Вероятностная теория фондовых бирж

The generator of almost all economic crises in modern history are financial crises, the trigger of which are exchange crashes. Currently, the situation is exacerbated and the risks are increasing. This is because the bulk of transactions are now made by computers. Working strictly according to algorithms aimed mainly at achieving quick results, they guarantee the absence of even minimal losses. They act almost synchronously, which can cause a chain reaction of collapse on the exchanges in isolation from the real state of affairs in the economy, and from the real value of assets. Meanwhile, regulators have no meaningful or reliable tools to monitor or manage any particularly volatile situations in the financial markets.

This is particularly valid in organized markets or exchanges where the prices of all global goods and assets are largely measured. All these management processes are currently reliant on tools using the analysis of accumulated historical experience and the use of empirical parametric models [Intriligator, 1971]. Therefore, overcoming the obvious stagnation in the development of theoretical finance is a long-overdue global task. The main challenge now is to overcome the near complete absence of a mathematical apparatus with which to describe the functioning of the exchange as an asset-pricing mechanism. Financial econometrics can do this qualitatively, but also required is the ability to calculate the temporal fine structure of the price and trade volume dynamics within short time intervals, such as during a single trading session.

Using a parallel with the physics theory of scattering, we can look at this differently. Econometrics focuses on solving the so-called «inverse problem», namely, the problem of extracting information from experimental data about the system under study. Conversely, we aim to solve a direct problem: the creation of a near-universal method of calculation from the first principles (ab initio) of the temporal exchange microstructures. These, with a characteristic time size of several temporal seconds, can be directly compared with the corresponding experimental fine structures of trading dynamics. This method could serve as a powerful tool for building a quantitative theory of exchanges.

We hope that in future, the probabilistic theory of exchanges developed in this study can serve as a basis for building a more general probabilistic financial theory. In doing so, a deeper understanding will be gained of how our global world of finance works.

It is obvious that organized markets are complex, multi-agent, non-equilibrium probabilistic systems, the description of which requires the application of adequate mathematical methods and apparatuses. The only suitable source of such methods and apparatuses is physics, where the experience of theoretical work with multiparticle systems with similar, formal structures has long been accumulated. In addition, quite a lot of experience has already been gathered in the application of the physical method in economics, namely, the use of formal methods and approaches of theoretical physics in solving economic problems.

In particular, probabilistic economic theory was developed [Kondratenko, 2005, 2015], a new theory of market economy. Initially, this theory was modeled on quantum mechanics with the derivation of economic equations of motion. Unfortunately, we are not yet able to accurately solve these equations for multi-agent markets. Because of this, a simpler version of the theory was later developed. It uses only the probabilistic method without solving equations of motion, namely, probability economics. It is used in this work as a basic theory for constructing probabilistic theory of exchanges. Although it contains no equations of motion, there is a mathematical apparatus that has proven very adequate and fruitful for describing exchange processes and structures.

To clarify, probability economics contains neither physics nor mechanics and, in particular, no quantum mechanics. This is an economic theory used to describe economic processes taking place on exchanges. This theory uses a mathematical apparatus which was created hundreds of years ago, and was previously used successfully to solve similar problems in physics. Probability economics has been developed in the spirit of both classical economic theory, and the physical method in economics. This variant has followed a figuratively similar evolutionary trajectory to the theories of Adam Smith to Karl Menger and onwards to Ludwig von Mises.

The works of these three authors have fostered my understanding of the essence and tasks of real economic science, as well as the desire to develop their ideas and concepts using the modern scientific probabilistic method of research. My primary task has been the creation of a mathematical apparatus adequate to the physical method and its use for the calculation of real economic systems. A similar process occurred during the creation and rapid rise of physical science, due to the creation of a powerful mathematical apparatus. It began with the discovery of the equations of motion and differential calculus.

