ESTIMATION OF PARAMETERS AND VERIFICATION OF STATISTICAL HYPOTHESES FOR GAUSSIAN MODELS OF STOCK PRICE

We construct models of asset prices on the Ukrainian stock market and analyse their applicability by checking appropriate statistical hypotheses using actual observed data. We also analyse the presence of jumps in the dynamics of different assets and estimate the Hurst coefficient for the logarithm of the price of the asset by two different methods.

We analyse data from the Ukrainian Stock Exchange, which focuses on the classical model of trading stocks.Since 2009, the Ukrainian Stock Exchange trades on all major types of securities (stocks, bonds, options, futures) and trading volumes have been steadily increasing.In this paper we investigate methods of parameter estimation and verification of statistical hypotheses for a Gaussian model of the Ukrainian Stock Exchange Index, which is calculated as the average price of 10 Ukrainian "blue chip" stocks (stocks of Ukraine's largest companies and leaders in their fields).For technical analysis we use data for received from the website of the Ukrainian Stock Exchange [3] for the period from 31 March 2014 to 28 August 2014 with observations taken every 5 minutes (a total of 8000 observations).A short description of the analysed data is presented in Table 1.Estimation of parameters and verification of statistical hypotheses for Gaussian models of stock price The following figure shows dynamics of the analysed asset with respect to its value at the beginning of the period analysed.

Analysis of the presence of jumps in the dynamics of the Ukrainian Stock Exchange Index
Financial markets sometimes generate a significant number of gaps in the price of certain shares, so-called jumps.Many practical and theoretical studies prove the existence of jumps and show their significant impact on financial management (in particular, risk-management and hedging of the portfolios of securities).The problem of jump identification is quite complex, because only discrete data are available to researchers.Recently, a variety of techniques and statistical tests, which allow the determination of jumps in the dynamics of stock prices, was developed.Some of them we apply to the assets that we investigate.The question of the presence of jumps is essential for the purposes of analysis of stock dynamics as they can significantly distort the results.
We build a model of the stock price on a fixed probability space ( ) -corresponds to the information that is available to market participants.If there are no jumps in the dynamic of stock price ( ) S t , we suggest it can be described by the standard lognormal model: where ( ) where ( ) J t is a counting process independent of ( ) W t , which is responsible for the appearance of jumps.
( ) Y t determines the size of jump at time t and does not depend on the previous jumps or ( ) is constant for all k .
We consider local dynamics of the process within the "window" of K consecutive observations that are used to determine the statistics defined below [1] provides proof of the following theorem, which allows us to build a statistical test for identification of jumps in the stock price dynamics by discrete observations.
where ξ has the distribution function ( ) ( )  otherwise we accept the alternative hypothesis.The parameters used for the purposes of our analysis of jumps in the stock dynamics are given in Table 2.
After all the necessary calculations, according to the test, we can see that there are jumps in dynamic of the Ukrainian Stock Exchange Index.More precisely, 49 jumps took place during the period analysed.As mentioned earlier, jumps can have a significant impact on the results of the analysis of the dynamics of the Ukrainian Stock Exchange Index.Thus, for the purposes of further analysis, corresponding values were excluded from the stock dynamics.The following figure shows the dynamics of the Ukrainian Stock Exchange Index during the analysed period after the exclusion of the jumps.

Fractional Brownian motion and the analysed model
As part of this paper we consider a stochastic model of the Ukrainian Stock Exchange Index based on fractional Brownian motion, which is described in detail below.
Fractional Brownian motion is a generalization of the Wiener process.However, unlike the case of the Wiener process, the increments of Fractional Brownian motion are correlated (and therefore dependent).Processes of this type were firstly considered in the work of Mandelbrot and Van Ness in 1968.
For fractional Brownian motion it is possible to define stochastic integrals, also known as fractional stochastic integrals.
We consider the following lognormal model of the Ukrainian Stock Exchange Index:

Definition of the Hurst coefficient and its evaluation using scalable rank
The Hurst coefficient is a measure of long-term memory, which is used in the analysis of time series.Below we provide a precise definition.Definition 3. The Hurst coefficient H is defined in terms of the asymptotic behavior of the rescaled range as a function of the time span of a time series as follows: where ( ) R n is the range and ( ) , where average.Then the estimator of the rescaled range can be found as Now an estimator of the Hurst coefficient can be found by applying a regression analysis to the following linear model where C is some constant.
We apply this method to actual data.Let { } be a time series which represents stock price at moments k t .Then the input data for the purposes of analysis is an auxiliary time series , which reflects the dynamics of changes in profitability in the lognormal model.Using the R software environment we made the necessary calculations and the conducted regression analysis.A scatter plot of corresponding regression model is presented below.Thus we see that the Hurst coefficient for the observed values of the Ukrainian Stock Exchange Index is greater than 1/2, indicating a positive long-term correlation between increments of Ukrainian Stock Exchange Index.Thus, after high values Ukrainian Stock Exchange Index increases are more likely, and vice versa.
This can be partially explained by influence on the stock market of various macroeconomic factors and the overall economic situation (we observe a serious long-term decline since 2008, resulting from the long-term global economic crisis).
However, given the fact that we consider a Ukrainian Stock Exchange Index with a high frequency of observations (in our case -every 5 minutes), we can see that the above factors have much less impact, and shortterm random factors have much more significant effect.

