Superhedging under Proportional Transaction Costs in Continuous Time
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arXiv
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| Format: | Preprint |
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2025
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| _version_ | 1866914168202854400 |
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| author | Almuzaini, Atiqah Ararat, Çağın Ma, Jin |
| author_facet | Almuzaini, Atiqah Ararat, Çağın Ma, Jin |
| contents | We revisit the well-studied superhedging problem under proportional transaction costs in continuous time using the recently developed tools of set-valued stochastic analysis. By relying on a simple Black-Scholes-type market model for mid-prices and using continuous trading schemes, we define a dynamic family of superhedging sets in continuous time and express them in terms of set-valued integrals. We show that these sets, defined as subsets of Lebesgue spaces at different times, form a dynamic set-valued risk measure with multi-portfolio time-consistency. Finally, we transfer the problem formulation to a path-space setting and introduce approximate versions of superhedging sets that will involve relaxing the superhedging inequality, the superhedging probability, and the solvency requirement for the superhedging strategy with a predetermined error level. In this more technical framework, we are able to relate the approximate superhedging sets at different times by means of a set-valued Bellman's principle, which we believe will pave the way for a set-valued differential structure that characterizes the superhedging sets. |
| format | Preprint |
| id |
arxiv_https___arxiv_org_abs_2511_18169 |
| institution | arXiv |
| publishDate | 2025 |
| record_format | arxiv |
| spellingShingle | Superhedging under Proportional Transaction Costs in Continuous Time Almuzaini, Atiqah Ararat, Çağın Ma, Jin Risk Management Optimization and Control Probability 26E25, 28B20, 60H10, 91G70, 93E20 We revisit the well-studied superhedging problem under proportional transaction costs in continuous time using the recently developed tools of set-valued stochastic analysis. By relying on a simple Black-Scholes-type market model for mid-prices and using continuous trading schemes, we define a dynamic family of superhedging sets in continuous time and express them in terms of set-valued integrals. We show that these sets, defined as subsets of Lebesgue spaces at different times, form a dynamic set-valued risk measure with multi-portfolio time-consistency. Finally, we transfer the problem formulation to a path-space setting and introduce approximate versions of superhedging sets that will involve relaxing the superhedging inequality, the superhedging probability, and the solvency requirement for the superhedging strategy with a predetermined error level. In this more technical framework, we are able to relate the approximate superhedging sets at different times by means of a set-valued Bellman's principle, which we believe will pave the way for a set-valued differential structure that characterizes the superhedging sets. |
| title | Superhedging under Proportional Transaction Costs in Continuous Time |
| topic | Risk Management Optimization and Control Probability 26E25, 28B20, 60H10, 91G70, 93E20 |
| url | https://arxiv.org/abs/2511.18169 |