Mathematical modeling of adaptive-rational methods of exchange rate forecasting
Financial markets have random processes with such complex dynamics that their identification and mathematical modeling to find patterns is often an impossible task. Even application of adaptive modelsThe mathematical modeling of the mechanisms and methods of reflecting the dynamics formed under the influence of the effects, the emergence of which is not detected in the data of the historical period, is therefore required. That is why it is necessary to mathematically simulate the mechanisms and methods of reflecting the dynamics formed under the influence of the effects, the possibility of whose occurrence in the future is not detected in the data of the historical period.
Directions for the development of an adaptive approach
The adaptive approach is developing in three directions. The first of these focuses on increasing the complexity of adaptive predictive models. The idea of the second direction is to improving the adaptive mechanism of forecasting models. In the third direction is implemented The approach of sharing adaptive principles and other forecasting methods, in particular, simulation modeling. At the same time, the optimality of the algorithm and the profitability trading systems is related to the quality of predictions.
The mathematical description of market relations can be regarded as a dynamic market model. In turn, the mathematical model allows theoretical methods to forecast market behavior - the dynamics of the market price - and, based on the forecasts, to form market transactions of optimal volume at a suitable moment in time. However, forecasting as an algorithm of actions should not be substituted by the use of market indicators or advisors, which give indirect and usually ambiguous information about the market price dynamics.
The set of analytical methods that make up a trading system allows you to work out the rules for buying or selling currencies. Trading systems based on one method are called indicatorsand the rules are called signals. Analytical methods include methods that use filtering or mathematical approximation of time series. In the technical analysis as the basic time series the series of price values for a certain period of time, trading volume or the number of open positions are used. Currency prices are the main object of technical analysis, so the choice of one or another indicator as a basis is not farfetched. What is a particular indicator? Indicator is a set of functions from one or more basic time series with a certain time "window".
Mathematical modeling for the search of patterns consists in the development and implementation of a holistic concept of adaptive-rational forecasting of financial markets, according to which the forecast should be built using actual data and taking into account subjective expectations based on the principle of adaptive distribution of confidence in the data of different nature. In the framework of the suggested conception unlike the existing ones, it is possible to construct the models which allow forming the most complete idea about reality of expected variants of anticipating dynamics of financial markets. Mathematical modeling by the adaptive-rational method of forecasting of financial markets is built in accordance with the law of necessary diversity and the principle of external addition and lays methodological foundation for reflection of anticipatory reality as a result of adaptive coordination of objective regularities and subjective expectations. In order to determine the adaptive component of forecast trajectories, a new class of adaptive models - models with The multilevel structure of the adaptive mechanism.
Multivariate forecast trajectory
The adaptive mechanism of these models provides identification of multi-trend processes, which significantly expands the possibilities of reflecting complex trend patterns, transformed into a "forecasting image of the future". In order to form the rational component of the forecast trajectories, the methodology of constructing alternative variants of the rational component on the basis of computational experiments carried out with the adaptive-simulation model and the approach to forecasting conditional subjective expectations using pseudo-selective populations are used. Rational component alternatives are ranked according to their likelihood of preference. The adaptive-rational mathematical model provides the construction of a multivariate forecast trajectory with probabilistic assessments of the degree of reality of these options. Its distinctive feature is ability to predict even those effects that are absent in the dynamics of the predicted process and predicting processes in the dynamics of which there are trend reversals. The peculiarity of the model is that its rational component is built into the feedback loop of the adaptive mechanism. This allows the model to be endowed with a new property, according to which feedback signals can be perceived with the opposite sign. Due to this property in the adaptive mechanism the lagged response is replaced by the expected response.
For practical use it is necessary to theoretically justify the adaptive-rational method of forecasting financial markets. As is known, modeling of forecast estimates of the future state of objects is most successful only when the model fully reflects both the nature of the management process and the specifics of the business environment. To clarify the nature of management it is necessary to determine its relationship with the characteristics of the external environment.
The objective aspect is related to two things. First of all, note that evolutionary changes that are sluggish are not always satisfying. Only rational actions can shorten the duration of the adaptation period. The second point is related to the existence of a rich theory and practice of using a rational approach to the real management of economic objects under conditions of certainty and risk. This approach is justified, but there are natural limits to its application in the form of the relativity of the level of knowledge at a particular point in time. Even those decisions that are taken under conditions of certainty contain elements of uncertainty associated with their implementation in the future. Despite the attractiveness of the rational approach, it cannot cope with the uncertainty of the future that always exists. The contradictory nature of this situation can be viewed from the perspective of the principle of external augmentation, which boils down to the fact that any management language is ultimately insufficient to meet its objectives, but this deficiency can be eliminated by incorporating "black box" into the control circuit.
