L^p-hedging problem by deriving an L^p-chaos expansion using iterated Stratonovich integrals and neural networks, achieving universal approximation results and closed-form strategies with short runtime.
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.