Non replication of options

In this paper we study the scarcity of replication of options in the two period model of financial markets with a finite set of states. Especially we study this problem in financial markets without binary vectors and in strongly resolving markets. We start our study by proving that a financial market does not have binary vectors if and only if for any portfolio, at lest one non trivial option is replicated. After this characterization we prove that in these markets, for any portfolio, at most m − 3 options can be replicated where m is the number of states, therefore for any portfolio, the number of the replicated options is between the natural numbers 1 and m− 3. Note that by the existing result of Baptista (2007), the set of non replicated options is of measure zero, and as it is known there are infinite sets with measure zero. In the sequel we generalize the definition of strongly resolving markets to a more general class of financial markets by considering the payoff matrix of primitive securities, not with respect to the usual basis of R, but with respect to the positive basis of the financial completion of the market. This allows us to generalize the result of Aliprantis-Tourky (2002) about the non-replication of options in a bigger class of financial markets. In this study, the theory of positive bases developed in [5] and [6] plays a central role. This theory simplifies and unifies the theory of options. JEL Classification : C600, D520, G190