In the abstract, the purpose of the black box is to formulate solutions in higher-order language that cannot be expressed in terms of the current control system. The black box problem can be solved in a variety of ways, the only important thing is that this external addition is of a different nature. Mechanisms of a different nature include adaptive mechanisms, which provide "soft" adjustment of decisions made within the framework of a rational management system. Control systems for real economic objects, built on the basis of combining two approaches - adaptive and rational - are called adaptive-rational. The need for such combination directly follows from the fundamental law of cybernetics - the law of necessary diversity, formulated by W.R. Ashby. The essence of the problem is that the possibilities of rational control due to limited knowledge do not always meet the requirements of this law, and, therefore, do not always provide effective control. In contrast to the rational, in The nature of adaptive control offers unlimited possibilities for a variety of appropriate responses on the diversity of the controlled object. Therefore, the idea of combining the two approaches, should be considered one of the components of the modern paradigm of management science and to use in models claiming a high level of adequacy.
In terms of economic theory
Interestingly, economic theory has not neglected the concepts of "adaptive" and "rational". The search for a concept explaining complex economic processes led science to the need to use the idea of adaptive behavior. On the basis of this idea the theory of adaptive expectations was developed, which was followed by the hypothesis of rational expectations. If we raise the question about sufficiency of using only one hypothesis when constructing a model adequately reflecting the real behavior of economic agents, the answer would probably be negative. In the behavior of even one agent it is possible to detect the orientation to both adaptive and rational expectations. It all depends on the length of the anticipatory period to which the expectations apply. For short periods, expectations are adaptive rather than rational, and vice versa for long periods. This reasoning is closely correlate with the idea of adaptive-rational management and lead to an important conclusion, the essence of which is that both management and the behavior of economic agents is a complex combination of adaptive and rational. And the level of the ratio of adaptive to rational is subject to change and in each case is determined by the conditions in which the agent is forced to act, as well as by the time of anticipation.
It becomes clear that the reliability of forecast estimates of the effects of control is in direct dependence on the extent to which they take into account the nature of the processes being forecasted. In other words, predictive models should reflect both adaptability and rationality of these processes. As concerns principles for building adaptive-rational models, the main principle, in fact, follows from the very name of the models; it is based on the idea of combining the adaptive and the rational. In the broad sense, adaptation is a process of adjustment, while rationality is something related to reason, so literally the term "adaptive-rational" can be interpreted as "adapting to reason. A legitimate question immediately arises: "How can this adaptation be technically accomplished?" In other words, the essence of the question is how the mechanism that implements the process of adaptation to the manifestations of reason should be arranged. The joint application of formalized procedures and expert evaluation is the most attractive possibility. It is the experts who concentrate "rationality" in their assessments that should become that mediated element of the adaptive-rational model, without which its substantive meaning is lost. Only with their help is it possible to solve the problem of incorporating rational expectations into predictive estimates.
Level of confidence in raw data
The next aspect requiring special consideration is level of confidenceThe variability of the data requires special approaches to their use in describing the dynamics of the future. The variability of the data requires special approaches to their use for describing the picture of the dynamics of the future. Obviously, for the moments of time close to the current one, the forecast estimates obtained by extrapolation have more confidence compared to the data of a subjective nature. Conversely, estimates of the distant future, based on the rational expectations of experts, usually have a higher degree of confidence than extrapolated forecast data. In fact, we are dealing with a situation where, over time, one set of data sort of loses its information value, while the other increases it. Therefore, the combined trajectory should be based on the principle of distributed trust in the data of different nature. The implementation of this principle implies the transition from a trajectory dominated by extrapolation estimates to the trajectory of rational expectations, carried out in accordance with the changes in the degree of confidence in the forecast estimates of different nature. Theoretically, different variants of algorithmic implementation of such transition are possible, but the approach based on adaptive modeling of the process of transition from one trajectory to another is the most acceptable.
Adaptive models have long been used to forecast economic processes. They are deservedly considered an effective tool short-term forecasting. However, the use of adaptive models to develop adaptive-rational forecasts imposes higher requirements for the ability of these models to adequately reflect the dynamics of the processes being forecasted. The high approximation accuracy achieved with the help of adaptive models creates a false idea of their high adequacy. Increasing the "real" adequacy of these models is associated with improving the structure and logic of building the adaptive mechanism through models with a multilevel structure of the adaptive mechanism, which, being a generalization of adaptive regression models, represent a new class of models. These models are used in the case of adaptive-rational forecasting of multitrend processes occurring, in particular, on heterogeneous (fractal) financial markets. Possible areas of improvement are defined within the concept of adaptive-rational forecasting of financial markets.
The adaptive-rational approach is used in situations where the use of traditional financial market forecasting apparatus is ineffective.
The possibility of using subjective information in adaptive-rational models is largely limited by the existing apparatus for processing expert data. The new possibilities of the developed apparatus for forming the adaptive and rational components are fully realized when they are used together in adaptive-rational models. The specifics of the joint use of these components allows us to build models, the application of which is focused on the solution of special classes of forecasting problems. For example, inclusion of the rational component in the feedback loop of the adaptive mechanism leads to models, with the help of which it is reasonable to predict the dynamics of processes with trend reversals.
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