A fundamental problem arising in Mathematical Finance is to construct arbitrage-free probability models from quoted prices of a finite number of futures and european-type derivatives, as has been achieved in (Davis and Hobson, 2007). This is not a parameter callibration procedure. If the data satisfy a certain condition, close to the no-arbitrage condition, then a no-arbitrage model can be constructed, including the probability space and associated processes. The problem can be cast within the framework of Markov's Moments Problems (MMP) (Krein & Nudelmann,1973). Moments apprear as prices corresponding to certain pay-off well defined functions. A central role in these problems is played by characterizing non-negative linear combinations of the functions as future pay-offs, and determining the present prices. Tchebycheff Systems (Karlin & Studden,1966) have been fruitfully applied for this purpose, in that their linear combinations exhibit managable properties of their zeroes together with their non-negativity. In the case of call-options, these nice properties are partially lost by the presence of 'interval zeroes', that is, whole intervals where their linear combinations may be zero. The full extent of the theory has to be adapted to this case by explicitly constructing its main objects and results. When so doing, the call-case reveals many properties which allow to establish the existence of a host of Markov transition- mean-preserving kernels, from which many probabilistic processes may be indicated and constructed. This material is not part of the classical literature, but owes its interpretation from the construction in Davis and Hobson's article.Additionally, quadratures of functions other than those originally present in the moments' definition, are possible, and precise lower and upper bounds may be given to them. Upper bounds for the integrals of convex functions are readily obtained, and form the nucleus of much that can be said in terms of Choquet's ordering of measure (P. A. Meyer,1966). Lower bounds for the same class of functions are more difficult to obtain exactly, since the sparseness of the strikes sets a limit to the precision we may aspire to exactly bound a large class of functions. Some steps in developing lower bounds for convex functions quadratures are indicated. Part of the MMP's Programme is to settle the question as to when is it possible to decare a full rectangle to consist of moments by testing just two of its vertices definining an off diagonal. This is another problem originally devised by Markov himself. This is done in the last section and has an interesting interpretions in terms of bid-ask prices.