Generic existence and robust nonexistence of numeraires in finite dimensional securities markets

A numeraire is a portfolio that, if prices and dividends are denominated in its units, admits an equivalent martingale measure that transforms all gai ns processes into martingales. We first supply a necessary and sufficient c ondition for the generic existence of numeraires in a finite dimensional se tting. We then characterize the arbitrage-free prices and dividends for whi ch the absence of numeraires survives any small perturbation preserving no arbitrage. Finally, we identify the cases when any small, but otherwise arb itrary, perturbation of prices and dividends preserves either the existence of numeraires, or their nonexistence under no arbitrage.