# Mathematical modeling and analysis of adaptive methods of exchange rate forecasting

*Mathematical forecasting methods can be developed on the basis of various functions, dynamic series and analytical dependencies. For mathematical modeling and forecasting of currency markets both price dynamics and its derivatives (indicator values, significant levels, etc.) and market macroeconomic indicators. In mathematical models of financial time series forecasting the price dynamics is used as an input information. However, information models of time series, which are descriptions of original objects using diagrams, graphs, formulas, drawings, etc., are handled differently. One of the most important types of information modeling is mathematical, when descriptions are formulated in the language of mathematics. Accordingly, the study of such models is carried out with the use of mathematical methods.*

Mathematically, the problem of predicting the exchange rate can be reduced to the problem of approximating multidimensional functions and, hence, to the problem of constructing a multidimensional mapping. Depending on the type of output variables, function approximation may take the form of: classification or regression. Consequently, in** forecasting models** of exchange rates can be distinguished two major subtasks: 1. constructing a mathematical model; 2. training expert networks that implement the solution of the problem. As a result of studying the subject area, a mathematical forecasting model should be developed, including a set of input variables; a method for forming input features and a method for training an expert system.

### Analytical dependencies

Let's look at the features **forecasting models exchange rate** based on 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 influencing the formation of the exchange rate. The model is based on the hypothesis that **purchasing power parity**. Further, in the process of considering real economic systems, new factors will be added, and the generalized model will select the main factors influencing the formation of the exchange rate.

Increasing the efficiency of short-term operations with currency is one of the important tasks in the activity of banks and other investors, who sell and buy different currencies in significant volumes, seeking to give movement to the available free reserves, to avoid losses from market fluctuations of exchange rates and to obtain additional profits. And **currency operations** are carried out with great speed through the Internet, as it is very important to enter the foreign exchange market with the offer before the competitors. All of this is part of the continuous process of forming an optimal structure of foreign exchange reserves.

The efficiency of foreign exchange operations substantially depends on the reliability of currency fluctuation forecasts. That is why short-term forecasting of exchange rates is of great practical importance for operational activities of banks and other investors. The question about possibility of application of statistical methods for this purpose seems to be relevant and natural. The problem **short-term** The main point to consider when forecasting currency rates using statistical models is that successful foreign exchange operations require one-day-ahead forecasts. As, for example, in the movie "Pi" mathematician Max Cohen has been trying for many years to find and decipher a universal numerical code according to which the rates of all listed shares. As he gets closer to the solution, the world around Max turns into a dark nightmare: he is pursued by powerful Wall Street analysts to discover the code of the universe's universe. On the brink of madness, Max must make a decisive choice between order and chaos and decide whether he is capable of coping with the powerful force his brilliant mind has now awakened. But this is fantasy. In reality, it is not hard work, but the course of thought that determines the investment return, with only adequate mathematical modeling to evaluate the effectiveness of an idea.

### Adaptive forecasting methods

It is difficult to draw a clear line separating adaptive forecasting methods from non-adaptive ones. Already forecasting by extrapolation of ordinary regression curves contains some element of adaptation, when with each new obtaining of actual data the parameters of regression curves are recalculated and refined. After a sufficiently long period of time, even the type of curve can be replaced. However, the degree of adaptation in this case is quite insignificant; in addition, over time, it decreases along with an increase in the total number of observation points and, accordingly, with a decrease in the specific weight of each new point in the sample.

The sequence of the adaptation process is as follows. Let the model be in a certain initial state, and a prediction is made on it. When one unit of time has elapsed (the simulation step), we analyze how far the result obtained by the model is from the actual value of the series. **Prediction error** is fed to the input of the system via feedback and is used by the model according to its logic to move from one state to another in order to better align its behavior with the dynamics of the series. The model should respond to changes in the series by compensating changes. Then a prediction is made for the next point in time, and the whole process repeats. Thus the adaptation is carried out interactively with each new actual point of the series. But what should be the rules for transition of the system from one state to another, what is the logic of the adaptation mechanism?

