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| Main Authors: | , |
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| Format: | Preprint |
| Published: |
2025
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| Subjects: | |
| Online Access: | https://arxiv.org/abs/2502.05942 |
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| _version_ | 1866910820073472000 |
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| author | Nkosi, Siboniso Confrence Mhlanga, Farai Julius |
| author_facet | Nkosi, Siboniso Confrence Mhlanga, Farai Julius |
| contents | Functional It^o calculus is based on an extension of the classical It^o calculus to functionals depending on the entire past evolution of the underlying paths and not only on its current value. The calculus builds on Follmer's deterministic proof of the It^o formula, see [3], and a notion of pathwise functional derivatives introduced by [5]. There are no smoothness assumptions required on the functionals, however, they are required to possess certain directional derivatives which may be computed pathwise, see [6, 9, 8]. Using functional It^o calculus and the notion of quadratic variation, we derive the functional It^o formula along with the Feynman-Kac formula for functional processes. Furthermore, we express the Greeks for path-dependent options as expectations, which can be efficiently computed numerically using Monte Carlo simulations. We illustrate these results by applying the formulae to digital options within the Black-Scholes model framework. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2502_05942 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Sensitivity analysis of path-dependent options in an incomplete market with pathwise functional Ito calculus Nkosi, Siboniso Confrence Mhlanga, Farai Julius Probability 60H07, 60G5, 60G51, 60H35 Functional It^o calculus is based on an extension of the classical It^o calculus to functionals depending on the entire past evolution of the underlying paths and not only on its current value. The calculus builds on Follmer's deterministic proof of the It^o formula, see [3], and a notion of pathwise functional derivatives introduced by [5]. There are no smoothness assumptions required on the functionals, however, they are required to possess certain directional derivatives which may be computed pathwise, see [6, 9, 8]. Using functional It^o calculus and the notion of quadratic variation, we derive the functional It^o formula along with the Feynman-Kac formula for functional processes. Furthermore, we express the Greeks for path-dependent options as expectations, which can be efficiently computed numerically using Monte Carlo simulations. We illustrate these results by applying the formulae to digital options within the Black-Scholes model framework. |
| title | Sensitivity analysis of path-dependent options in an incomplete market with pathwise functional Ito calculus |
| topic | Probability 60H07, 60G5, 60G51, 60H35 |
| url | https://arxiv.org/abs/2502.05942 |