Analytical methods of mathematical modeling of the exchange rate
Mathematical methods of forecasting can be developed on the basis of: exponential functions, power functions, dynamic series and analytical dependencies [1]. Let us consider the peculiarities of the model of exchange rate forecasting on the basis of analytical dependencies. This model is built on the basis of the analysis of the mechanism of exchange rate formation. The type of formula in this case will depend on the nature and type of interacting factors affecting the formation of the exchange rate. The model is based on the hypothesis of purchasing power parity. Ideally, the model defines the "exchange rate" function by means of domestic and foreign money supply and real national products in constant prices. Further, in the process of considering real economic systems, new factors will be added and those given above will be adjusted. And the generalized model will select the main factors affecting the formation of the exchange rate.
Influence Factors
In order to obtain an effective forecasting model it is necessary to carefully analyze the nature of the impact of the factors affecting the formation of the exchange rate. An important role here is played by the availability of statistical data related to the factors in question and the dynamics of the exchange rate. In this case the analysis of the influence of some factors on the exchange rate is rather complicated. Theoretically, forecasting on the basis of this model may be the most accurate. But for THIS It takes a lot of analytical work to find the factorsThe main objective of the report is to identify the regularities in their dynamics, the nature of the effect on the exchange rate in various possible situations. Accounting for time lags between the beginning of changes in the factors determining the dynamics of the exchange rate, to account for the simultaneous impact of possible different changes in several factors on the change of the exchange rate. Such work can be done only by large state structures (organizations) or financial concerns. The analysis of the known developments in the field of forecasting of economic processes on a macro-level has shown that in most cases the only possible solutions are construction of linear econometric macroeconomic models which are based on the hypotheses of "efficient market" and "rational expectations". However, the existing contradictions and imperfections in these hypotheses are often the reason for erroneous results. Besides, these models do not provide the account of the changes in the structure of the influence of the fundamental indicators and the account of the attitude of the market participants to the uncertainties and contradictions in the development of the market situations. This determines the expediency of developing other approaches and methods to the problems of mid-term forecasting of exchange rates and fundamental analysis of macroeconomic dynamics. They should provide the solution of these tasks taking into account the existing uncertainty in their development, associated with both the complexity of the structure of relations of fundamental macroeconomic indicators, the non-stationary nature of the influence of some fundamental indicators on others, and with the subjective perception and assessments of the participants of macroeconomic dynamics.
Artificial Intelligence and Neural Networks
More opportunities for predicting time series are provided by artificial intelligence models and neural networks, the development of which has received much attention in recent years [2, 3]. It is believed that with the help of neural networks it is possible to approximate with high accuracy an arbitrary continuous function f(x1 , x2 , ..., xn). But here another question is open, what mechanism of formation (learning) of structure and memory of artificial intelligence and neural networks, providing acceptable accuracy, speed of adaptation of network and computational costs. The complexity of solving this problem often limits the area of application of these models. This makes reasonable the use of statistical adaptive models of recurrent prediction in terms of obtaining short-term forecasts provided that they are supplemented with modeling procedures which allow reflecting such aspects specific to decision-making in economic systems as accounting for changes in the structure of relations in the simulated processes of macroeconomic dynamics, changes in subjective expectations of market participants as to the dynamics of development of the analyzed macroeconomic situations. At the same time, taking into account the fact that exchange rates should not be considered as random variables, but as non-stationary random processes with sufficiently complex dynamics of development, then from the whole variety of known approaches to the assessment of emerging currency risks one should single out complex methodsThe methods are based on the combination of statistical measurements of risk with the scenario methods of analysis of resistance to crisis phenomena in the international currency market FOREX. However, these methods are currently still insufficiently developed and require development. This problem is solved with the use of intelligent decision support systems (DSS), information processing in which is based on the methods of system analysis of heterogeneous, diverse data of considerable volume, which leads to the impossibility of direct application of known methods and techniques.
