Market with Markov jump volatility V: Market completion with derivatives

The final, fifth, part of the series is devoted to a replenishment procedure of the market with a Markov jump regime change. The market includes a riskless bank deposit with a known nonrandom interest rate and a set of underlying risky assets. The instantaneous interest rates and volatilities of the assets are the functions of the hidden regime change factor described by some finite state Markov jump process. The purpose of this article is to complete the investigated market. It means the market enlargement by a set of auxiliary financial instruments. The point is that any contingent claim declared in the market can be replicated with some self-financing portfolio containing the original and auxiliary instruments. For the market completion, it is enough to include European-style derivatives built on underlying risky assets already on the market. In this case, the number of added derivatives coincides with the number of market modes. The problem of replenishing the market has a nonunique solution and the article compares the proposed replenishment method with the existing one.