The high degree of uncertainty in stock markets significantly complicates the process of forecasting the dynamics of financial instruments. This problem is important both for states and investment companies, as well as for other market participants who need to make long-term investment decisions based on preventive measures to reduce the impact of financial crisis risks on their activities.

To date, the main difficulties in predicting numerical series are:

- lack of stationarity of the process;
- lack of stationarity of the process;
- small prediction interval
- ignoring the strategies of individual market participants

Below is an overview of the most practically used tools for predicting financial instruments.

Regression-type forecasting methods are required to investigate the correlation between more than two variables. Well-known regression models include the following.

**Simple linear regression**.

The model is based on the hypothesis that there is a discrete external factor X*(t)*, which dominates the influence of the whole analyzed process Z*(t)*, with a linear type of relationship. The following equation describes this dependence:

where a*0* and a*1* are regression coefficients; εt is the model error.

In order to obtain prediction values Z*(t) *for time interval *t*, it is necessary to have values X*(t) *for the same time interval *t*, but in reality this is not often the case.

**Multiple regression (multilple regression).**

n reality, numerous discrete external factors X*1(t)*,…,X*s(t)* have a significant impact on the process Z*(t)*. Then the equation of the model will be as follows:

However, this method also has a weighty weakness. Namely, the future value of the process *Z(t) *can be calculated only after determining the future values of all factors X*1*, X*2(t)*, …, X*s(t)*, which is very rarely possible in reality.

**Nonlinear regression.**

This model is based on the hypothesis of a function that confirms the existence of a correlation between the initial process Z*(t) *and some external factor X*(t)*. The function of such a model has the form:

In order to build this type of model, first of all we should calculate the parameters of function *A*. Then let us make an assumption, according to which:

On this basis, we define the parameters *A = [a0. a1]. *Real processes according to which the type of the relationship that combines the process Z*(t)* and the external factor X*(t)* has already been determined in advance are very rare. It is due to this fact that the use of non-linear regression models is extremely limited, and such methods are used relatively infrequently in the practice of stock market activity.

The most widely used fairly are the Dickey-Fuller tests. The method for evaluating the time series for integrability is represented by the formula:

The main idea of this method is as follows: it is necessary to check the stationarity of the process, i.e. to confirm or refute the corresponding hypothesis, and also alternately to check its order differences, with a trend to increase. Tests such as the DF-test contain a significant drawback, namely: they do not provide for the probable residual autocorrelation. However, if autocorrelation is still present in the residuals, the results of the least squares method may become unreliable. The right half of the equation can be fitted with new variables, namely lag values that are carried over from the left half. Then the equation will have the form:

This test is called the general Dickey-Fuller test (ADF-test). It is important to emphasize that it is the most productive and the most common of the simple integrability tests.

**Autoregressive forecasting models.**

A form of autoregression is considered extremely useful for the purpose of determining existing time series in reality. In an autoregressive model, 𝑍*(𝑡)* is expressed as a finite linear set of previous values of this process and momentum, which is called “white noise”. The equation of such a function is represented as:

where Z*(t) is an *autoregressive process of order *p*, denoted as AR*(p)*, where *C *is a real constant φ*1* , …, φp are coefficients, ε*t *is the model error. In order to determine φ*i *and *C*, it is reasonable to apply the maximum likelihood method or the standard method of least squares. The general model is often referred to as *ARMA*.

The *ARIMA(p,d,q) *model has several modifications, one of which is *ARIMAX(p,d,q)*, whose equation is represented as:

where *a1*, …, *as* are coefficients of external factors are coefficients of external factors X*1(t)*, …, X*s (t)*. Often Z*(t) *is defined as the result of calculation by the *MA(q) *model. After that, further predictive value of Z*(t) is *determined by the autoregressive model. It also contains the regressors of external factors X*1(t)*, …, X*s(t)*, as a clarifying addition.

