Arbitrage and investment opportunities

Abstract

We consider a model in which any investment opportunity is described in terms of cash flows. We don't assume that there is a numeraire, enabling investors to transfer wealth through time; the time horizon is not supposed to be finite and 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 available 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.