  This paper deals with applications of coherent risk measures to pricing in
incomplete markets. Namely, we study the No Good Deals pricing technique based
on coherent risk. Two forms of this technique are presented: one defines a good
deal as a trade with negative risk; the other one defines a good deal as a
trade with unusually high RAROC. For each technique, the fundamental theorem of
asset pricing and the form of the fair price interval are presented. The model
considered includes static as well as dynamic models, models with an infinite
number of assets, models with transaction costs, and models with portfolio
constraints. In particular, we prove that in a model with proportional
transaction costs the fair price interval converges to the fair price interval
in a frictionless model as the coefficient of transaction costs tends to zero.
Moreover, we study some problems in the ``pure'' theory of risk measures: we
present a simple geometric solution of the capital allocation problem and apply
it to define the coherent risk contribution. The mathematical tools employed
are probability theory, functional analysis, and finite-dimensional convex
analysis.
