Exploring the relationship between the hedging strategies based on coherent risk measures and the martingale probabilities via optimization approach

An application of the duality theory of linear optimization leads to the well known arbitrage pricing theorems of financial mathematics, namely, the equivalence between the absence of arbitrage and the existence of an equivalent martingale probability measure. The prices of contingent claims can then be calculated based on the set of martingale probability measures. Especially, in the incomplete market which has more than one equivalent probability measures, an interval for the noarbitrage price is obtained rather than a single value. In this thesis, we address a problem of pricing contingent claims in a discrete model of the incomplete market by extending the hedging concept. A narrower no-arbitrage interval of the contingent claim price is obtained by replacing the traditional no-risk condition with a new idea associated with coherent risk measures. The price interval can be calculated by solving a pair of linear programs where the decision variables vary over a subset of martingale probability measures which is uniquely characterized by a given coherent risk measure. Some computational results are also reported, showing that the no-arbitrage interval may turn into a single point if an adequate coherent risk measure is employed. Such a single value is considered as a fair price of the contingent claim since the seller and the buyer face to the same risk if the contingent claim is traded at that price.