The probabilistic method has long been applied at the empirical level in economic research by using the basic formulas of probability theory. The use of the probabilistic method in economics used an analogy with quantum mechanics of physical multiparticle systems [Kondratenko, 2005, 2015], and broadly pushed forward the framework of ideas and conceptions about the modern economic world. It gave rise to a new, probabilistic style of scientific economic thinking and created a new, dynamic probabilistic picture of the modern economic world. This veered away from the traditional static ideas of the economic mainstream, including neoclassical economic theory. This monograph solves the problem of this approach’s practical application to specific economic systems, or exchanges. There is enough input data in the form of supply and demand quotations for quantitative study, as well as enough relevant, experimental data in the form of market prices and trade volumes to verify the theory.

Probability economics is built in terms of probability distributions. These are usually accepted in various scientific fields; primarily in physics, in areas such as statistical and quantum physics. Emphasized here is that probability distributions create the basis and language of the probabilistic method used to study complex dynamical systems. The way in which real markets could be quantitatively described using methods of probability economics was demonstrated earlier [Kondratenko, 2005, 2015] using examples of small model commodity economies. Our work has succeeded in achieving the following goal: to create the foundation of the exchanges theory. Based on probability economics, it overcomes the disadvantages of the modern theory of finance described earlier, and results are in good agreement with the respective experimental exchange data.

The microscopic theory developed in this work is devoted to the study of various exchange structures and processes at the level of exchange agents, and more precisely, at the level of actions of individual exchange agents. By this we mean the mechanisms of formation of exchange microstructures, such as temporal price and trade volume fine structure which depending on the quotations of market agents at each particular time. This theory gives a microscopic view of exchanges and exchange phenomena.

Our book will show how probability economics gives a reasonably accurate, microscopic description of the exchange phenomena. Regularities and patterns are derived from the detailed structures and mechanisms of work, found in the formation of prices and trade volumes.

We emphasize that for the purposes of this study there is no difference between stock, commodity and other exchanges. The theory being developed is equally suitable for describing any exchange, so henceforth we will use the term “stock exchanges”, or simply “exchanges”. As the main example for the study, several days of intraday dynamics occurring between 2013 and 2020 were chosen of Sberbank shares, Brent crude oil futures («Brent futures») and US dollars on the Moscow Exchange (MOEX) in Moscow.

The quotations of exchange agents used in this case are available on the MOEX website for a small fee, so all the numerical results of this work and its conclusions can be easily verified by other researchers. In addition, the intraday dynamics of Brent futures for several days of 2020 on the Intercontinental Exchange Futures Europe (ICE) in London were similarly studied.

The book is written in the form of a research report, as it presents the results of practical application of the original economic theory. This was developed earlier by the author, specifically to quantitatively solve the direct problem for the exchange markets that were described earlier. As far as we know, there are no other documented studies of this type. For this reason, the book does not provide a detailed overview of the history of the issue, and references are made only to works whose results were used in the development of this theory. Moreover, hundreds of books and articles on econophysics and physical economics have been published relatively recently [Mantegna et al., 1999; Chernavsky D. et al., 2002; Farmer et al., 2005; Richmond et al., 2013; Ippoliti and Cheng (eds.), 2017], as well as articles by David Orrell, for example [(Orrell, 2020]. Together, the latter can be seen as an excellent modern overview of the application of theoretical physics methods used to describe economic phenomena.

Bear in mind that when basing a new economic model that resembles of one from physics, it is useful to employ the latter during the initial stage to help name and define new concepts. We have done so in this book by establishing parallels between the economic many-agent systems, and many-particle systems from physics. This applies, for example, to such terms as microscopic and macroscopic theories, direct and inverse problems, equations of motion, etc. Time will tell which of these new concepts and terms will take root in economics.

In conclusion, we summarize the monograph with a subjective assessment of the results obtained and the conclusions of the study. This monograph presents the basics of the probabilistic theory of exchanges, based on probabilistic economic theory using agent quotations provided by exchanges. By its nature, this exchange theory is microscopic, so its analytical and numerical methods make it possible to calculate and describe various exchange microstructures and microprocesses.

