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The Dalang Morton Willinger Theorem under cone constraints

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Date
2003
Link to item file
http://halshs.archives-ouvertes.fr/halshs-00151469/en/
Dewey
Economie Financière
Sujet
Cone constraints; Dalang–Morton–Willinger theorem; Arbitrage; Short sale constraints
Journal issue
Journal of Mathematical Economics
Volume
39
Number
1/2
Publication date
2003
Article pages
111-126
URI
https://basepub.dauphine.fr/handle/123456789/560
Collections
  • CEREMADE : Publications
Metadata
Show full item record
Author
Napp, Clotilde
Type
Article accepté pour publication ou publié
Abstract (EN)
The Dalang–Morton–Willinger theorem [Stochastics Stochastic Rep. 29 (1990) 185] asserts, for a discrete-time perfect market model, that there is no arbitrage if and only if the discounted price process is a martingale with respect to an equivalent probability measure. The financial market is supposed to be perfect in the sense that there is no transaction cost, no imperfection on the numéraire, no short sale constraint, no constraint on the amounts invested, etc.In this note, we explore the same issue in the presence of such imperfections, more precisely, in the presence of polyhedral convex cone constraints. We first obtain a generalization of the Dalang–Morton–Willinger theorem [Stochastics Stochastic Rep. 29 (1990) 185]: we prove that under polyhedral convex cone constraints, absence of arbitrage is equivalent to the existence of a discount process such that, taking this process as a deflator, the net present value of any available investment opportunity is nonpositive.We then apply this general result to specific market imperfections fitting in the convex cone framework, like short sale constraints, solvability constraints, constraints on the quantities, amounts or proportions invested. We improve a result of Pham–Touzi [J. Math. Econ. 31 (2) (1999) 265]. We show that our model enables to deal with financial markets with possible imperfections on the numéraire (like different borrowing and lending rates, or more general convex cone constraints involving the numéraire).

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