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Fair pricing
and hedging under small perturbations of the numéraire on a
finite probability space
William Busching, Delphine Hintz, Oleksii Mostovyi and Alexey Pozdnyakov
Vol. 15 (2022), No. 4, 649–668
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Abstract |
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We consider the problem of fair pricing and hedging in the sense of Föllmer and Schweizer (1989) under small perturbations of the numéraire. We show that for replicable claims, the change of numéraire affects neither the fair price nor the hedging strategy. For nonreplicable claims, we demonstrate that is not the case. By reformulating the key stochastic control problem in a more tractable form, we show that both the fair price and optimal strategy are stable with respect to small perturbations of the numéraire. Further, our approach allows for explicit asymptotic formulas describing the fair price and hedging strategy’s leading order correction terms. Mathematically, our results constitute stability and asymptotic analysis of a stochastic control problem under certain perturbations of the integrator of the controlled process, where constraints make this problem hard to analyze. |
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Keywords
fair pricing, Föllmer–Schweizer decomposition, numéraire,
stability, asymptotic analysis, conditional fair price
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Mathematical Subject Classification
Primary: 60G42, 60J74, 90C31, 91G20, 93E20
Secondary: 91G10
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Milestones
Received: 10 August 2021
Revised: 7 January 2022
Accepted: 19 January 2022
Published: 7 January 2023
Communicated by Robert B. Lund
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Authors |
| William Busching | |
| Bowdoin College Brunswick, ME United States |
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| Delphine Hintz | |
| Bethany Lutheran College Mankato, MN United States |
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| Oleksii Mostovyi | |
| Department of Mathematics University of Connecticut Storrs, CT United States |
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| Alexey Pozdnyakov | |
| University of Connecticut Storrs, CT United States |
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