Pricing with coherent risk

This is the first of two papers dealing with applications of coherent risk measures to basic problems of financial mathematics. In this paper, we study applications to pricing in incomplete markets. We prove the fundamental theorem of asset pricing for the no good deals (NGD) pricing technique based on coherent risks. The model considered includes static and dynamic models as well as models with infinitely many assets, and models with transaction costs. In particular, we prove that in a dynamic model with proportional transaction costs the fair price interval converges to the fair price interval in the frictionless model as the coefficient of transaction costs tends to zero. Moreover, we study some problems in the “pure” theory of risk measures. In particular, we introduce the notion of a generator that opens the way for geometric constructions. Based on this notion, we give a simple geometric solution of the capital allocation problem.