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

Belenky Alexander; Egorova Lyudmila Gennadyevna

Working paper

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

Математические методы анализа решений в экономике, бизнесе и политике. WP7. Издательский дом ВШЭ, 2016. No. 02.

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.

Priority areas: economics

Language: English

Full text (PDF, 641 Kb)

Text on another site

Keywords: linear programming stock exchange

Publication based on the results of: Development and research of new mathematical models in socio-economic and political spheres(2016)

An Approach to Forming and Managing a Portfolio of Financial Securities by Small and Medium Price-Taking Traders in a Stock Exchange

Belenky A., Egorova L. G. In bk.: Advances in Intelligent Systems and Computing. Iss. 359: Modelling, Computation and Optimization in Information Systems and Management Sciences. Switzerland: Springer, 2015. P. 257-268.

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.

Added: Jun 1, 2015

Effectiveness of Different Trading Strategies for Price-takers

Egorova L. G. In bk.: Procedia Computer Science. 2nd International Conference on Information Technology and Quantitative Management, ITQM 2014. National Research University Higher School of Economics (HSE) in Moscow (Russia) on June 3-5, 2014. Vol. 31. Amsterdam: Elsevier, 2014. P. 133-142.

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.

Added: Jun 12, 2014

Компьютерное моделирование установления равновесия на рынке ценных бумаг

Малашкин А. В. В кн.: Математические и инструментальные методы экономики: теория и практика Сборник материалов международного научного e-симпозиума Россия, г. Москва, 29-30 сентября 2014 г. Киров: МЦНИП, 2014. С. 162-169.

Computer simulation of equilibrium prices for the stock exchange

Added: Jan 29, 2015

Две модели принятия решений участником торгов на фондовой бирже по формированию и изменению своего инвестиционного портфеля

Belenky A., Egorova L. G. Математические методы анализа решений в экономике, бизнесе и политике. WP7. Издательский дом ВШЭ, 2015. No. WP7/2015/02 .

Two mathematical models formalizing the decision-making process by a trader on developing and changing her investment portfolio in a stock exchange are presented. According to the first model the trader can correctly predict future values of financial securities of her interest. In this case, the problem of finding optimal strategies of investing in these financial securities is reduced to solving a linear programming problem. Under the second model, by means of linear inequalities of a balance type, the trader can estimate the area in which the values of the whole spectrum of the above financial securities may change. In this case, the same problem is formulated as an antagonistic game, analogous to the game with nature, with a nonlinear payoff function. It is proven that saddle points in this game can be found by solving linear programming problems forming a dual pair.

Added: May 31, 2015

The Effectiveness Of Different Trading Strategies For Price-Takers

Egorova L. G. Financial Economics. FE. Высшая школа экономики, 2014. No. WP BRP 29/FE/2014 .

Simulation models of the stock exchange are developed to explore the dependence between a trader’s ability to predict future price movements and her wealth and probability of bankruptcy, to analyze the consequences of margin trading with different leverage rates and to compare different investment strategies for small traders. We show that in the absence of margin trading the rate of successful predictions should be slightly higher than 50% to guarantee with high probability that the final wealth is greater than the initial and to assure very little probability of bankruptcy, and such a small value explains why so many people try to trade on the stock exchange. However if trader uses margin trading, this rate should be much higher and high rate leads to the risk of excessive losses.

Added: Apr 21, 2014

Исследование операций. Линейное программирование и стохастические модели.

Каштанов В. А., Зайцева О. Б. М.: КУРС: ИНФРА-М, 2016.

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

Added: Nov 13, 2016

Математические и инструментальные методы экономики: теория и практика Сборник материалов международного научного e-симпозиума Россия, г. Москва, 29-30 сентября 2014 г

Edited by: Н. Н. Карабутов Киров: МЦНИП, 2014.

Mathematical and computer simulation of economic processes.

Added: Jan 29, 2015

Динамика инвестиционной политики в Китае

Никитина С. В. Сборники конференций НИЦ Социосфера. 2011. № 21. С. 6-15.

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.

Added: Nov 15, 2012

Is it so bad that we cannot recognize black swans?

