### 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 discusses a new approach to developing tools for quantitatively analyzing the financial behavior of small and medium price-taking traders each possessing abilities to predict share price values for a set of financial securities traded in a stock exchange. Tools for forming and managing a trader’s portfolio of securities from this set are proposed. Particularly, it is shown that when the trader can treat share price values from the portfolio as random variables with known (to her) distributions, an optimal portfolio composition is found by solving a linear programming problem. Otherwise, this optimal composition is found as the trader’s equilibrium strategy in an antagonistic two-person game with the stock exchange being the other player. In this game on polyhedra of disjoint player strategies, described by systems of linear equations and inequalities of a balance kind, calculating saddle points is reduced to solving linear programming problems forming a dual pair.

We introduce simulation models of stock exchange to explore which traders are successful and how their strategies influence to their wealth and probability of bankruptcy. The results of our experiments show that there is a critical level of agent’s experience (or luck) such that agents with this or higher level almost sure will survive on the market on the long run. This critical level is just slightly higher 1/2 and such small value explains why so many people try to trade on the stock exchange. But if trader uses margin trading, the critical level is much higher and shows the risk of excessive losses.

Computer simulation of equilibrium prices for the stock exchange

Contents of the book is divided into 2 parts of deterministic and stochastic models of Operations Research.

The first part of "Deterministic models of Operations Research" - is the base section, in which the emphasis is on linear programming.

The second part - "Stochastic models of Operations Research" includes a model of reliability and queuing models. This is original material.

The textbook can be useful to students of undergraduate and graduate programs in areas of training in "Applied Mathematics", "Applied Mathematics and Computer Science", "Information systems and technologies", as well as graduate students and science teachers who are interested in the problems of optimization in stochastic models

Mathematical and computer simulation of economic processes.

The article examines the experience of China's investment policy aimed at creating favorable conditions to attract investment, particularly foreign direct investments, to the most important country's industries. In recent years, this policy (the establishment of free economic zones, trade liberalization, the establishment of an appropriate legislative framework, state support for investors) has brought noticeable positive results, but with the beginning of the global financial crisis allowed to avoid the most painful consequences. This experience taking into account all its particularities can be useful for our economy.

We present processes on stock exchange as two random processes one of which reflects the regular regime of economy and the other one–crises. If regular processes are correctly recognized with the probability slightly higher than 1/2, this gives positive average gain to the player. We believe that this very phenomenon lies on the basis of unwillingness of people to expect crises permanently and to try recognizing them.

The manual is devoted to the mathematical theory and methods of optimization applied to administrative decisions in economy. Volume 1 described approaches to mathematical modeling of management problems in economy and methods of mathematical programming tasks solution. Besides strict mathematical proofs, there are directing reasons, which is sometimes enough for understanding. There are many economic examples and exercises with detailed solutions. Readers are supposed to know the bases of the mathematical analysis and linear algebra, though necessary data from these courses in a concise form are provided in appendices.

The paper investigates the optimization methods used by an investor working on the Russian stork market. The efficient sets, corresponding for the two different states of the market (with «moderate» and «rapid» growth rates), are build. The paper denies the necessity of the «deep» diversification of the portfolio on the Russian stork market. Some recommendations concerning the investment portfolio management are formulated.