The autoregressive model with conditional heteroscedasticity by T.P. Borreslev is essentially a residuals model for the *AR(p) *method. We first define an *AR(p) *model for the required series. Then we refer to the assumption that the model error has components:

where σ*t* is the standard deviation dependent on the time index; υ*t* is a random variable with a normal distribution, the mean value is 0 (zero), and the standard deviation is 1 (one).

The value of the standard deviation is calculated by the formula:

where β*0* , …, β*q *and γ*1*, …, γ*p *are coefficients. The *GARCH(p,q) *model includes 2 indicators: *p *– the order of autoregression of residuals squares; *q *– the number of prior estimates of residuals. The use of this method is especially widespread in finance, because it is successfully applied when it is necessary to model volatility. There are many modifications of the *GARCH* method, such as *NGARCH*, *EGARCH *and others. All of them include the assumption of uncorrelated normal distribution of residuals. If at least one of these assumptions is incorrect, then there is a risk that the forecast intervals will be wrong.

**Prajakta S.K. exponential smoothing (ES) models.**

These are to some extent filter models, through which the terms of the original series pass, and the result is the current values of the exponential average, which is given by the equation:

where S*(t) *is the value of exponential mean at time *t*; ε*t* is white noise; α is the smoothing parameter, 0 < α *< *1; initial conditions are defined as S*(1) = *Z*(0)*. Here each next smoothed value S*(t) *is the weighted average between the previous smoothed value S*(t-1) *and the previous value of the time series Z*(t)*.

The model shows adequate results on a small forecasting horizon, because it overlooks the trend and changes of seasonal nature. On the other hand, the mentioned factors can be taken into account by using:

- Holt’s model, which is based on a linear trend
- Holt-Winters model, which takes into account both seasonality and the multiplicative exponential trend
- the Theil-Wage model using additive linear trend and seasonality data

The advantages of these models are the relative simplicity of design and analysis, as well as uniformity, which allows us to order the calculation process and make comparisons. The key disadvantage of this class of models is their inflexibility. Nevertheless, it is this class of forecasting models that is the most widespread in the case when there is a need to calculate for a long-term period.

**Maximum likelihood sampling model (MMSP).**

This model can indeed be effective, but only in a limited range of applications. It is important to clarify that it is inappropriate to apply it to a number of exchanges, including FOREX, because in this case, according to studies, it works ineffectively.

**Model on Neural Networks ***(ANN).*

Neurons use different types of connections, so in the scientific literature it is common to distinguish three types of networks: recurrent networks, single-layer networks, and multilayer networks. The main advantage of such models is the absence of linearity, because they can connect current and future indicators by non-linear dependence. The advantages of neural networks traditionally include high adaptability, scalability (because the structure of ANN, built on the principle of parallelism, allows to accelerate the calculations) and uniformity. And yet, the ANN models are fraught with numerous weaknesses: the ambiguity of the algorithm for choosing an appropriate architecture; the opacity of the modeling process. Separately, it is necessary to emphasize the difficulty of satisfying the requirement of the training sample, which implies consistency, hence the difficulties in determining the feasibility of using a particular algorithm, as well as the high cost and resource intensity of the training process.

**A model on Markov chains **is an effective way to predict stock prices, but to get better results you need to create a sufficiently large intervals, a small period of time.

In essence, Markov theory is only a simplified model of a complex decision-making process. The structure of the Markov chain and the state transition probabilities are fundamental factors in determining the type of relationship between the current and future values of the process under analysis.

The strength of these models is the relative simplicity of analysis and modeling. The weakness of the Markov chain models, however, is that they cannot be used to model processes with long memory characteristics.

**The classification-regression tree (CART) model.**

This method develops a model of processes influenced by continuous external and categorical factors. If external factors are continuous, it is reasonable to use regression trees. Conversely, for categorical factors, it is better to use the classification type of branching. There are also mixed *CART* models, which, if necessary, allow to take into account all the factors mentioned.

