Probabilistic Theory of Stock Exchanges

To my beloved parents Vasily Denisovich and Alexandra Vasilievna Kondratenko with appreciation and gratitude

PREFACE

This is the third monograph in a series devoted to the development of probabilistic economic theory [Kondratenko, 2005, 2015, 2021]. When the first book was being prepared for publication, the publisher asked the author how soon, in the author’s opinion, this new theory would gain widespread acceptance among economists. His answer was: at the very least 20 years, for the simple reason that it is a priori not enough to put forward another alternative economic theory – first, it is necessary to show its benefits as compared to the others, second – to prove that it is correct, and third – to showcase how it can be used in practice to describe experimental data and develop predictions of economic dynamics. This monograph addresses all three of these issues: it shows how this theory is better than others; it proves, by the example of important organized markets such as stock exchanges, that it is correct; and, based on the analytical and numerical results obtained, it offers a method for describing and forecasting economic dynamics.

It is explained below why stock markets were chosen as the main object of study. It is well known that specialized activities of goods exchange or trade between different producers and different countries plays an eminent role in the growth of the welfare of individuals and states as a whole in the early stages of its development, and then in the subsequent rapid formation and development of capitalism in the world. In this process, the most important place was first occupied by ordinary markets, which facilitated and thus accelerated the process of exchange of some goods or values for others, and then by organized markets, the highest form of which are, in particular, stock exchanges: commodity, financial, currency and others, which form and represent in their entirety the stock exchange economy. This exchange economy is now effectively using all the most advanced methods and tools of electronic trading, including artificial intelligence and algorithmic trading using computers; while the extensive use of the Internet makes the exchange economy very fast, virtual and truly global. It is the high speed of information exchange and transactions that distinguishes the new virtual exchange economy from the traditional real economy. But this specific feature of exchange trading can carry additional risks for both the exchange and the real economy, the tasks of which are still rather superficially understood in the economic academic community. In today's global economic world, the role of stock exchanges has become so significant that it would not be an exaggeration to say that the entire global economy is gradually becoming exchange-based. For sure, it would be more accurate to talk about the transformation of the global economy into a financial economy, but this monograph will be focused on studying only the role of exchanges in the economy, so the term “exchange economy” will be used. The main purpose of exchanges in today's economy is to determine the prices of all traded assets, including various money (currencies), to facilitate their trade and provide financing for the global economic activity. But it is also important that the situation in exchanges is the most universal indicator of the situation in the entire global economy. The paradox of this situation in this global financial world is that there is clearly no correlating situation in the world of theoretical finance. An adequate theory of organized financial markets still does not exist, which means that an adequate theory of the global market real economy is also out of question. This situation gives rise to certain risks of the emergence and uncontrolled development of negative trends in financial markets, which can lead to large-scale financial and then economic crises. And that’s what we regularly observe in real life: the generator of almost all economic crises in modern history are financial crises triggered by a stock market crash.

Currently, the situation is aggravating and the risks are increasing due to the fact that the bulk of transactions are now carried out by computers, working strictly according to algorithms, aimed mainly at achieving quick results without even minimal losses, and acting almost simultaneously, which can cause a chain collapse at stock exchanges uncoupled from the actual situation in the economy and the real value of assets. Meanwhile, the regulators do not have any meaningful and reliable tools for controlling and managing particularly explosive situations in financial markets, especially in organized markets or stock exchanges, where the prices of all global commodities and assets are largely determined. Such regulating agencies currently use accumulated historical experience and empirical parametric models to develop their management processes [Intriligator, 2002]. For these reasons, overcoming the apparent stagnation in the development of theoretical finance is a long overdue global task. The main challenge here and now is the almost complete absence of a mathematical apparatus with a potential to be used not only describe the exchange functioning as an asset pricing mechanism, which is done by financial econometrics at a qualitative level, but also accurately calculate the temporal fine price structure and the temporal fine structure of the trade volume during short time intervals, for example, during one trading session. Using the analogy with the scattering theory in physics, this can be formulated the other way around. Econometrics solves the so-called inverse problem, namely, extracting information about the studied system from the experimental data. The present study, however, aims at solving a direct problem – creating a fairly universal method for ab initio calculation of the exchange time microstructures with a characteristic size of a few seconds, which can be directly compared with the corresponding experimental fine structures of time trading dynamics, a method that could serve as a powerful tool for building a general theory of exchanges. We hope that the probabilistic theory of stock exchanges developed in this study can serve as a basis for building a general probabilistic financial theory and a deeper understanding of how the global world of finance works.

