Finitely Additive Equivalent Martingale Measures

Patrizia Berti (), Luca Pratelli () and Pietro Rigo ()
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Patrizia Berti: Universita’ di Modena e Reggio-Emilia
Luca Pratelli: Accademia Navale
Pietro Rigo: Universita’ di Pavia

Journal of Theoretical Probability, 2013, vol. 26, issue 1, 46-57

Abstract: Abstract Let L be a linear space of real bounded random variables on the probability space $(\varOmega ,\mathcal{A},P_{0})$ . There is a finitely additive probability P on $\mathcal{A}$ such that P∼P 0 and E P (X)=0 for all X∈L if and only if cE Q (X)≤ess sup (−X), X∈L, for some constant c>0 and (countably additive) probability Q on $\mathcal{A}$ such that Q∼P 0. A necessary condition for such a P to exist is $\overline{L-L_{\infty}^{+}}\cap L_{\infty}^{+}=\{0\}$ , where the closure is in the norm-topology. If P 0 is atomic, the condition is sufficient as well. In addition, there is a finitely additive probability P on $\mathcal{A}$ such that P≪P 0 and E P (X)=0 for all X∈L if and only if ess sup (X)≥0 for all X∈L.

Keywords: Arbitrage; De Finetti’s coherence principle; Equivalent martingale measure; Finitely additive probability; Fundamental theorem of asset pricing; 60A05; 60A10; 28C05; 91B25; 91G10 (search for similar items in EconPapers)
Date: 2013
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DOI: 10.1007/s10959-010-0337-0

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