In everyday life we are often forced to make decisions involving risks and
perceived opportunities. The consequences of our decisions are affected by the
outcomes of random variables that are to various degrees beyond our control. Such
decision problems arise, for instance, in financial and insurance markets.
Throughout this paper, we consider a valuation method for contingent claims
taking control of shortfall risk into account in the framework of complete
market models. After giving a general form of the valuation, we shall deal
with models whose underlying assets are described by diffusion processes,
and obtain a result for American type claims
As a matter of sound investment practice, FHLBanks should be able to measure the price
sensitivity for individual securities and for the entire portfolio. In general, techniques used to
measure the risk of individual securities are also appropriate for the entire portfolio.
To estimate portfolio sensitivity, FHLBanks generally use, at a minimum, duration. Because of
the presence of options in most portfolios, duration may not be an effective risk measure.
We resolve these issues as follows. We show that a nonincreasing returns to scale (nrs) model is
usually appropriate when modeling rational choice among investors. We show when multiple risk
and return measures can justiﬁably be combined and identify some suitable measures. We show
we need a nonlinear model to justify the assumption of convexity and to model diversiﬁcation.
We develop a method to approximate a solution to this model as accurately as needed using a
sequence of linear models.
Coherent measures of risk come up again and again in our discussion.