The numéraire portfolio in semimartingale financial models

By Ioannis Karatzas, Constantinos Kardaras. Finance and Stochastics, 2007

381 citations recorded by the source corpus.

Abstract We study the existence of the numéraire portfolio under predictable convex constraints in a general semimartingale model of a financial market. The numéraire portfolio generates a wealth process, with respect to which the relative wealth processes of all other portfolios are supermartingales. Necessary and sufficient conditions for the existence of the numéraire portfolio are obtained in terms of the triplet of predictable characteristics of the asset price process. This characterization is then used to obtain further necessary and sufficient conditions, in terms of a no-free-lunch-type notion. In particular, the full strength of the “No Free Lunch with Vanishing Risk” (NFLVR) condition is not needed, only the weaker “No Unbounded Profit with Bounded Risk” (NUPBR) condition that involves the boundedness in probability of the terminal values of wealth processes. We show that this notion is the minimal a-priori assumption required in order to proceed with utility optimization. The fact that it is expressed entirely in terms of predictable characteristics makes it easy to check, something that the stronger NFLVR condition lacks.

Subfields: Finance, Management Science and Operations Research

View Paper: https://www.semanticscholar.org/paper/3b6b3cfc4615c59a32a52d25343cdacc9303fb1e

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Citations in the corpus

Approximating the numéraire portfolio by naive diversification (2010, background): “…concerned with the existence of the NP, which has been studied in many papers including Long (1990), BajeuxBesnainou and Portait (1997), Becherer (2001), Platen (2002), Bühlmann and Platen (2003), Platen (2006), Platen and Heath (2006), Karatzas and Kardaras (2007) and Kardaras and Platen (2008).”
Duality theory for portfolio optimisation under transaction costs (2014, background): “19 in [18] the existence of an optimal strategy to the frictionless utility maximisation problem (2.”
A new perspective on the fundamental theorem of asset pricing for large financial markets (2014, background): “Karatzas [11] in the context of classical small financial markets.”
Analysis of continuous strict local martingales via h-transforms (2007, background): “This is the spirit taken in Platen [31], Platen and Heath [32], Fernholz and Karatzas [12], Fernholz, Karatzas, and Kardaras [13], and Karatzas and Kardaras [22].”
Pathwise stochastic integrals for model free finance (2013, background): “Indeed, (NA1) is equivalent to the existence of an unbounded utility function such that the maximum expected utility is finite [23,25].”
Optimal arbitrage under model uncertainty. (2011, neutral): “We refer the reader to Delbaen and Schachermayer (1995b) for the origin of the resulting NUPBR concept, and to Karatzas and Kardaras (2007) for an elaboration of this point in a different context, namely, the existence and properties of the numéraire portfolio.”
M-6-On Minimal Market Models and Minimal Martingale Measures (2010, neutral): “I n the above most general form, they are due to [18], but variants and prec ursors can already be found in [21], [3] and [5].”
On the martingale property of certain local martingales (2009, background): “6 in Karatzas and Kardaras [35], Example 1 in Section 4.”
Finitely Additive Probabilities and the Fundamental Theorem of Asset Pricing (2009, positive): “For checking the existence of the latter, there exists a necessary and sufficient criterion in terms of the predictable characteristics of the discounted asset-price process, as was shown in [16].”