# Article

 Title: Notes on free lunch in the limit and pricing by conjugate duality theory (English) Author: Henclová, Alena Language: English Journal: Kybernetika ISSN: 0023-5954 Volume: 42 Issue: 1 Year: 2006 Pages: 57-76 Summary lang: English . Category: math . Summary: King and Korf [KingKorf01] introduced, in the framework of a discrete- time dynamic market model on a general probability space, a new concept of arbitrage called free lunch in the limit which is slightly weaker than the common free lunch. The definition was motivated by the attempt at proposing the pricing theory based on the theory of conjugate duality in optimization. We show that this concept of arbitrage fails to have a basic property of other common concepts used in pricing theory – it depends on the underlying probability measure more than through its null sets. However, we show that the interesting pricing results obtained by conjugate duality are still valid if it is only assumed that the market admits no free lunch rather than no free lunch in the limit. (English) Keyword: free lunch Keyword: free lunch in the limit Keyword: fundamental theorem of asset pricing Keyword: incomplete markets Keyword: arbitrage pricing Keyword: multistage stochastic programming Keyword: conjugate duality Keyword: finitely-additive measures MSC: 49N15 MSC: 90C15 MSC: 91B26 MSC: 91B28 idZBL: Zbl 1249.90180 idMR: MR2208520 . Date available: 2009-09-24T20:14:00Z Last updated: 2015-03-28 Stable URL: http://hdl.handle.net/10338.dmlcz/135699 . Reference: [1] Castaing C., Valadier M.: Convex Analysis and Measurable Multifunctions.(Lecture Notes in Mathematics 580.) Springer–Verlag, Berlin 1977 Zbl 0346.46038, MR 0467310, 10.1007/BFb0087688 Reference: [2] Delbaen F., Schachermayer W.: A general version of the fundamental theorem of asset pricing.Math. Ann. 300 (1994), 463–520 Zbl 0865.90014, MR 1304434, 10.1007/BF01450498 Reference: [3] Dybvig P. H., Ross S. A.: Arbitrage.In: A Dictionary of Economics, Vol. 1 (J. Eatwell, M. Milgate, and P. Newman, eds.), Macmillan Press, London 1987, pp. 100–106 Reference: [4] Eisner M. J., Olsen P.: Duality for stochastic programs interpreted as L.P. in $L_p$-space. SIAM J. Appl. Math. 28 (1975), 779–792 MR 0368742, 10.1137/0128064 Reference: [5] Föllmer H., Schied A.: Stochastic Finance.An Introduction in Discrete Time. Walter de Gruyter, Berlin 2002 Zbl 1126.91028, MR 1925197 Reference: [6] Henclová A.: Characterization of arbitrage-free market.Bulletin Czech Econom. Soc. 10 (2003), No. 19, 109–117 Reference: [7] Hewitt E., Stromberg K.: Real and Abstract Analysis.Third edition. Springer Verlag, Berlin 1975 Zbl 0307.28001, MR 0367121 Reference: [8] King A.: Duality and martingales: A stochastic programming perspective on contingent claims.Math. Programming, Ser. B 91 (2002), 543–562 Zbl 1074.91018, MR 1888991, 10.1007/s101070100257 Reference: [9] King A., Korf L.: Martingale pricing measures in incomplete markets via stochastic programming duality in the dual of $L^\infty$.SPEPS, 2001–13 (available at http://dochost.rz.hu-berlin.de/speps) Reference: [10] Naik V.: Finite state securities market models and arbitrage.In: Handbooks in Operations Research and Management Science, Vol. 9, Finance (R. Jarrow, V. Maksimovic, and W. Ziemba, eds.), Elsevier, Amsterdam 1995, pp. 31–64 Reference: [11] Pennanen T., King A.: Arbitrage pricing of American contingent claims in incomplete markets – a convex optimization approach.SPEPS, 2004–14 (available at http://dochost.rz.hu-berlin.de/speps) Reference: [12] Pliska S. R.: Introduction to Mathematical Finance, Discrete Time Models.Blackwell Publishers, Oxford 1999 Reference: [13] Rockafellar R. T.: Conjugate Duality and Optimization.SIAM/CBMS monograph series No. 16, SIAM Publications, Philadelphia 1974 Zbl 0296.90036, MR 0373611 Reference: [14] Rockafellar R. T., Wets R. J.-B.: Nonanticipativity and $Ł^1$-martingales in stochastic optimization problems.Math. Programming Stud. 6 (1976), 170–187 MR 0462590, 10.1007/BFb0120750 Reference: [15] Rockafellar R. T., Wets R. J.-B.: Stochastic convex programming: Basic duality.Pacific J. Math. 62 (1976), 173–195 Zbl 0339.90048, MR 0416582, 10.2140/pjm.1976.62.173 Reference: [16] Rockafellar R. T., Wets R. J.-B.: Stochastic convex programming: Singular multipliers and extended duality.Pacific J. Math. 62 (1976), 507–522 Zbl 0346.90057, MR 0489880, 10.2140/pjm.1976.62.173 Reference: [17] Rockafellar R. T., Wets R. J.-B.: Stochastic convex programming: Relatively complete recourse and induced feasibility.SIAM J. Control Optim. 14 (1976), 574–589 Zbl 0346.90058, MR 0408823, 10.1137/0314038 .

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