Technical and fundamental analysis
Traditionally for Forecasting changes in exchange ratesThe analysis of financial markets is based on two types of analysis: technical and fundamental [1], which are caused by fluctuations of supply and demand balance on currency assets, connected with the movement of capital on foreign trade and investment deals. Most methods of technical analysis are based on short-term statistical evaluation of time series, which represents the dynamics of changes in currency rates at the time periods, preceding the analysis. As a rule, these methods do not take into consideration the influence of factors influencing exchange rates, which greatly limits the possibility of getting effective forecasting assessments. At the same time, a significant impact on changes in exchange rates, particularly EUR/USD, is caused by regular publications by Reuters, Dow Jones and Bloomberg on various macroeconomic fundamental indicators characterizing the state of development of the economies and financial systems of the U.S. and the Eurozone.
These messages act as events that can adjust the perceptions of international currency market participants about the current dynamics of national economies, financial markets and the influence of monetary regulation mechanisms on them, and, consequently, about the current value of certain currency assets, which leads to a change in the balance of supply and demand for them and, accordingly, to the changes in exchange rates. Obviously, these circumstances indicate that the key role in the analysis and forecasting of the international foreign exchange market belongs to fundamental analysis.
The exchange rate, which we will choose the dollar exchange rate, and commodity markets are thought to move in opposite directions. The rise in the dollar counteracts inflation, which ultimately causes commodity prices to fall. In turn, falling commodity prices cause interest rates to fall and bond prices to rise. And rising bond prices boost the stock market. A drop in the dollar causes the exact opposite effect, namely a rise in inflation (higher commodity prices), lowering bond and stock prices. The peak in the bond market against the backdrop of an economic recovery serves as a signal that the economy is moving from a state of normal, non-inflationary growth to a phase of "unhealthy" growth. Investors sell bonds due to an increase in inflationary pressures and fears of a subsequent interest rate hike. After a while, rising interest rates begin to put bearish pressure on the stock market, and it also turns downward. When rising inflationary pressures cause interest rates to peak, investors' desire to buy dollars begins to reverse. Commodity markets also begin to turn downward due to a possible subsequent production slowdown. Further, as economic growth slows, the need for goods and money decreases, inflationary pressures subside, and commodity prices begin to fall. As commodity prices and interest rates fall, the bond market begins to rise. Gradually, the stock market turns to follow. After that, the commodities market also moves into the growth phase, and inflationary pressures begin to form. Investors' desire to buy the dollar arises again. This example shows a close connection between exchange rates and processes of macroeconomic dynamics, which makes it necessary to consider them together when solving the task of forecasting the development of foreign exchange markets.
"Efficient Market" and "Rational Expectations"
What the above models have in common is that they are based on the hypotheses of "efficient market", "rational expectations" and the linear paradigm. The "efficient market" hypothesis goes back to the early 1970s and serves the important function of arguing for the use of linear regression equations in the analysis of capital markets. This hypothesis asserts, that the capital market is created by the mistakes of many. Investors in this case are assumed to be rational: they know collectively what information is important and what information is not. Then, after systematizing this information and assessing risks, the collective consciousness of the market finds, for example, an equilibrium price. The concept of equilibrium is widespread. For example, it is assumed that supply is always equal to demand. Exogenous factors that perturb the system may throw it out of equilibrium, but the system responds to these perturbations and returns to its equilibrium position in a linear fashion. In this case, the system reacts immediately because investors believe that today's price change depends only on today's unexpected news, yesterday's news is not very significant, today's profits have no relation to yesterday's, i.e., they are independent quantities. Then, if a sufficiently large number of price changes are accumulated, in the limit their probability distribution becomes normal, which allows us to use regression equations for modeling.
