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Consistent Price Systems and Arbitrage Opportunities of the Second Kind in Models with Transaction Costs

Lépinette, Emmanuel; Kabanov, Yuri (2012), Consistent Price Systems and Arbitrage Opportunities of the Second Kind in Models with Transaction Costs, Finance and Stochastics, 16, 1, p. 135-154. http://dx.doi.org/10.1007/s00780-010-0144-6

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Type
Article accepté pour publication ou publié
Date
2012
Journal name
Finance and Stochastics
Volume
16
Number
1
Publisher
Springer
Pages
135-154
Publication identifier
http://dx.doi.org/10.1007/s00780-010-0144-6
Metadata
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Author(s)
Lépinette, Emmanuel

Kabanov, Yuri
Abstract (EN)
In contrast with the classical models of frictionless financial markets, market models with proportional transaction costs, even satisfying usual no-arbitrage properties, may admit arbitrage opportunities of the second kind. This means that there are self-financing portfolios with initial endowments laying outside the solvency region but ending inside. Such a phenomenon was discovered by M. R´asonyi in the discrete-time framework. In this note we consider a rather abstract continuous-time setting and prove necessary and sufficient conditions for the property which we call No Free Lunch of the 2nd Kind, NFL2. We provide a number of equivalent conditions elucidating, in particular, the financial meaning of the property B which appeared as an indispensable “technical” hypothesis in previous papers on hedging (super-replication) of contingent claims under transaction costs. We show that it is equivalent to another condition on the “richness” of the set of consistent price systems, close to the condition PCE introduced by R´asonyi. In the last section we deduce the R´asonyi theorem from our general result using specific features of discrete-time models.
Subjects / Keywords
Consistent price systems; No Free Lunch; Arbitrage; Transaction costs; Martingales; Set-valued processes
JEL
G13 - Contingent Pricing; Futures Pricing
G11 - Portfolio Choice; Investment Decisions

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