Obviously, organized markets are complex multiagent non-equilibrium probabilistic systems, and their description requires adequate mathematical methods. At the moment, the only suitable source of such methods is, of course, physics, which has long accumulated experience in theoretical work with similar multiparticle systems. Besides, quite a lot of experience has already been accumulated with regard to the physical methods application in economics, i.e., the use of formal methods and approaches of theoretical physics to solve economic problems. In particular, in the works of A.V. Kondratenko [2005, 2015] a new theory of market economy – Probabilistic Economic Theory – was developed. Initially, this theory was based on quantum mechanics with the derivation of economic equations of motion, which have not yet been solved for multiagent markets. Subsequently, a simpler version of the theory was developed using the probabilistic method, namely – probability economy or probabilistic economics, which was used as the basic theory to develop the probabilistic theory of the exchange in this paper, although it does not have motion equation, but has a mathematical framework, which proved very adequate and viable for the description of exchange processes and structures. To be on the safe side and avoid misunderstandings, let us clarify that there is literally no physics in this theory, no mechanics, let alone quantum mechanics. It is an economic theory to describe the economic processes that take place in the stock exchange. It simply adapts to economics mathematical tools that were created hundreds of years ago and were previously successfully used to solve similar problems in physics.

Probabilistic economics can be seen as a theory developed in the manner of general ideology as a physical method in economics, as well as in classical economic theory, understanding it as a version of the economic theory development, which can be figuratively described by an evolutionary trajectory: Adam Smith – Karl Menger – Ludwig von Mises. The works of these three great economists gave rise to the author's understanding of the essence and purpose of real economic science and the desire to develop their ideas and concepts by using the modern scientific probabilistic method of research. The primary task then was to create a mathematical apparatus or body equivalent to the physical method and use it to calculate specific real economic systems similar to how the same process worked in the history of the development and rapid rise of physical science, due to the creation of a powerful mathematical body, starting with the discovery of the equations of motion and differential calculus.

The probabilistic method has long been used in economic research at the empirical level using the basic formulas of probability theory. The use of this method in economics, by analogy with the quantum mechanics of physical systems [Kondratenko, 2005, 2015], has widely expanded the scope of our ideas about the modern economic world, generated a new probabilistic style of scientific economic thinking and created a new probabilistic dynamic picture of the modern economic world, alternative to traditional static ideas of mainstream economics, including neoclassical economics. This monograph essentially solves the problem of practical application of this approach to specific economic systems, namely stock exchanges, since we have enough input data for their quantitative study, i.e., supply and demand quotations, as well as relevant experimental data represented by market prices and trade volumes for theory verification.

Probabilistic economics, as well as many probabilistic theories in various fields of science, primarily in physics, for example, in statistical and quantum physics, is developed in terms of probability distributions. We emphasize that probability distributions are the very thing that forms the basis and language of a probabilistic, scientifically substantiated method for studying complex dynamic systems.

The possibilities of probabilistic economics methods to accurately describe real markets have been demonstrated earlier [Kondratenko, 2005, 2015] using examples of small model commodity economies. In this work the author, on the basis of probabilistic economics, aimed to create the foundations of the stock exchange theory, capable of filling the gaps described above in the modern theory of finance, the results of which will be in good agreement with the experimental stock exchange data. This goal was achieved. Let us specify that it is clear from general considerations that the microscopic theory developed in this study is devoted to the study of various stock exchange structures and processes at the level of exchange agents, and even more precisely – at the level of actions of individual exchange agents. The main purpose of microscopic theory is to describe the process of these structures’ formation based on the specific actions of the agents. First of all, this study will cover the mechanisms of exchange prices and trade volumes formation based on the quotations of market agents (in a particular period of time). It can be figuratively said that this theory gives a microscopic view of the stock exchange and stock exchange phenomena.

The book will show that probabilistic economics provides an adequate, fairly accurate micro- and macroscopic description of the stock exchange, namely, detailed structures and mechanisms of its operation, which were studied to derive certain patterns in the work of stock exchanges, in particular patterns in the prices and trade volumes formation.

We emphasize that for the purposes of this study we made no difference between stock, commodity, currency and other exchanges; the theory developed is equally suitable for describing any exchange where assets are traded, so for brevity in this book we will talk about stock exchanges, or simply exchanges.