But markets are seldom so well organized. Quite often, when investors least expect it, there is an exponential superreaction to one impact or another in the market. This suggests that market processes are nonlinear, and a large number of specialists are aware of the relationship of such a superreaction to reality ? But if markets are nonlinear dynamic systems, then the use of linear regression analysis can lead to erroneous results, which necessitates a review of the assumptions that underlie the current theory of capital markets. We should also pay attention to the following contradictions in the "efficient market" and "rational expectations" hypotheses, which claim that market expectations of investors are accurate and unintended.
However, investors are often inclined to self-righteous predictionswhich can be a reason to ignore a certain amount of market information. In particular, investors may not react to emerging trends until those trends are well established. Then they make a decision, which is conditioned by the accumulated, but previously ignored information, thus reducing its efficiency. Otherwise, investors are not always rational, they are full of biases in their subjective assessments and may be confident in their predictions much more than is justified by the available information ?6? A similar conclusion is drawn by G. Soros ?7?. He expresses an opinion about a more complicated nature of market processes, the concept of market equilibrium and formulates the provisions of the reflexivity theory which he has developed. J. Soros argues that market developmentThe opposite is also true: fundamental factors are determined by the market, i.e. by the behavior of market participants, their estimates and expectations. At the same time, the ability to make a correct assessment of the development of market situations depends on the ability to anticipate the prevailing expectations of market participants rather than on the ability to predict changes in the real world.
The latter can be achieved by solving problems of fundamental analysis and forecasting macroeconomic dynamics and exchange rates as part of the creation of the appropriate Decision support systems (DSS).
The solution of the problem of recognition of the state of uncertainty in the market participants, reflecting their attitude to the development of the interaction of economic processes, and the adjustment of forecast models, are still quite complex and little-studied elements in the organization of forecast modeling of macroeconomic processes. This circumstance is due to the fact that one of the factors influencing the peculiarities of the construction of procedures for obtaining forecast assessments of the development of complex systems is the type of structured problem areas, for which these procedures are developed. The non-triviality of discrete-sequence forecasting is due to the fact that, in contrast to well-algorithmic interpolation procedures, forecasting requires extrapolation of data on the past into the future. At the same time, it is necessary to take into account an unknown pattern in the phenomenonThe underlying process that generates discrete sequences. A large number of studies are devoted to the development of mathematical prediction models, but the most common are methods based on the probabilistic-statistical apparatus. Their use requires a significant amount of experimental data, which cannot always be collected in the conditions of events that took place relatively recently.
Neuro-fuzzy approach
Recently, there has been a renewed interest in the use of artificial neural networks. They are regarded as close to the human brain universal models that are trained to recognize unknown patterns. But, as in the case of probabilistic-statistical methods, training neural networks requires a large sample of experimental data. In addition, a trained neural network does not allow for a clear interpretation.
Another approach to prediction combines experimental data about the process with expert-linguistic information about patterns that can be seen in existing data is data mining system. The use of expert-linguistic regularities, which are formalized by means of fuzzy logic, allows to build a prediction model in conditions of small experimental samples. This approach is ideologically quite close to the so-called neuro-fuzzy approach, which combines the learning abilities of neural networks and the easy interpretability of fuzzy rules. However, it does not use a neural network to train a prediction model, but directly tunes fuzzy rules using existing experimental data.
Taking into account the above-mentioned features, it is proposed to solve the problem of fundamental analysis and forecasting of the processes of macroeconomic dynamics and exchange rates in the direction of creating of multilevel modeling systems. They should provide a combination of the functionality of adaptive statistical forecasting of exchange rates and macroeconomic dynamics processes in the short-run at the lower level and simulation modeling and structural analysis at the second level. Thus, where the situational assessment of the influence of some fundamental indicators on others in the chains of their cause-and-effect relations, as well as the identification of the emergence of structural instability in them with the assessment of the propagation of wave impulses and the corresponding adjustment of statistical forecasting models are carried out. These models should also reflect the consideration of changing subjective assessments of market participants with regard to the dynamics of development of the analyzed macroeconomic situations.
