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Working paper

Two approaches to modeling the interaction of small and medium price-taking traders with a stock exchange by mathematical programming techniques

The paper presents two new approaches to modeling the interaction of small and medium price-taking traders with a stock exchange. In the framework of these approaches, the traders can form and manage their portfolios of financial instruments traded on a stock exchange with the use of linear, integer, and mixed programming techniques. Unlike previous authors’ publications on the subject, besides standard securities, the present publication considers derivative financial instruments such as futures and options contracts. When a trader can treat price changes for each financial instrument of her interest as those of a random variable with a known (for instance, a uniform) probability distribution, finding an optimal composition of her portfolio turns out to be reducible to solving an integer programming problem. When the trader possesses no particular information on the probability distribution of the above-mentioned random variable for financial instruments of her interest but can estimate the areas to which the prices of groups of financial instruments are likely to belong, a game-theoretic approach to modeling her interaction with the stock exchange is proposed. In antagonistic games modeling the interaction in this case, finding the exact value of the global maximin describing the trader’s guaranteed financial result in playing against the stock exchange, along with the vectors at which this value is attained, is reducible to solving a mixed programming problem. Finding the value of an upper bound for this maximin (and the vectors at which this upper bound is attained) is reducible to finding a saddle point in an auxiliary antagonistic game on disjoint polyhedra, which can be done by solving linear programming problems forming a dual pair.