Calculations of this kind, first performed in this study, are also published for the first time in this monograph. Particular attention has been paid to the calculation of market prices and trade volumes of various assets (Sberbank shares, futures for Brent crude oil, US dollars) on the MOEX and ICE (Brent crude oil futures) during one trading session, along with a detailed comparison of the theoretical results with the corresponding experimental data. This comparison demonstrates a good agreement between the theory and experiment, which allows us to assert that the main scientific problem of this study is solved in the monograph. We demonstrate that probabilistic economic theory finds its experimental confirmation and thereby acquires a solid experimental justification. This radically distinguishes it from several other economic theories that have a heuristic or empirical character.

Another important task is also solved here, namely, the economic mechanism underlying the formation of market prices and trade volumes. This is described in detail and serves as a bridge connecting the microscopic economic world with the macroscopic economic world. The formation process of the macrocosm from the microcosm is hereby demonstrated. We show how the action and time dynamics of the exchange market as a whole are formed from the actions of exchange agents. A new, universal system of stock indices of assets, exchanges and the global system of exchanges has also been developed.

Similarly, a strategy has been developed for digitalization, forecasting and management of a country’s and the world’s economy. This is based on digital platforms used to accumulate the plans of economic agents, processing them using the formulas of probabilistic economic theory. If implemented, these will in turn improve the quality of public economic administration of the country, and the world.

This study demonstrates the importance and significance of stock exchanges as experimental economic laboratories, aimed primarily at testing models, evaluating model parameters and, ultimately, verifying existing and new economic theories. If we look at probabilistic economic theory, it was predominantly developed using the business world as an experimental laboratory, where the significant business experience of the author was formalized applying mathematical apparatus from theoretical physics. In the present study, using MOEX and ICE as experimental laboratories, probabilistic theory of stock exchanges has been developed by the author.

Our work consistently reveals great prospects in the further use of exchanges as powerful modern experimental economic laboratories. Some 300-500 years ago theoretical physics arose from the science of the solar system, and similarly, modern economic science will arise from the development of exchanges theory. It will consist of a closely interacting probabilistic economic theory and experimental stock exchange economics. Meeting all the generally accepted standards of the natural sciences, it will remain a humanitarian and social science.

Acknowledgments

Participants of the project «Quantum Finance Investments» of Investment Company EXCELLENCE Vitaly Martynovich and Maria Makarkina made a great contribution to the success of the project. The computer platform «QUANTUM FINANCE» was developed mainly by Vitaly Martynovich and Maria Makarkina, which they implemented in the C# language. This was used to perform calculations of exchange structures by applying the methods of probabilistic economic theory. Maria Makarkina also provided significant assistance in the preparation of this monograph for publication. I sincerely thank them both for their fruitful, long-term cooperation.

I respond with gratitude to Dmitry Sviridenko, who took on the important responsibility as executive editor of the monograph.

Thanks also go to the reviewers of the monograph Sergey Parinov and Yuri Perevyshin for the difficult work done at a highly professional level in reviewing the manuscript that presented the new theory.

I am grateful to the Alexander von Humboldt Foundation (Alexander von Humboldt Stiftung), which, many years ago, provided a scholarship that allowed him to see firsthand in West Germany how developed market economies work and how financial markets function within them.

I am grateful to the Moscow Exchange and the Intercontinental Exchange Futures Europe, which provided us with access to historical data and online quotations.

I also express gratitude to the Investment Companies FINAM and Interactive Brokers for the excellent implementation of the intermediary broker functions with the MOEX and ICE exchanges, respectively.

I sincerely thank the first reader of the manuscript of the book Konstantin Gluschenko for valuable comments and recommendations. Taking them into account made the material of the book more understandable for readers who adhere to orthodox economic views.

