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

Posted: 13 Feb 2001

See all articles by Bruno Girotto

Bruno Girotto

Università degli Studi di Trieste

Fulvio Ortu

Bocconi University - Department of Finance

Abstract

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.

Suggested Citation

Girotto, Bruno and Ortu, Fulvio, Generic Existence and Robust Nonexistence of Numeraires in Finite Dimensional Securities Markets. Mathematical Finance, Vol. 10, Issue 4, October 2000. Available at SSRN: https://ssrn.com/abstract=242877

Bruno Girotto (Contact Author)

Università degli Studi di Trieste ( email )

Piazzale Europa 1
Department of Mathematics
34127 Trieste
Italy

Fulvio Ortu

Bocconi University - Department of Finance ( email )

Via Roentgen 1
Milano, MI 20136
Italy

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