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

TL;DR

This paper investigates how small changes in the numéraire affect fair pricing and hedging strategies, revealing stability for replicable claims and providing explicit formulas for non-replicable claims.

Contribution

It introduces a reformulation of the stochastic control problem that clarifies the stability and asymptotic behavior of prices and strategies under numéraire perturbations.

Findings

Replicable claims' prices and strategies are unaffected by numéraire changes.

Non-replicable claims exhibit sensitivity, but their leading order corrections are explicitly characterized.

The approach simplifies the analysis of stochastic control problems with perturbations.

Abstract

We consider the problem of fair pricing and hedging under small perturbations of the num\'eraire. We show that for replicable claims, the change of num\'eraire affects neither the fair price nor the hedging strategy. For non-replicable 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 num\'eraire. 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.