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We consider the problem of fair pricing and hedging under small perturbationsof the num\'eraire. We show that for replicable claims, the change ofnum\'eraire affects neither the fair price nor the hedging strategy. Fornon-replicable claims, we demonstrate that is not the case. By reformulatingthe key stochastic control problem in a more tractable form, we show that boththe fair price and optimal strategy are stable with respect to smallperturbations of the num\'eraire. Further, our approach allows for explicitasymptotic formulas describing the fair price and hedging strategy's leadingorder correction terms. Mathematically, our results constitute stability andasymptotic analysis of a stochastic control problem under certain perturbationsof the integrator of the controlled process, where constraints make thisproblem hard to analyze.

Fair pricing and hedging under small perturbations of the num\'eraire on a finite probability space