Generic Existence and Robust Nonexistence of Numéraires in Finite Dimensional Securities Markets

Bruno Girotto and Fulvio Ortu

Mathematical Finance, 2000, vol. 10, issue 4, 429-442

Abstract: A numéraire 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 numéraires in a finite dimensional setting. We then characterize the arbitrage‐free prices and dividends for which the absence of numéraires 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 numéraires, or their nonexistence under no arbitrage.

Date: 2000
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