Aleskerov F. T., Egorova L. G. Economics Letters. 2012. Vol. 117. No. 3. P. 563-565.

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.

Added: Oct 15, 2012

Методы оптимальных решений, издание 3-е, исправленное и дополненное

Соколов А. В., Токарев В. В. Т. 1: Общие положения. Математическое программирование.. М.: Физматлит, 2012.

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.

Added: Nov 25, 2013

Исследование возможностей диверсификации рисков портфельного инвестирования на российском фондовом рынке

Петров С. С., Ошарин А. М. Аудит и финансовый анализ. 2007. № 2. С. 183-188.

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.

Added: Dec 12, 2012

Распознавание биржевых процессов как пуассоновского потока событий двух типов: модели с поощрением и обучением

Egorova L. G. Математические методы анализа решений в экономике, бизнесе и политике. WP7. Высшая школа экономики, 2011. No. 02.

The behavior on a stock exchange can be represented as a reaction to the flow of events of two types entering the financial market – “regular” events and “crises”. Broker’s wealth depends on how successfully she identifies which event – crisis or not – occurs at the moment. It was shown in [1] that successful identification of the ‘regular’ events a little more than in half of the cases allows the player to have a nonnegative average gain. We expand the model in [1] by introducing the possibility of a player to learn on her behavior and to receive the award for the “correct” behavior. It turns out that the ability to analyze actions and learn from them sometimes allows the player successfully identify regular events even less than in half of the cases to obtain a nonnegative expected payoff.

Added: Dec 25, 2012

Экономика и управление: проблемы и перспективы развития. Сборник научных статей по итогам международной научно-практической конференции г.Волгоград 15-16 ноября 2010 г.

Ч. 1. Волгоград: Волгоградское научное издательство, 2010.

Added: Jan 18, 2013

Экономическая активность населения России в 1990-е гг.

Гимпельсон В. Е. Проблемы рынка труда. WP3. Высшая школа экономики, 2002. № 01.

Added: Mar 26, 2013

Обновляемый образовательный ресурс "Экономическая школа НИУ ВШЭ"

2012.

Added: Feb 22, 2013

Влияние выбора целей и инструментов политики денежных властей на уязвимость экономик

Трунин П. В., Дробышевский С. М., Евдокимова Т. В. М.: Издательский дом «Дело» РАНХиГС, 2012.

Added: Mar 26, 2013

Правовые институты в России 2000-2007 годах: взгляд со стороны бизнеса

Яковлев А. А. Общественные науки и современность. 2008. № 4. С. 21-37.

Added: Sep 22, 2012

На океанских волнах. Россия в мировом кризисе

Уринсон Я. М. Вестник Европы. 2014. № 38-39.

Added: Feb 5, 2018

Применение различных вариантов правила передачи голосов

Карпов А. В., Вольский В. И. Полития: Анализ. Хроника. Прогноз. 2011. № 2. С. 162-174.

Added: Jun 24, 2011

Институт минимальной заработной платы в странах ОЭСР

Вишневская Н. Т. Проблемы рынка труда. WP3. Высшая школа экономики, 2005. № 03.

The paper examines the institute of minimum wage in developed and transition economies and in a number of the developing countries. First of all the institutional mechanism of minimum wage fixing is considered. One of the sections explores the dynamics of absolute and relative levels of minimum wage. The special attention is paid to the impact of the institute of minimum wage on the labour market. The author considers the mechanism of transmission of the minimum wage increases on the employment and unemployment dynamics. The paper also contains the result of the empirical research. The experience of many countries witnesses that large increases in minimum wage levels lead to the stagnation of the employ-ment, especially of the disadvantaged groups. The negative effect is larger for the companies with higher share of labour costs and more active use of unqualified labour, that is small businesses and agricultural enterprises. One of the main conclusions is that the minimum wage is not an effective tool of the poverty reduction as the majority of the recipients live in households of average and upper average income.

Added: Oct 13, 2012

Численность и состав российской бюрократии: между советской номенклатурой и госслужбой гражданского общества

Гимпельсон В. Е. Проблемы рынка труда. WP3. Высшая школа экономики, 2002. № 05.

Added: Mar 26, 2013