Chaotic Hedging with Iterated Integrals and Neural Networks
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arXiv
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| Format: | Preprint |
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2022
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| _version_ | 1866912852764262400 |
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| author | Neufeld, Ariel Schmocker, Philipp |
| author_facet | Neufeld, Ariel Schmocker, Philipp |
| contents | 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. |
| format | Preprint |
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arxiv_https___arxiv_org_abs_2209_10166 |
| institution | arXiv |
| publishDate | 2022 |
| record_format | arxiv |
| spellingShingle | Chaotic Hedging with Iterated Integrals and Neural Networks Neufeld, Ariel Schmocker, Philipp Mathematical Finance Machine Learning Probability Computational Finance 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. |
| title | Chaotic Hedging with Iterated Integrals and Neural Networks |
| topic | Mathematical Finance Machine Learning Probability Computational Finance |
| url | https://arxiv.org/abs/2209.10166 |