As a conclusion, we provide a summary of the monograph with a subjective evaluation of the results obtained and the conclusions drawn in the study. The book outlines the basics of the probabilistic theory of stock exchanges, built on the basis of probabilistic economic theory using agent quotations provided by stock exchanges. By its nature, this theory of exchanges is microscopic, so its analytical and numerical methods make it possible to calculate and describe various exchange microstructures and microprocesses. The calculations of this type were performed for the first time in this study and are also published for the first time. The main attention was paid to the calculation of market prices and trade volumes of various assets (Sberbank shares, Brent oil futures, American dollars) on the Moscow Exchange and Intercontinental Exchange Futures Europe (Brent oil futures) during one trading session, as well as a detailed comparison of the calculation results with the experimental data. This comparison demonstrates a good agreement between theory and experiment, which allows us to state that the monograph has fulfilled the main scientific purpose of the proposed study, namely, to show that the probabilistic economic theory finds experimental confirmation and, thus, acquires solid experimental justification.

This sets it apart from other economic theories of a heuristic or empirical nature. Another important purpose has also been fulfilled, namely to describe in detail the economic mechanism underlying the formation of market prices and trade volumes and serving as a bridge connecting the microscopic and the macroscopic economic world, thus demonstrating the process of macrocosm formation from microcosm, namely how the actions of exchange agents form the action and temporal dynamics of the market as a whole.

A new, universal system of exchange indices of assets, stock exchanges and the global system of exchanges has been developed. By analogy, the strategy of digitalization, forecasting and management of economies on the basis of digital platforms was developed for accumulating plans of economic agents and processing them using the formulas of probabilistic economic theory, which, if implemented, will in turn improve the quality of public administration of the economy of individual countries and the world as a whole.

The monograph demonstrates the importance and significance of stock exchanges as experimental economic laboratories, aimed primarily at testing models, evaluating parameters of models and, ultimately, verifying the existing and new economic theories. While the construction of probabilistic economic theory as an «empirical» laboratory was to a certain extent backed up by the business, the practical 25-year experience of management of which was formalized by the mathematical body of theoretical physics, the development of probabilistic theory of stock exchanges in the study was based on MOEX and ICE as experimental economic laboratories. Step by step, the monograph reveals the enormous prospects for the further use of stock exchanges as powerful up-to-date experimental economic laboratories, which allow us to argue that just as theoretical physics emerged from the science of the solar system some 300–500 years ago, in the near future a modern economic science, consisting of closely interacting theoretical economics and experimental economics, will emerge from the development of stock exchange theory based on exchange experiments that will match all generally accepted standards for physical sciences, while remaining a human and social science.

Acknowledgments

The content of the book is a summary of the results of the project "Quantum Finance Investments of EXCELLENCE Investment Company" in Novosibirsk.

Project participants Vitaly Martynovich and Maria Makarkina contributed greatly to the project's success. Thus, the computer platform QUANTUM FINANCE for calculations of exchange structures using the probabilistic economic theory methods was developed by Vitaliy Martynovich and Maria Makarkina and implemented in C#. Maria Makarkina also provided substantial assistance in preparing this monograph for publication. The author sincerely thanks them for the fruitful cooperation for many years.

The author considers it his duty to thank Dmitry Sviridenko, who undertook the important work of the responsible editor of the monograph, and the reviewers of the monograph, Sergey Parinov and Yuri Perevyshin, for the challenging work of reviewing the manuscript with the new theory at a high professional level.

The author is grateful to the Alexander von Humboldt Foundation (Alexander von Humboldt – Stiftung), which provided a scholarship that enabled the author many years ago to witness for the first time how developed market economies work and how financial markets function in West Germany.

The author is grateful to Moscow Exchange and Intercontinental Exchange Futures Europe for providing access to historical data and online quotations.

The author is also grateful to the investment companies FINAM and Interactive Brokers for their excellent broker-intermediary functions with the IB and ICE exchanges, respectively.

The author sincerely thanks the first reader of the book manuscript, Konstantin Gluschenko, for critical comments, consideration of which made the material of the book more understandable for readers who hold fundamentally different orthodox economic views.

In conclusion, the author would like to take the opportunity, unfortunately, very late, to pay back some of his old debts.

First, the author would like to thank Vladimir Evstigneev for his informal but very informative and useful review of the author’s first paper on probabilistic economic theory in the collection of papers of the Russian Academy of Sciences and State Administration [Kondratenko, 2005] and Ksenia Kondratenko for her help in preparing the manuscript of this paper. Second, the author notes the important role that Professor George Judge of the University of California (Berkeley) played in this research by approving and enthusiastically supporting that very first paper 15 years ago, for which the author is immensely grateful.

Editor-in-chief

DSc in Physics and Mathematics Professor D.I. Sviridenko

Reviewers:

DSc S.I. Parinov

PhD Y.N. Perevyshin

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.

