Abstract
In the modern version of arbitrage pricing theory suggested by Kabanov and Kramkov the fundamental financially meaningful concept is an asymptotic arbitrage. The “real world” large market is represented by a sequence of “models” and, though each of them is arbitrage free, investors may obtain non-risky profits in the limit. Mathematically, absence of the asymptotic arbitrage is expressed as contiguity of envelopes of the sets of equivalent martingale measures and objective probabilities. The classical theory deals with frictionless markets. In the present paper we extend it to markets with transaction costs. Assuming that each model admits consistent price systems, we relate them with families of probability measures and consider their upper and lower envelopes. The main result concerns the necessary and sufficient conditions for absence of asymptotic arbitrage opportunities of the first and second kinds expressed in terms of contiguity. We provide also more specific conditions involving Hellinger processes and give applications to particular models of large financial markets.
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References
Bouchard B., Chassagneux J.-F.: Representation of continuous linear forms on the set of làdlàg processes and the pricing of American claims under transaction costs. Electron. J. Probab. 14, 612–632 (2009)
Campi L., Schachermayer W.: A super-replication theorem in Kabanov’s model of transaction costs. Financ. Stoch. 10(4), 579–596 (2006)
Denis E., Kabanov Y.: Consistent price systems and arbitrage opportunities of the second kind in models with transaction costs. Financ. Stoch. 14, 625–667 (2010)
De Vallière D., Denis E., Kabanov Yu.: Hedging of American options under transaction costs. Financ. Stoch. 13(1), 105–119 (2009)
Föllmer H., Schachermayer W.: Asymptotic arbitrage and large deviations. Math. Financ. Econ. 1(3–4), 213–249 (2008)
Guasoni, P., Lépinette, E., Rásonyi, M.: The fundamental theorem of asset pricing under transaction costs. Financ. Stoch. (forthcoming). http://papers.ssrn.com/s013/papers.cfm?abstract_id=1784440
Huberman G.: A simple approach to arbitrage pricing theory. J. Econ. Theory 28(1), 183–191 (1982)
Klein I., Schachermayer W.: Asymptotic arbitrage in non-complete large financial markets. Theory Probab. Appl. 41(4), 927–934 (1996)
Klein I.: A fundamental theorem of asset pricing for large financial markets. Math. Financ. 10, 443–458 (2000)
Klein I.: Free lunch for large financial markets with continuous price processes. Ann. Appl. Probab. 13/4, 1494–1503 (2003)
Kabanov Y., Kramkov D.: Large financial markets: asymptotic arbitrage and contiguity. Theory Probab. Appl. 39(1), 222–229 (1994)
Kabanov Y., Kramkov D.: Asymptotic arbitrage in large financial markets. Financ. Stoch. 2(2), 143–172 (1998)
Kabanov Y., Safarian M.: Markets with transaction costs. Mathematical theory. Springer, Berlin (2009)
Kabanov Y., Stricker C.: Hedging of contingent claims under transaction costs. In: Sandmann, K., Schönbucher, Ph. (eds) Advances in finance and stochastics, pp. 125–136. Springer, New York (2002)
Mbele Bidima Martin, L. D., Rásonyi, M.: On long-term arbitrage opportunities in Markovian models of financial markets. Ann. Oper. Res. 1–16 (2011). doi:10.1007/s10479-011-0892-5
Ross S.A.: The arbitrage theory of asset pricing. J. Econ. Theory 13(1), 341–360 (1976)
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Lepinette, E., Ostafe, L. Asymptotic arbitrage in large financial markets with friction. Math Finan Econ 6, 313–335 (2012). https://doi.org/10.1007/s11579-012-0077-2
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DOI: https://doi.org/10.1007/s11579-012-0077-2