Conditions for Exact Hedging in an Unconstrained Regime-Switching Market Model

Abstract

In an earlier contribution “Unconstrained hedging within a regime-switching market model” (Sixth Indian Control Conference, Hyderabad, December 18-20, 2019) the authors address the problem of unconstrained hedging in a financial market model which includes regime-switching, in the sense that the basic sources of randomness in the market model are a standard multidimensional Brownian motion, together with an independent finite-state Markov chain (the latter process models so-called regime-switches, which are occasional “large-scale” random changes in the market parameters, as opposed to the persistent “small-scale” changes in the market parameters which are driven by the Brownian motion). Under these conditions the market model is “incomplete”, and the best that one can do is establish existence of a least initial wealth along with an investment strategy for which the corresponding wealth process almost-surely majorizes - but generally does not equal - the contingent claim at close of trade (in this case the claim is said to be “super-hedged”). The goal of the present work is to complement this result and introduce natural conditions on the regime- switching model under which there exists a least initial wealth and an investment strategy such that the corresponding wealth almost-surely equals the contingent claim at close of trade (so that the claim is “exactly hedged”). Our motivation is primarily in the works of Cvitanic and Karatzas (1993) and El Karoui and Quenez (1995) who address the case where incompleteness in the market model arises from portfolio constraints rather than regime-switching.