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Finance and Stochastics

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01.07.2013

Dynamic no-good-deal pricing measures and extension theorems for linear operators on L ^∞

verfasst von: Jocelyne Bion-Nadal, Giulia Di Nunno

Erschienen in: Finance and Stochastics | Ausgabe 3/2013

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Abstract

In an L ^∞-framework, we present majorant-preserving and sandwich-preserving extension theorems for linear operators. These results are then applied to price systems derived by a reasonable restriction of the class of applicable equivalent martingale measures. Our results prove the existence of a no-good-deal pricing measure for price systems consistent with bounds on the Sharpe ratio. We treat both discrete- and continuous-time market models. Within this study we present definitions of no-good-deal pricing measures that are equivalent to the existing ones and extend them to discrete-time models. We introduce the corresponding version of dynamic no-good-deal pricing measures in the continuous-time setting.

Vorheriger Artikel Equilibrium model with default and dynamic insider information

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We recall that even though the σ-algebra \(\mathcal{F}_{t}\) is (up to P-null sets) countably generated, the space \(L^{\infty}(\mathcal{F}_{t})\) is not separable in general. For example, if Ω is the space of continuous functions C([0,t];ℝ^d) equipped with the sup-norm, then Ω is a separable topological space, and its Borel σ-algebra \(\mathcal{F}_{t}\) is countably generated, but the corresponding Banach space \(L^{\infty}(\mathcal{F}_{t})\) equipped with the ess-sup-norm is not a separable topological space. Indeed, if Ω is a Polish space, \(\mathcal{F}_{t}\) its Borel σ-algebra, and \(L^{\infty}(\mathcal{F}_{t})\) is separable, then Ω is at most countable.

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Titel: Dynamic no-good-deal pricing measures and extension theorems for linear operators on L ∞
verfasst von: Jocelyne Bion-Nadal
Giulia Di Nunno
Publikationsdatum: 01.07.2013
Verlag: Springer-Verlag
Erschienen in: Finance and Stochastics / Ausgabe 3/2013
Print ISSN: 0949-2984
Elektronische ISSN: 1432-1122
DOI: https://doi.org/10.1007/s00780-012-0195-y

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