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Large Financial Markets and Asymptotic Arbitrage with Small Transaction Costs

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  • Irene Klein
  • Emmanuel Lepinette
  • Lavinia Ostafe

Abstract

We give characterizations of asymptotic arbitrage of the first and second kind and of strong asymptotic arbitrage for large financial markets with small proportional transaction costs $\la_n$ on market $n$ in terms of contiguity properties of sequences of equivalent probability measures induced by $\la_n$--consistent price systems. These results are analogous to the frictionless case. Our setting is simple, each market $n$ contains two assets with continuous price processes. The proofs use quantitative versions of the Halmos--Savage Theorem and a monotone convergence result of nonnegative local martingales. Moreover, we present an example admitting a strong asymptotic arbitrage without transaction costs; but with transaction costs $\la_n>0$ on market $n$ ($\la_n\to0$ not too fast) there does not exist any form of asymptotic arbitrage.

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Paper provided by arXiv.org in its series Papers with number 1211.0443.

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Date of creation: Nov 2012
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Handle: RePEc:arx:papers:1211.0443

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  1. Black, Fischer & Scholes, Myron S, 1973. "The Pricing of Options and Corporate Liabilities," Journal of Political Economy, University of Chicago Press, University of Chicago Press, vol. 81(3), pages 637-54, May-June.
  2. Ross, Stephen A., 1976. "The arbitrage theory of capital asset pricing," Journal of Economic Theory, Elsevier, Elsevier, vol. 13(3), pages 341-360, December.
  3. Paolo Guasoni & Miklós Rásonyi & Walter Schachermayer, 2010. "The fundamental theorem of asset pricing for continuous processes under small transaction costs," Annals of Finance, Springer, Springer, vol. 6(2), pages 157-191, March.
  4. Y.M. Kabanov & D.O. Kramkov, 1998. "Asymptotic arbitrage in large financial markets," Finance and Stochastics, Springer, Springer, vol. 2(2), pages 143-172.
  5. Yuri M. Kabanov & Günter Last, 2002. "Hedging under Transaction Costs in Currency Markets: a Continuous-Time Model," Mathematical Finance, Wiley Blackwell, Wiley Blackwell, vol. 12(1), pages 63-70.
  6. J. Kallsen & J. Muhle-Karbe, 2010. "On using shadow prices in portfolio optimization with transaction costs," Papers 1010.4989, arXiv.org.
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