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Mathematics and Financial Economics

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09.06.2020

Arbitrage-free modeling under Knightian uncertainty

verfasst von: Matteo Burzoni, Marco Maggis

Erschienen in: Mathematics and Financial Economics | Ausgabe 4/2020

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Abstract

We study the Fundamental Theorem of Asset Pricing for a general financial market under Knightian Uncertainty. We adopt a functional analytic approach which requires neither specific assumptions on the class of priors \(\mathcal {P}\) nor on the structure of the state space. Several aspects of modeling under Knightian Uncertainty are considered and analyzed. We show the need for a suitable adaptation of the notion of No Free Lunch with Vanishing Risk and discuss its relation to the choice of an appropriate technical filtration. In an abstract setup, we show that absence of arbitrage is equivalent to the existence of approximate martingale measures sharing the same polar set of \(\mathcal {P}\). We then specialize our results to a discrete-time financial market in order to obtain martingale measures.

Vorheriger Artikel Asset pricing in a pure exchange economy with heterogeneous investors

Nächster Artikel Properly discounted asset prices are semimartingales

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We say that that \(\mathcal {A}\subset \mathbb {L}^0\) is monotone if \(Y\le X\) and \(X\in \mathcal {A}\) implies \(Y\in \mathcal {A}\).

If \(\mathcal {P}\ll P'\) for some \(P'\in \mathcal {M}_1\), the Halmos Savage Lemma (see [28]) implies that there exists a probability P which is equivalent to \(\mathcal {P}\).

This situation is reminiscent of the example of the two call options with different strikes but same price given in [24].

With a slight abuse of notation, \(\mathfrak {Q}_{app}\approx \mathcal {P}\) means that the whole collection of probabilities belonging to some approximate separating class is equivalent to \(\mathcal {P}\).

More precisely is \(\mathrm {q.s.}\) equal to a continuous function

\(\lambda \) is the Lebesgue measure on \(\mathbb {R}\).

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Titel: Arbitrage-free modeling under Knightian uncertainty
verfasst von: Matteo Burzoni
Marco Maggis
Publikationsdatum: 09.06.2020
Verlag: Springer Berlin Heidelberg
Erschienen in: Mathematics and Financial Economics / Ausgabe 4/2020
Print ISSN: 1862-9679
Elektronische ISSN: 1862-9660
DOI: https://doi.org/10.1007/s11579-020-00267-w

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