A numeraire is a portfolio that, when securities, prices and dividends are expressed in its units, admits an equivalent martingale measure transforming any gain process into a martingale. We show that the set of equivalent martingale measures of a numeraire is one-to-one with a subset of Arrow-Debreu state prices, which becomes the whole set if and only if the numeraire is self-financing. Hence our result extends those (e.g. Harrison and Kreps (Journal of Economic Theory, 1979, 20, 381-408) Dothan (Prices in Financial Markets, Oxford Univ. Press, New York, 1990)) stated for specific self-financing numeraires. We also identify markets admitting self-financing numeraires, and characterize completeness, in terms of equivalent martingale measures, without requiring that specific securities be traded.
Numeraires, equivalent martingale measures and completeness in finite dimensional securities markets
Ortu, Fulvio
1997-01-01
Abstract
A numeraire is a portfolio that, when securities, prices and dividends are expressed in its units, admits an equivalent martingale measure transforming any gain process into a martingale. We show that the set of equivalent martingale measures of a numeraire is one-to-one with a subset of Arrow-Debreu state prices, which becomes the whole set if and only if the numeraire is self-financing. Hence our result extends those (e.g. Harrison and Kreps (Journal of Economic Theory, 1979, 20, 381-408) Dothan (Prices in Financial Markets, Oxford Univ. Press, New York, 1990)) stated for specific self-financing numeraires. We also identify markets admitting self-financing numeraires, and characterize completeness, in terms of equivalent martingale measures, without requiring that specific securities be traded.I documenti in IRIS sono protetti da copyright e tutti i diritti sono riservati, salvo diversa indicazione.