In conclusion, after a very long delay, I would like to pay off some my old debts. Firstly, to express gratitude to Vladimir Evstigneev for the informal, but very informative and useful review of my first book on the topic of probabilistic economic theory [Kondratenko, 2005]. Ksenia Kondratenko also deserves a big thank you for her help in preparing the manuscript of this book. Secondly, I note the important role played in this study by the professor of California University (Berkeley) George Judge, who applauded and supported enthusiastically that book many years ago, for which I am immensely grateful to him.

Anatoly V. Kondratenko Russia, Novosibirsk, September 2021

All rights reserved. No part of the book or whole book may be reproduced or transmitted in any form or by any means without the written permission of the author.

Часто используемые символы

(FREQUENTLY USED SYMBOLS)

Г – агентная ширина (agent width)

МБ – Московская Биржа (Moscow Exchange, MOEX)

BRENT – фьючерсы на нефть марки Brent (Brent oil futures)

C – нормировочные константы (normalization constants)

D – спрос (demand)

D (t, p, q) – вероятностная рыночная функция спроса (probabilistic market

demand function)

D0(t) – рыночная функция полного спроса (total market demand function)

ICE – Intercontinental Exchange Futures Europe

F(t, p, q) – вероятностная рыночная функция сделок (probabilistic market

deal function)

M (t)– количество котировок предложения (number of supply quotations)

MTV(t) – вероятностный объем торгов на рынке (probabilistic market trade

volume)

N(t) – количество котировок спроса (number of demand quotations)

P – цена (price)

p – независимая переменная цены (independent price variable)

pM – вероятностная рыночная цена (probabilistic market price)

q – независимая переменная количества (independent quantity

variable)

qM – вероятностное рыночное количество (probabilistic market quantity)

Q – количество (quantity)

PQ – цена и количество (price and quantity)

S – предложение (supply)

SBER – акции Сбербанка (Sberbank shares)

S&D – предложение и спрос (supply and demand)

S(t, p, q) – вероятностная рыночная функция предложения (probabilistic

market supply function)

S0(t) – рыночная функция полного предложения (total market supply

function)

t – независимая переменная времени (independent time variable)

TV(t) – объем торгов (trade volume)

T – время (time)

USD/RUB – фьючерсы на американские доллары, торгуемые на МБ за русские рубли

Введение

"Если в наше время успехи на поприще естественных наук встречают всеобщую и радостную признательность, тогда как наша наука обращает на себя так мало внимания и значение ее вызывает столь сильное сомнение именно в тех кругах общества, для которых ей следовало бы служить основой практической деятельности, то причина этого не должна вызывать недоумения со стороны людей беспристрастных. Еще ни одна эпоха не ставила хозяйственных интересов выше, еще никогда потребность в научном основании хозяйственной деятельности не была так развита и не чувствовалась так глубоко; еще никогда практические деятели не обладали таким умением пользоваться успехами науки на всех поприщах человеческой деятельности. Поэтому тот факт, что практические деятели, не заботясь о развитии, достигнутом нашей наукой, прибегают в своей хозяйственной деятельности к собственному опыту, объясняется не легкомыслием или неспособностью их точно так же, как и высокомерным отказом от более глубокого понимания тех фактов и отношений, которые определяют успех их деятельности, понимания, доставляемого им истинной наукой. Причина этого бросающегося в глаза равнодушия заключается не в чем ином, как в настоящем положении самой науки, в бесплодности делавшихся до сих пор попыток постичь ее эмпирические основания".

Менгер Карл [2005]

"Эмпирические основания экономической науки являются определенно неадекватными. Наши знания о существенных фактах в области экономики несравненно меньше, чем знания, которыми мы располагали в физике к тому моменту, когда была достигнута ее математизация. В самом деле, решающий перелом, который произошел в физике в XVII в. (особенно в механике), был возможен единственно благодаря предшествующему развитию астрономии. Он опирался на несколько тысячелетий систематических научных астрономических наблюдений, достигших апогея в таком несравненном наблюдателе, как Тихо Браге. Ничего подобного в экономической науке не происходило. В физике было бы абсурдным ожидать появления Кеплера и Ньютона без Тихо, и нет никаких оснований надеяться на более легкое развитие в экономике".

