Complication and Optimization in Additive Markets

In this study, the geometric Additive market models are considered. In general, these market models are incomplete, that means: the perfect replication of derivatives, in the usual sense, is not possible. In this study, it is shown that the market can be completed by new artificial assets which are called ¿power-jump assets¿ based on the power-jump processes of the underlying Additive process. Then, the hedging portfolio for claims whose payoff function depends on the prices of the stock and the power-jump assets at maturity is derived. In addition to the previous completion strategy, it is also shown that, using a static hedging formula, the market can also be completed by considering portfolios with a continuum of call options with different strikes and the same maturity. What is more, the portfolio optimization problem is considered in the enlarged market. The optimization problem consists of choosing an optimal portfolio in such a way that the largest expected utility of the terminal wealth is obtained. For particular choices of the equivalent martingale measure, it is shown that the optimal portfolio consists only of bonds and stocks.