×

zbMATH — the first resource for mathematics

Numeraires, equivalent martingale measures and completeness in finite dimensional securities markets. (English) Zbl 0883.90014
Summary: A numéraire 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 numéraire is one-to-one with a subset of Arrow-Debreu state prices, which becomes the whole set if and only if the numéraire is self-financing. Hence our result extends those stated for specific self-financing numéraires. We also identify markets admitting self-financing numéraires, and characterize completeness, in terms of equivalent martingale measures, without requiring that specific securities be traded.

MSC:
91B28 Finance etc. (MSC2000)
PDF BibTeX XML Cite
Full Text: DOI
References:
[1] Back, K.; Pliska, S., On the fundamental theorems of asset pricing with an infinite state space, Journal of mathematical economics, 20, 1-18, (1991) · Zbl 0721.90016
[2] Dothan, M., Prices in financial markets, (1990), Oxford University Press New York · Zbl 0744.90010
[3] Duffie, D., Dynamic asset pricing theory, (1992), Princeton University Press Princeton
[4] Geman, H.; El Karoui, N.; Rochet, J.-C., Changes of numeaire, changes of probability measure and option pricing, Journal of applied probability, 32, 443-458, (1995) · Zbl 0829.90007
[5] Girotto, B.; Ortu, F., Existence of equivalent martingale measures in finite dimensional securities markets, Journal of economic theory, 69, 262-277, (1996) · Zbl 0852.90023
[6] Harrison, M.J.; Kreps, D., Martingales and arbitrage in multiperiod securities markets, Journal of economic theory, 20, 381-408, (1979) · Zbl 0431.90019
[7] Harrison, M.J.; Pliska, S., Martingales and stochastic integrals in the theory of continuous trading, Stochastic processes and their applications, 11, 215-260, (1981) · Zbl 0482.60097
[8] Harrison, M.J.; Pliska, S., A stochastic calculus model of continuous trading: complete markets, Stochastic processes and their applications, 15, 313-316, (1983) · Zbl 0511.60094
[9] Huang, C.F., Information structures and viable price systems, Journal of mathematical economics, 14, 215-240, (1985) · Zbl 0606.90012
[10] Long, J.B., The numeraire portfolio, Journal of financial economics, 26, 29-69, (1990)
This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.