🤖 AI Summary
This paper addresses the $L^p$-hedging problem ($p geq 1$) for financial derivatives by introducing a novel $L^p$-Wiener–Itô chaos expansion framework based on iterative Stratonovich integrals. Unlike classical orthogonal expansions, this approach applies to general exponentially integrable functionals driven by continuous semimartingales, relaxing orthogonality constraints to enhance approximation universality. The integrands are parameterized via stochastic neural networks and optimized using $L^p$ approximation theory and numerical training, enabling arbitrary-precision $L^p$-norm approximation of any $p$-integrable derivative. When $p = 2$, the method recovers minimum-variance hedging as a special case. The resulting hedging strategy admits closed-form expressions, facilitating real-time, low-overhead, high-accuracy approximate dynamic replication. The key contribution is the first extension of chaos decomposition to non-orthogonal, exponentially integrable continuous semimartingale settings—unifying theoretical optimality with computational tractability.
📝 Abstract
In this paper, we extend the Wiener-Ito chaos decomposition to the class of continuous semimartingales that are exponentially integrable, which includes in particular affine and some polynomial diffusion processes. By omitting the orthogonality in the expansion, we are able to show that every $p$-integrable functional of the semimartingale, for $p in [1,infty)$, can be represented as a sum of iterated integrals thereof. Using finitely many terms 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. In particular, for $p = 2$, we recover the optimal hedging strategy in the sense of quadratic hedging. Moreover, since the hedging strategy of the approximating option can be computed in closed form, we obtain an efficient algorithm to approximately replicate any sufficiently integrable financial derivative within short runtime.