The obvious strengths of these models are:

- high speed and transparency of the tree learning process, which, for example, distinguishes them from
*ANN*models - scalability, due to which there is a fast processing of large data sets and opens up the possibility to use categorical external factors

Weaknesses of *CART* models:

- opacity of the tree structure formation process
- lack of uniformity
- the impossibility of unambiguous choice of time and stage of cessation of further branching (growing) of a tree

**The Support Vector Method (SVM) **is actively used in the electric power industry to model the future dynamics of electricity costs. The model is based on classification in such a way that the original time series are converted into a high-dimensional space. As a result, in the training phase it becomes possible to unambiguously identify the external factors whose future values will need to be referenced when allocating the Z*(t) *forecasts to subclasses.

**The transfer function (TF) model **is used to predict the process Z*(t)*, considering the external factor X*(t)*. The dependence of the future value is defined as:

where *B *is a shift operator: *B1 *Z*(t) = *Z*(t – 1), …, Bk *Z*(t) = *Z*(t – k)*. The time segment η*(t) *characterizes the perturbation from the outside. Then the function η*(B) *has the form:

The coefficients of the function α*i *define the relationship between the processes Z*(t) *and X*(t) *as dynamic.

**Models based on multi-agent systems.**

The agent-based approach is applicable when it is necessary to analyze multicomponent self-organizing processes, which are typical for a number of application areas and are characterized as complex. However, the process of developing such a system is non-trivial and even difficult, because the agent has the properties of independence and autonomy from the overall system. The agent tends to perform purposeful actions, contact with other agents, make decisions, adapt to the environment, make movements, etc.

**Models based on econometrics.**

Econometric models and their approaches differ from traditional econometric methods in the extensive use of graphical drawings. But in practice it turns out that in some situations this can lead to errors, as this approach does not allow us to detect the peculiarities of the data under study. For some reasons, a number of scientists note the Lux-Marchesi model as the most effective. It presents three categories of stock market participants:

- representatives of fundamental analysis, who buy stocks when their price falls below the level determined by long-term factors
- Representatives of technical analysis (or “pessimists”) who sell stocks when the quotes increase in order to fix profits
- Technical analysts (or “optimists”) who buy stocks only when they are rising

This model is based on the concept of statistical physics about the interaction of elements under the influence of internal conditions of the system.

The model composes the probabilities of movement of representatives of one category of market participants to other groups, with the income from the implementation of strategies forming the functions of transitions. The dynamics of quotations depends on the ratios of demand of all three categories of market participants. This model implies that the firm market position can occur after the significant fluctuations when the number of followers of the technical analysis decreases and the number of supporters of the fundamental analysis increases in its turn. The Lux-Marchesi model is based on the stability of the number of shares available in the market.

Since the late 1980s, many scientists have sought to find a solution to the problem of predicting the dynamics of financial instruments in stock markets. In spite of this and in spite of the large number of works on this subject, very few real projects that use the multi-agent approach in market modeling have been found.

In fact, there is no single clearly defined approach in any of the types of models and methods analyzed. But it is also inefficient to design and create multi-agent models for every specific task. The reason for this is that the model being presented may be governed by a number of heuristics that cannot be formally justified, nor allowed to be validated until the simulation run stage. In the course of the simulation process, inference methods, actions, and the system of mutual information exchange between agents can be refined on a regular basis.

A solution to this problem could be the development of simulation modeling. However, this goal is not unreasonably considered to be even more complex and costly, in particular in the framework of agent-based modeling.

According to some authors, optimization methods, graph theory and systems based on neural networks can still be considered quite effective. But the same neural network aspect cannot be considered fully optimal and take into account all necessary factors.

As a result, the existing solutions indicate that this problem is either partially solved, or there is no practical implementation. There is an obvious need to use neural network technology to model complex strategies of individual agents, to eliminate pseudo-random, manipulative events and, as a consequence, to increase the predictive ability.

The strengths and weaknesses of each of the models under consideration have been analyzed above. It was noted that at the current stage of development of stock dynamics estimation and forecasting the most widely used methods are neural network models (ANN), as well as autoregressive models (ARIMA).