TITLE:
7BThe 2-Dimensional Lattice-Subspaces in Finite-State Finance
AUTHORS:
Christos E. Kountzakis
KEYWORDS:
Lattice-Subspace, Positive Projection, Positive Extension
JOURNAL NAME:
Journal of Mathematical Finance,
Vol.4 No.4,
August
28,
2014
ABSTRACT: The main result of this
paper is the construction of a strictly positive extension of any no-arbitrage
price system defined on an incomplete market of
any dimension and for any (finite) number of states of the world. This is
proved by using the component functionals of the
positive basis of the 2-dimensional lattice-subspace, which is spanned by the
riskless asset and any of the state-discriminating payoffs, which are
generically existent in the span of the primitive, non-redundant assets of the
incomplete market, whose time-period 1 payoffs are x1, x2,···, xn, respectively. This
result is also valid in the case where the market is complete. Both of these
results lead to a new statement of both of the Fundamental Theorems of Asset
Pricing in the finite-state case.