Abstract:
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.
Keywords:capital allocation, coherent risk measure, extreme measure, generator, no good deals, RAROC, risk contribution, risk-neutral measure, support function, tail V@R, transaction costs, weighted V@R.
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