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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