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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
Abstract

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.

Keywords
fair pricing, Föllmer–Schweizer decomposition, numéraire, stability, asymptotic analysis, conditional fair price
Mathematical Subject Classification
Primary: 60G42, 60J74, 90C31, 91G20, 93E20
Secondary: 91G10
Milestones
Received: 10 August 2021
Revised: 7 January 2022
Accepted: 19 January 2022
Published: 7 January 2023

Communicated by Robert B. Lund
Authors
William Busching
Bowdoin College
Brunswick, ME
United States
Delphine Hintz
Bethany Lutheran College
Mankato, MN
United States
Oleksii Mostovyi
Department of Mathematics
University of Connecticut
Storrs, CT
United States
Alexey Pozdnyakov
University of Connecticut
Storrs, CT
United States