SPPR
The creation of the DSS for the tasks of fundamental analysis and forecasting of macroeconomic dynamics and exchange rates is associated with the development of its components, such as:
- Structuring the processes of international capital flows, the development of national economies and financial markets with the formation of a set of fundamental macroeconomic indicators that characterize the above processes, as well as the scheme of their cause-effect relations;
- representation of processes related to changes in one or another fundamental indicator in the form of time series using both numerical and qualitative form of setting time series members in the form of symbolic chains of fuzzy variables, allowing to solve the subsequent problems of obtaining forecast assessments of trends taking into account subjective expectations of market participants, which, as a rule, are connected with qualitative characteristics of dynamic processes being assessed;
- assessment of the interaction of the processes associated with many fundamental indicators, in order to select for further analysis in the studied set of cause-effect relations only those relations of fundamental indicators, where there is a real influence of one fundamental indicator on the trend of another, which ensures the compression of the conceptual scheme of the macrosystem under consideration;
- structural analysis of interactions of processes of macroeconomic dynamics, allowing to recognize the existing chains of influence of fundamental indicators in the system of links of macroeconomic processes of international capital movement, development of national economies and financial markets, and accordingly allowing to form the structure of links of parameters in the models of forecasting of trends of fundamental indicators. The structural analysis of interactions of processes of macroeconomic dynamics also creates the basis for recognition of occurrence and tracking of distribution of crisis situations, which may arise due to irrational behavior of active elements of the considered macrosystem, and also allows to identify fundamental indicators, which trend may change its character in the future due to formation of the impulse disturbances wave in the chains of influence of fundamental indicators;
- evaluation of the market participants' subjective expectations concerning the development dynamics of the analyzed macroeconomic situations in the system of the fundamental indicators connections, which allows to provide the complex forecasting assessments of the fundamental indicators trends with the account of the formalisms introduced into the analysis and forecasting models, reflecting this or that degree of the market participants' uncertainty concerning the development of the analyzed situations.
Forecasting exchange rates and macroeconomic dynamics, which is based on the adaptive statistical models of recurrent prediction with the addition of the latter by the modeling procedures, allows to reflect such aspects, which are characteristic for decision-making in economic systems, as the consideration of changes in the structure of relations in the simulated processes of macroeconomic dynamics, subjective expectations of the market participants concerning the dynamics of the analyzed macroeconomic situations. That is necessary in the system of the fundamental analysis and forecasting of exchange rates, because at that the account of tendencies in the development and interaction of factors, influencing an exchange rate, is provided and, in its turn, it considerably strengthens the opportunities of the forecasting system. Therefore, analytical methods of mathematical modeling of an exchange rate is such a system of deep data processing (data mining), which simplifies the process of research and decision-making. It allows the analyst to perform an interactive analysis of the data in order to discover hidden, previously unknown rules and patterns of great practical importance.
Literature
1. Sobolev V.V.. Currency Dealing in Financial Markets / Yu. - Novocherkassk, 2009. - 442 с.
2. Fuzzy sets in control models and artificial intelligence / Edited by D.A. Pospelov.-M.:Nauka.-1986.-312 p.
3. Romanov V.P. Intellectual information systems in economics: Tutorial. / Ed. by N.P. Tikhomirov. - Moscow: publishing house Examen, 2003. - 496 с.
4. Mishkin F. Economic theory of money, banking and financial markets: Textbook for universities / Per. with English D.V. Vinogradov ed. by M.E. Doroshenko. - M.: Aspect Press, 1999. - 820 с.
5. Peters E. Fractal Analysis of Financial Markets: Applying Chaos Theory to Investment and Economics. - Moscow: Internet Trading, 2004. - 304 с.
6. Sobolev V.V.. Financiers/ Yuzhno-Russian State Technical University (NPI). - Novocherkassk, 2009. - 315 с.
7. 7. Soros J. Alchemy of Finance: Per.s Engl. - M.: Infra-M, 1996. - 416 с.