Generic Existence and Robust Nonexistence of Numeraires in Finite Dimensional Securities Markets
Posted: 13 Feb 2001
A numeraire is a portfolio that, if prices and dividends are denominated in its units, admits an equivalent martingale measure that transforms all gains processes into martingales. We first supply a necessary and sufficient condition for the generic existence of numeraires in a finite dimensional setting. We then characterize the arbitrage-free prices and dividends for which the absence of numeraires survives any small perturbation preserving no arbitrage. Finally, we identify the cases when any small, but otherwise arbitrary, perturbation of prices and dividends preserves either the existence of numeraires, or their nonexistence under no arbitrage.
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