FREQUENTLY USED SYMBOLS

Г– agent width

MOEX – Moscow Exchange

BRENT – Brent oil futures

C – normalization factors

D – demand

D(t, p, q) – probabilistic market demand function

D⁰(t) – total market demand function

ICE – Intercontinental Exchange Futures Europe

F(t, p, q) – probabilistic market deal function

M(t)– number of supply quotations

N(t) – number of demand quotations

MTV(t) – probabilistic market trade volume

P – price

p – independent price variable

p^M – probabilistic market price

q – independent quantity variable

q^M– probabilistic market quantity

Q – quantity

PQ – price and quantity

S – supply

SBER – Sberbank shares

S D – supply and demand

S(t, p, q) – probabilistic market supply function

S⁰(t) – total market supply function

t – independent time variable

TV(t) – trade volume

T – time

USD/RUB – USD/RUB futures at MOEX: U.S. dollars are traded for Russian rubles (₽).

INTRODUCTION

"The impartial observer can have no doubt about the reason our generation pays general and enthusiastic tribute to progress in the field of the natural Sciences, while economic Science receives little attention and its value is seriously questioned by the very men in society to whom it should provide a guide for practical action. Never was there an age that placed economic interests higher than does our own. Never was the need of a scientific foundation for economic affairs felt more generally or more acutely. And never was the ability of practical men to utilize the achievements of Science, in all fields of human activity, greater than in our day. If practical men, therefore, rely wholly on their own experience, and disregard our Science in its present state of development, it cannot be due to a lack of serious interest or ability on their part. Nor can their disregard be the result of a haughty rejection of the deeper insight a true Science would give into the circumstances and relationships determining the outcome of their activity. The cause of such remarkable indifference must not be sought elsewhere than in the present state of our Science itself, in the sterility of all past endeavors to find its empirical foundations. The reason for this conspicuous indifference is none other than the present state of science itself, the fruitlessness of hitherto attempts to comprehend its empirical foundations".

Carl Menger [2007].

"Next, the empirical background of economic science is definitely inadequate. Our knowledge of the relevant facts of economics is incomparably smaller than that commanded in physics at the time when the mathematization of that subject was achieved. Indeed, the decisive break which came in physics in the seventeenth century, specifically in the field of mechanics, was possible only because of previous developments in astronomy. It was backed by several millennia of systematic, scientific, astronomical observation, culminating in an observer of unparalleled caliber, Tycho de Brahe. Nothing of this sort has occurred in economic science. It would have been absurd in physics to expect Kepler and Newton without Tycho, – and there is no reason to hope for an easier development in economics".

John Von Neumann and Oskar Morgenstern [1970]

The founding fathers of the Austrian school of economics established economic theory on a solid empirical footing in their day, which predetermined its successful development for many years to come. But the current levels of rigor of the underlying concepts and assumptions of these theories, as well as the quantitative description of real economic processes and phenomena, not to mention the quality of economic forecasts, are clearly insufficient for developing an evidence-based management of countries’ economies and achieving sustainable development of the global economy. There is a huge gap between the modern requirements that the society in its wide understanding presents to the economic science and the ability of this science to meet such requirements. This, as 150 years ago, generates a negative, at best ironic, public attitude to economic science, existing in the form of a number of often mutually exclusive theories, such as neoclassical economics and the Austrian school of economics (below often just Austrian economics), whose adherents give contradictory estimates, forecasts and recommendations. It has now come to the realization that empirical foundations alone are clearly insufficient for the verification of adequate models and approaches to economics. It is time to make a strict selection among all existing theories and currents of economic thought by means of their experimental verification in order to further develop economic theory, capable of providing a quantitative description of economic processes and phenomena at a high scientific level, comparable with the level of research in the natural sciences. As a result of this selection, economic theory will get a solid experimental foundation and become a unified economic science, like physics and other natural sciences, rather than a stream of ten parallel currents competing for financial resources, represented by neoclassical economics, Keynesianism, Marxism, etc.

To be clear, let us emphasize once again that all currently wide known economic theories, including neoclassical economics as the mainstream, are essentially heuristic or, at best, empirical theories built upon observation of the economic activities of market agents and the state, as well as on collecting various economic facts and their subsequent verbal generalization into a set of formulated principles for the economic activities of people and enterprises as well as the economic policies of the state. Not surprisingly, therefore, the theories of finance derived from them are extremely limited in their ability to quantitatively describe the temporal fine structure of the dynamics of ordinary and even more so of organized markets both because of our limited knowledge of the general economic laws governing the functioning of markets and because of an almost complete lack of a mathematical body which could be used to calculate at the microscopic level the temporal fine structure of markets in small time intervals, such as one trade session, and then to discover new patterns of how the markets work, using detailed comparison of the obtained results with the experimental data.