Джон Фон Нейман и Оскар Моргенштерн [1970]

Отцы-основатели австрийской экономической школы поставили в свое время экономическую теорию на прочную эмпирическую базу, что предопределило ее успешное развитие на многие годы вперед. Но существующих на данный момент уровней строгости основополагающих концепций и допущений этих теорий, а также количественного описания реальных экономических процессов и явлений, не говоря уже о качестве экономических прогнозов, явно недостаточно для построения научно обоснованного управления экономиками стран и достижения устойчивого развития глобальной экономики. Имеется огромная пропасть между современными требованиями, предъявляемыми обществом в широком смысле этого слова к экономической науке, и ее возможностями соответствовать этим требованиям. Это, как и 150 лет назад, порождает негативное, в лучшем случае ироничное, отношение в обществе к экономической науке, существующей в виде ряда часто взаимоисключающих теорий, например неоклассической экономики и австрийской экономической школы (ниже часто просто австрийской экономики), адепты которых дают противоречивые оценки, прогнозы и рекомендации. В настоящее время пришло осознание того факта, что одних эмпирических оснований явно недостаточно для верификации адекватных моделей и подходов к экономике. Пришло время провести строгий отбор среди всех существующих теорий и течений экономической мысли путем их экспериментальной верификации с целью дальнейшего развития экономической теории, способной давать количественное описание экономических процессов и явлений на высоком научном уровне, сравнимом с уровнем исследования в естественных науках. В результате этого отбора экономическая теория получит прочный экспериментальный фундамент, станет единой экономической наукой, как, например, физика и другие естественные науки, а не потоком десятка конкурирующих за финансовые ресурсы параллельных течений в форме неоклассики, кейнсианства, марксизма и т.д.

Для ясности еще раз подчеркнем, что все широко известные в настоящее время экономические теории, включая в качестве мейнстрима неоклассическую экономику, являются, по сути, эвристическими, а в лучшем случае, эмпирическими теориями, построенными на наблюдении за хозяйственной деятельностью агентов рынка и государства, а также на сборе разнообразных экономических фактов и последующем их словесном обобщении в виде набора сформулированных принципов экономической деятельности людей и предприятий, а также экономической политики государства. Поэтому неудивительно, что вытекающие из них теории финансов являются крайне ограниченными в своей способности количественно описывать временную тонкую структуру динамики обычных и тем более организованных рынков как вследствие нашего ограниченного знания общих экономических законов, управляющих функционированием рынков, так и вследствие практически полного отсутствия математического аппарата, с помощью которого можно было бы на микроскопическом уровне рассчитывать временную тонкую структуру рынков в небольших временных интервалах, например в течение одной торговой сессии, а потом путем детального сравнения полученных результатов с экспериментальными данными выявить новые закономерности функционирования рынков.

В данной книге мы нацелены на последовательное преодоление указанных проблем в рамках вероятностной экономики по следующей программе действий: разработка математического аппарата для расчетов ab initio (из первых принципов) временной тонкой структуры организованных финансовых рынков или бирж, формулировки закономерностей в функционировании финансовых рынков, полученных на основе сравнения теоретических результатов с экспериментальными данными, предоставляемыми биржами, и в конечном итоге формулировки закономерностей, управляющих этими рынками, на строгом математическом языке. Полученные таким образом закономерности могут быть использованы для получения уравнений движения, описывающих временную динамику рыночных экономических систем, другими словами, уравнений, описывающих эволюцию экономик. Таким образом, целью настоящего исследования является создание такой теории организованных рынков, которая имела бы под собой прочное экспериментальное обоснование. В то время, как в случае успеха данного предприятия можно было бы утверждать, что построенная ранее вероятностная экономическая теория также получила надежный экспериментальный фундамент.