**In essence, this question is solved by each researcher intuitively. The logic of the adaptation mechanism is set a priori, and then tested empirically. **When building a model, we inevitably endow it with innate properties and, at the same time, for greater flexibility, must take care of the mechanisms of conditioned reflexes, which are learned or lost with a certain inertia. Their totality constitutes the logic of the adaptation mechanism. Due to the simplicity of each individual model and the limited background information, often represented by a single series, one cannot expect any single adaptive model to be suitable for predicting any series, any variation in behavior.** Adaptive models **are flexible enough, but one should not count on their universality. Therefore, when building and explaining specific models, it is necessary to take into account the most probable regularities of development of a real process, and to correlate dynamic properties of a series with model capabilities. It is necessary to include in the model those adaptive properties, which are sufficient for the model to follow the real process with a given accuracy.

At the same time, one cannot hope for a successful **model self-adaptation**The number of parameters increases the system's sensitivity, which leads to its swinging and deterioration of the forecasts obtained from it. Thus, when building an adaptive model we have to choose between a general and a private model, and, after weighing their advantages and disadvantages, give preference to the one, from which we can expect the smallest forecasting error. That is why it is necessary to have a certain stock of specialized models, diverse in their structure and functional properties. To compare possible alternatives, a criterion of model usefulness is needed. In spite of the fact that in the general case such criterion is a matter of controversy, in case of short-term forecasting the mean square of the forecasting error is usually a recognized criterion. The quality of a model is also judged by the presence of autocorrelation in the errors. In more advanced systems, the trial-and-error process is carried out by analyzing both sequential in time and parallel (competing) modifications of the model [2].

### Short-term exchange rate forecasting

Information about the dynamics of exchange rates gives the impression of a chaotic movement: falling and rising rates alternate with each other in some random order. Even if there is a rising trend over a long period of time, it is easy to see on the graph that this trend makes its way through complex movements **currency rate time series**. The direction of the series changes all the time under the influence of irregular and often unknown forces. The object under study is fully exposed to the elements of the world market, and there is no accurate information about the future movement of the rate. It is necessary to make a forecast. At the same time it is absolutely clear that **to predict even the sign of the rate increase** **it's very difficult**. This is usually done by experts, who analyze the current conjuncture and try to single out factors, regularly connected with the movement of the rate (fundamental analysis). When building formal models they also try to highlight the range of significant factors and use them to construct an indicator, but neither practical experts, nor formal methods give good and steady results. We believe this can be explained, first of all, by the fact that even if there really is any circle of factors influencing the exchange rate in a stable way, their influence is reliably hidden by an imposed random component and controlling influences central banks.

As a result, these factors and their impact are quite difficult to isolate. **Therefore, it is necessary to consider short-term forecasting of the exchange rate as essentially a task of forecasting the sequential movement of an isolated time series, the reason for which is mainly the mass behavior of small and large financial players in the foreign exchange market, who carry out the bulk of financial transactions with the currency. **This approach can be attributed to technical analysis. Of course, an individual participant of the currency game is free to change his strategy quite arbitrarily. And yet we can assume that the behavior of the whole mass of participants through the supply and demand ratio, influencing the exchange rate, has a certain dominant logic in the current period of time, which can be found through the law of large numbers. For example, when the rate of a currency falls, it may be bought with the expectation of its further rise. And such a mass demand for a currency does lead to its appreciation. Or vice versa, if confidence in the currency falls and its further depreciation is expected, then mass supply prevails and the exchange rate falls even lower. Note that with this simplified approach, the dynamics of the time series itself can be read as a chronological record of the mass behavior of currency market participants. This makes it possible to build a model based on the series itself, without attracting additional information, and to use all the reasoning about the mass behavior of market participants only for qualitative interpretation. **If we could find at least short-term regularities in the dynamics of the series, which are realized with a probability of more than 50%, it would give reason to count on success.** Then it would be possible to apply statistical methods for predicting rates, capturing more or less stable relationships of successive time series events [2,4].

