Arbitrage and Investment Opportunities

Abstract

We consider a model in which all investment opportunities are described in terms of cash flows. We don't assume that there is a numéraire, the time horizon is not supposed to be finite, the investment opportunities are not specifically related to the buying and selling of securities on a financial market. In this quite general framework, we show that the assumption of no-arbitrage is essentially equivalent to the existence of a "discount process" under which the "net present value" of any investment is nonpositive. Since most market imperfections, such as short sale constraints, convex cone constraints, proportional transaction costs, no borrowing or different borrowing and lending rates, etc., can fit in our model for a specific set of investments, we then obtain a characterization of the no-arbitrage condition in these imperfect models, from which it is easy to derive pricing formulae for contingent claims. Compared with existing results, our approach allows to consider markets with no numéraire or with a numéraire that is subject to constraints. Besides, we introduce a notion of no-free lunch which is less restrictive than the usual one. Last, we characterize the assumption of no-arbitrage (or no-free lunch) for more general investment opportunities, which enables us to consider investments that are not necessarily related to a market model and, more interestingly, to generalize the results obtained for imperfect markets and to obtain them all in a unified way.