In this article we study the completion by options of a two-period security market in which the space of marketed securities is a subspace X of R m . Although there are important results about the completion (by options) Z of X, the problem of the determination of Z in its general form is still open. In this paper we solve this problem by determining a positive basis of Z. This method of positive bases simplifies the theory of security markets and also answers other open problems of this theory. In the classical papers of this subject, call and put options are taken with respect to the riskless bond 1 of R m . In this article we generalize this theory by taking call and put options with respect to different risky vectors u from a fixed vector subspace U of R m . This generalization was inspired by certain types of exotic option in finance. (2000): 46B40, 46A35, 91B28, 91B30 Mathematics Subject Classification

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In this article we give a new criterion for the existence of a bounded base for a cone P of a normed space X. Also, if P is closed, we give a partial answer to the problem: is 0 a point of continuity of P if and only if 0 is a denting point of P ? The above problems have applications in the theory of Pareto efficient points.

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 m , 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

Keywords:Base of a cone Numeraire asset Coherent and convex risk measures Dual representation of risk measures This work is devoted to the study of coherent and convex risk measure on non-reflexive Banach spaces. An extension of dual representation and continuity results which hold in the case of reflexive spaces is established in this paper for the class of non-reflexive Banach spaces. This study also relies on the fact that the riskless bond is replaced by some numeraire asset which defines a base on the cone of the spot-price functionals. Non-reflexive Banach spaces and risk measuresMost common cases of normed and Banach spaces used in risk measures' theory are the L p (Ω, F , μ) spaces in which 1 p ∞ and μ is here a probability measure on the measurable space (Ω, F ). In most of the cases, the probability space (Ω, F , μ) is supposed to be a non-atomic space. L p spaces in these cases are actually Banach spaces and moreover Banach lattices (see also in [2, Th. 12.5]). As it is well known, if 1 < p < ∞, then the space L p (Ω, F , P ) is reflexive (see also in [2, Th. 12.27]). Also, L 1 and L ∞ are not in general reflexive. For this question, we may repeat the statement of the corollaryHence the cases of non-reflexive spaces which usually arise in risk measures' theory and applications are the L 1 -spaces and the L ∞ -spaces. As it is also quoted in [18, p. 476], the cause for considering an L p -space where p ∈ [1, ∞) as a model of financial positions or actuarial risks is that the distributions of the random variables in these models may allow for the presence of fat-tails, or else for unbounded possible values. We remind that the distributions having fat-tails are those for which the exponential moments do not exist, or else if X is a random variable with positive outcomes and > 0, while the cumulative distribution function of it is F X (x), x 0, then the integral ∞ 0 e x dF X (x) is not finite for any > 0. Also, in[18] the case of normal or stable distributions is mentioned. The case of L 1 is proposed as a model also in [15, p. 237] among other L p spaces with p ∈ [1, ∞). About stable distributions, authors in [18] append to [25, Chapter 7]. The central moments E μ (X k ) of the distributions having fat-tails just like Pareto distribution, don't exist for every k. For example if X follows a Pareto distribution, the moments' existence depends on the parameter a in the probability density function f (x) = aθ a (x+θ) a+1 , x 0 of the Pareto distribution. For k > a the moment E μ (X k ) does not exist (see also in [24, p. 255]).In the paper [18] dual representation and continuity results are proved in the case of L p spaces, which include the case of the non-reflexive L 1 . Continuity results as it is also mentioned in [18, p. 475], are applied in questions of robustness and approximation. Also in [15] and [13], continuity and dual representation results are proved for L 1 respectively, if the numeraire asset is 1 and the partial ordering of the space is the usual (componentwise) one. In [9], convex risk me...