# Article

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Keywords:
free lunch; free lunch in the limit; fundamental theorem of asset pricing; incomplete markets; arbitrage pricing; multistage stochastic programming; conjugate duality; finitely-additive measures
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.
References:
[1] Castaing C., Valadier M.: Convex Analysis and Measurable Multifunctions. (Lecture Notes in Mathematics 580.) Springer–Verlag, Berlin 1977 DOI 10.1007/BFb0087688 | MR 0467310 | Zbl 0346.46038
[2] Delbaen F., Schachermayer W.: A general version of the fundamental theorem of asset pricing. Math. Ann. 300 (1994), 463–520 DOI 10.1007/BF01450498 | MR 1304434 | Zbl 0865.90014
[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
[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 DOI 10.1137/0128064 | MR 0368742
[5] Föllmer H., Schied A.: Stochastic Finance. An Introduction in Discrete Time. Walter de Gruyter, Berlin 2002 MR 1925197 | Zbl 1126.91028
[6] Henclová A.: Characterization of arbitrage-free market. Bulletin Czech Econom. Soc. 10 (2003), No. 19, 109–117
[7] Hewitt E., Stromberg K.: Real and Abstract Analysis. Third edition. Springer Verlag, Berlin 1975 MR 0367121 | Zbl 0307.28001
[8] King A.: Duality and martingales: A stochastic programming perspective on contingent claims. Math. Programming, Ser. B 91 (2002), 543–562 DOI 10.1007/s101070100257 | MR 1888991 | Zbl 1074.91018
[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)
[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
[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)
[12] Pliska S. R.: Introduction to Mathematical Finance, Discrete Time Models. Blackwell Publishers, Oxford 1999
[13] Rockafellar R. T.: Conjugate Duality and Optimization. SIAM/CBMS monograph series No. 16, SIAM Publications, Philadelphia 1974 MR 0373611 | Zbl 0296.90036
[14] Rockafellar R. T., Wets R. J.-B.: Nonanticipativity and $Ł^1$-martingales in stochastic optimization problems. Math. Programming Stud. 6 (1976), 170–187 DOI 10.1007/BFb0120750 | MR 0462590
[15] Rockafellar R. T., Wets R. J.-B.: Stochastic convex programming: Basic duality. Pacific J. Math. 62 (1976), 173–195 DOI 10.2140/pjm.1976.62.173 | MR 0416582 | Zbl 0339.90048
[16] Rockafellar R. T., Wets R. J.-B.: Stochastic convex programming: Singular multipliers and extended duality. Pacific J. Math. 62 (1976), 507–522 DOI 10.2140/pjm.1976.62.173 | MR 0489880 | Zbl 0346.90057
[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 DOI 10.1137/0314038 | MR 0408823 | Zbl 0346.90058

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