In this case the following task is set. Firstly, to find out the applicability of any statistical methods, the purpose of which is to describe recurring events or situations characterized by relatively stable relationships, for short-term exchange rate forecasting. Secondly, if statistical methods are applicable for the solution of the given problem, then establish their most perspective class, indicate characteristic features of these methods, pay special attention to the simplest of them. Thirdly, show practical results by example. Let us note that a lot of attention has always been paid to the issues of forecasting of exchange rates. Let us mention, for example, the work of Clive W.J. Granger and Oscar Morgenstern (Granger Clive W.J., Morgenstern Oscar. Predictability of stock market prices. Massachusetts, 1970), which examines the dynamics of stock prices and provides an extensive bibliography. This monograph actually concludes that if there is any **correlation** in a series of this kind, then it is most likely that it is present between adjacent rate increases. However, the question arises whether we are trying to predict completely random fluctuations in exchange rates. The answer to this question is found in a special study [5].

### Modern Forecasting

A new view on the role of forecasting has established itself as an obligatory element of the decision-making process. A logical consequence of the increased role of forecasting was the increased requirements for the validity and reliability of forecast estimates. However, the level of compliance of the apparatus of modern forecasting with these new requirements remains excessively low. Even the use of adaptive models, with the help of which it is possible, as a rule, to achieve the necessary level of adequacy in describing the processes being forecasted, only partially solves the problem of increasing the reliability. The modern economy generates processes with such complex dynamics that the identification of its regularities by the apparatus of modern forecasting is often an insoluble task. **Improvement of this apparatus, first of all, needs new ideas and new approaches, on the basis of which it is possible to implement mechanisms and ways of reflecting the dynamics formed under the influence of effects, the possibility of the appearance of which in the future is not detected in the data of the historical period. **A clear contradiction arises, overcoming which will contribute to the formation of a new view of the **Forecasting as a proactive reflection in a probabilistic environment **representations of the process under study in the form of a trajectory built on the basis of objective trends and subjective expectations.

Within the framework of economic forecasting, the development of the adaptive approach takes place in three directions. The first of them is mainly focused on **complications** adaptive predictive models. The idea of the second direction is to** improving **the adaptive mechanism of forecasting models. The third direction implements the approach **sharing** adaptive principles and other methods of forecasting, in particular, simulation modeling. The development of adaptive-simulation models is devoted to the works of V.V. Davnis [3].

*Market development is determined by fundamental factorsbut the opposite is also true - the fundamentals are determined by the market*i.e. the behavior of market participants, their evaluations and expectations. At the same time, the ability to give 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 [6, 7]. Therefore, the ideas of the development of the mathematical apparatus of forecasting do not sufficiently take into account the properties of the activity of economic systems, which reduces even with high interpolation accuracy the level of plausibility of forecast estimates. At the same time, forecasts based only on subjective information are focused on the prediction of qualitative characteristics, and therefore their use is possible only in special cases. This brings to the fore the problem of constructing forecasts based on the combination of extrapolation and subjective estimates. Studies in this area were conducted, but the analysis of the results of these studies showed the predominance of creative character in them, which indicates, in fact, the initial level of development of the problem of building combined forecasts.

### Literature

1. Sobolev V.V.. Currency Dealing in Financial Markets / Yu. - Novocherkassk, 2009. - 442 с.

2. Lukashin Y. P. P. Adaptive methods of short-term forecasting of time series: Textbook. - Moscow: Finance and Statistics, 2003. - 416 с.

3. Davnis V.V., Tinyakova V.I. Adaptive models: analysis and forecasting in economic systems. - Voronezh: Publishing house of Voronezh State University, 2006.

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. Lukashin Y.P. On the Possibility of Short-Term Forecasting of Currency Rates with the Help of Simplest Statistical Models // Bulletin of Moscow State University. -1990. - Ser. 6. Economy. -№ 1.-С. 75-84.

6. Sobolev V.V.. Financiers / South-Russian State Technical University (NPI).-Novocherkassk, 2009.-315 p.

7. 7. Soros J. Alchemy of Finance: Per.s Engl. - M.: Infra-M, 1996. - 416 с.