A bounded arbitrage strategy for a multiperiod model of a financial market in discrete time. (Ukrainian, English) Zbl 1199.91277

Mishura, Yu. S.; Shelyazhenko, P. S.; Shevchenko, G. M.

Teor. Jmovirn. Mat. Stat. 77, 122-131 (2007); translation in Theory Probab. Math. Stat. 77, 135-146 (2008).

A natural question that arises when modeling of financial markets is whether or not an arbitrage strategy exists. Conditions for the existence of arbitrage strategies are studied in many papers. It is known for models with discrete time that the existence of an arbitrage strategy is related to the existence of an equivalent martingale measure for the price process. This case is discussed in detail by J. M. Harrison and S. R. Pliska [Stochastic Processes Appl. 11, 215–260 (1981; Zbl 0482.60097)] and R. C. Dalang, A. Morton and W. Willinger [Stochastics Stochastics Rep. 29, No. 2, 185–201 (1990; Zbl 0694.90037)]. F. Delbaen and W. Schachermayer [Math. Ann. 300, No. 3, 463–520 (1994; Zbl 0865.90014)] consider arbitrary continuous semimartingales in continuous time. It is proved in the last paper that if there is no arbitrage strategy, then an absolutely continuous local martingale measure exists. Later Y. Kabanov and C. Stricker [“Remarks on the true no-arbitrage property”, Manuscript, Laboratoire de Mathématiques de Besançon, (2003)] generalized this result and dropped the assumption of the continuity of the semimartingales. The classical notion of arbitrage strategy is discussed in the books A. N. Shiryaev [Essentials of stochastic finance. Transl. from the Russian by N. Kruzhilin, Singapore: World Scientific (1999; Zbl 0926.62100)] and H. Föllmer and A. Schied [Stochastic finance. An introduction in discrete time. 2nd revised and extended ed., Berlin: de Gruyter (2004; Zbl 1126.91028)] The notion of arbitrage strategy is extended in the paper by Yu. S. Mishura [Prykl. Stat., Aktuarna Finans. Mat. 2003, No. 1-2, 49–54 (2003; Zbl 1150.91381)]. In contrast to the case of large markets, small financial markets may have an analogue of arbitrage strategy for which the profit is bounded in a certain sense since the market itself is bounded. Such an arbitrage strategy is called bounded or \(\varepsilon\)-arbitrage strategy. The mathematical tools used to study such strategies are different from the classical techniques, since the markets without \(\varepsilon\)-arbitrage strategies may have a classical arbitrage strategy. The basic notions of \(\varepsilon\)-arbitrage strategies are introduced in the last paper. An analogue of the first fundamental theorem of financial markets is also proved for the case of a bounded arbitrage strategy and for a single-period model with nonrandom initial data.
In the paper under review results similar to the classical arbitrage theory for multiperiod models of financial markets in discrete time are proved. An analogue of the classical Dalang-Morton-Willinger theorem (see, for example, R. C. Dalang et al. [loc. cit.]) is proved for the case of a bounded arbitrage strategy and for a multiperiod model. This result provides several conditions that are equivalent to the nonexistence of a bounded arbitrage strategy in the case of multiperiod models of financial markets. The difference between single-period and multiperiod models is discussed.

Reviewer: Mikhail P. Moklyachuk (Kyïv)

MSC:

91G80

Financial applications of other theories

Keywords:

financial market; multiperiod model; bounded arbitrage strategy

Citations:

Zbl 0482.60097; Zbl 0694.90037; Zbl 0865.90014; Zbl 0926.62100; Zbl 1126.91028; Zbl 1150.91381