In this book we are committed to consistently overcome the above problems within the framework of probabilistic economics according to the following program of actions: developing a mathematical apparatus for calculating ab initio (from the first principles) the temporal fine structure of organized financial markets or exchanges, determining patterns in the functioning of financial markets obtained by comparing theoretical results with experimental data provided by exchanges, and eventually determining the patterns governing these markets, in the strict mathematical language. The patterns thus obtained can be used to derive motion equations that describe the temporal dynamics of market economic systems, in other words, equations that describe the evolution of economies. Thus, the purpose of this study is to create a theory of organized markets that has a sound experimental foundation. Meanwhile, if this venture proves successful, it could be argued that the previously constructed probabilistic economic theory also received a solid experimental foundation.

Using the analogy with the theory of scattering, we can say that probabilistic economics is aimed at solving the direct problem of economics, namely, based on some general principles, to calculate the results of economic activity or economic experiments and to compare them directly with the corresponding experimental data, which will allow to obtain a reliable interpretation of experimental data. At the same time, econometric studies solve the opposite problem – to extract information about the properties of the studied economic system from the experimental data using the help of mathematical methods. In order to avoid misunderstandings, we emphasize that everything that is stated in this monograph, and everything that is asserted in it, unless specifically stated, concerns only the direct problem of economics.

All of the above can be phrased somewhat differently. At the present time, there are two main problems of economic science in the theory of organized markets.

Problem 1 is the almost complete absence of a mathematical body that would allow us to conduct full-fledged theoretical calculations of the details of the exchange markets temporal dynamics at the microscopic level on a small-time scale, for example within 1 hour or one trading session.

Problem 2 is the lack of an experimental basis for economic science in the sense understood in natural sciences: data from systematic theoretical calculations should quantitatively coincide with corresponding experimental data with a satisfactory degree of accuracy.

When solving the above problems, we intend to rely on the physical method of economic research, the main feature of which is the aspiration to find and formulate economic laws in the form of equations, to use to the full extend the mathematical body to perform sufficiently accurate quantitative calculations and constantly rely on experiment in verification of hypotheses, theories and concrete numerical results. This method makes it possible to overcome the shortcomings of the simple empirical method, which currently prevails in economic research, based on a logical analysis of experimental data, as, for example, in the Austrian economic school or in econometric analysis of price dynamics, and to achieve the same level of scientific rigor as in natural sciences, above all physics. Without physical methods of research, i.e., without reliance on experiment, the further development of economic theory is impossible; otherwise, it will long remain in its infancy, in other words, it will remain a kind of protoscience in comparison, say, with natural sciences, above all physics.

First, we will very briefly express a subjective opinion about the state of modern economic science, so that the reader could understand the logic of the research undertaken in this work and its main objectives, and, ultimately, the value of the results obtained. We will formulate our opinion in the form of two statements.

Statement one. In our view, all old and new, widely known economic theories, including neoclassical economics, Marxist and Keynesian theories, the Austrian economic school and other currents of economic thought are, in fact, either heuristic or, at best, empirical theories with neither clear unambiguous experimental results, nor rigorous mathematical theories that allow ab initio calculations on the dynamics of specific real market systems whose results coincide with the corresponding experimental results of these markets work with a reliable level of accuracy. Moreover, proponents of even the most logically advanced empirical economic theory, namely the Austrian economic school, argue [Von Mises, 2005; De Soto, 2009] that neither experimentation nor even the use of the mathematical apparatus to describe economic phenomena and market processes is possible in principle. On this basis they categorically denounce all attempts to use the achievements of physics and mathematics for development of the quantitative economic theory. In our opinion, the current situation in economics is not absolute; it only repeats the similar situation that existed in physics 300–600 years ago before the works of Nicholas Copernicus, Isaac Newton, Galileo Galilei and other physicists and mathematicians of the new era in physics. What is the main reason for economics to lag behind physics for so long in this respect? John von Neumann and Oskar Morgenstern provide an excellent answer to this question in the quote from their book given in the epigraph. The reason was hidden in an objective factor, namely in the very absence of the possibility to rely on experiment in economics, at least in the form of systematic long-term observations of the cyclic motion of the planets of the solar system, as was done in physics. At present such an opportunity is provided to us by electronic exchanges with their digital platforms and big data that can be used, in general, for the verification and development of economic theories.