Estimation of the Hurst coefficient in the analysed model using quadratic variations
For the analysed model the Hurst coefficient can be estimated using the property of self-similarity of fractional Brownian motion.For this purpose we use the following theorem, which is a consequence of results of Dozzi, Mishura and Shevchenko [5].
Theorem 2. Let ( ) B t be a fractional Brownian motion with Hurst coefficient H that is observed at moments , 0, , Based on this theorem we can prove the following proposition.
By the definition of process ( ) Where in the last inequality we used property (4) of fractional Brownian motion from Proposition 1.It follows from Theorem 2 that the last sum is The first two sums go to zero faster than allows us to build a simple estimator for the Hurst coefficient H , which is derived from observations of process ( ) and based on the ratio of quadratic variations

H
Obviously the consistency of this estimator is a direct consequence of Proposition 1.
We also applied this estimation procedure to the actual values of the Ukrainian Stock Exchange Index for the period analysed, and determined that the estimate of the Hurst parameter is 2 0.5236 ˆ212.

H =
The two different estimation methods that we considered give almost identical results.For further analysis we use estimator 2 Ĥ .Now we can consider the problem of testing the hypothesis of whether the observed dynamics of the Ukrainian Stock Exchange Index correspond to the analysed model.

Presentation of the Wiener process as a fractional stochastic integral
Fractional Brownian motion can be presented in the form of a stochastic integral from a Wiener process and vice versa.In particular, in [4] Ilkka Norros, Esko Valkeila and Jorma Virtamo considered kernel, fractional stochastic integrals which make it possible to convert fractional Brownian motion into a Wiener process.This transformation is described in detail below.
Consider the following function ( The following theorems are proved in [4]  ( ) and further the transformation of the obtained time series: ( ) By Theorem 4, for sufficiently large N , the corresponding sums converge to integral transformations and , 1, , where W is a Wiener process.Thus, when the null hypothesis holds, the following regression model is a standard Gaussian regression: , 0, , 1 where i ε is a sequence of independent normal variables.
100 Estimation of parameters and verification of statistical hypotheses for Gaussian models of stock price Thus in order to confirm the null hypothesis, it is sufficient and necessary to confirm the hypothesis of normality of residuals in the above-mentioned regression model.For these purposes, we will use the Jarque-Bera test (see [6]- [7]).Note that as it was shown in the previous section, if the Jarque-Bera test shows that the residuals are normally distributed, then ( ) Y t satisfies the analysed model.Thus, for a large number of observations the probabilities of Type I and Type II errors for our test coincide with the corresponding probabilities of the Jarque-Bera test for residuals.After conducting all necessary transformations, we had 20 observations of i   .The actual value of the statistic is less than the threshold value, and we accept the null hypothesis.Thus we can conclude that the proposed model can be used to describe the dynamics of assets on the Ukrainian Stock Exchange, at least in the short term perspective.

Conclusions
The problem of researching the Ukrainian stock market has recently become more important and urgent.Compared to the stock markets of the economically developed countries of Europe and America, where the corresponding mathematical apparatus has been developing since the middle of the twentieth century, the Ukrainian market is comparatively young.The purpose of this paper is to make the first steps towards conducting the necessary analysis.
In this paper we analysed the presence of jumps in the dynamic of assets on the stock market and statistically confirmed their existence.The results are important and can be used for a wide range of purposes.In particular, it is necessary to consider the effect of jumps during the calculation of the fair price of derivatives on the assets of the Ukrainian Stock Exchange.For the purposes of further analysis, jumps were excluded from consideration.
Also in this study we evaluated the Hurst coefficient for the logarithm of the price of the analysed assets by two different methods.In particular, we showed that the assets on the market might have long-term memory, but given a large number of random factors affecting the value of an asset at every moment, the estimated value of the Hurst coefficient appears to be close enough to 1 2 .The problem considered in this paper is extremely relevant to modern Ukrainian realities, because a developed and open stock market is one of the necessary factors for the economic development of a country.The results of this work form a basis for further analysis.

Figure 1 .
Figure 1.Dynamics of Ukrainian Stock Exchange Index during the analysed period we accept the null hypothesis,

Figure 2 .
Figure 2. The histogram of time gaps between two consecutive jumps in the dynamics of the Ukrainian Stock Exchange

Figure 3 .
Figure 3.The dynamics of Ukrainian Stock Exchange index during the period analysed, with jumps removed.
n is the variance of the first n observations.This definition allows us to build a simple estimation procedure for the Hurst coefficient based on the observed values of the time series { } it is necessary to calculate rescaled range of the time series{ the auxiliary time series

L
and i l and built the corresponding regression model.In order to apply the Jarque-Bera test, we calculated the value of the Jarque-Bera statistic for the number of observations 1 S and K are the coefficients of skewness and kurtosis of of the residuals of the regression model.A Histogram of the residuals of the regression model is presented in the following figure.

Figure 5 .
Figure 5. Histogram of the residuals in the analysed regression model

Table 2 .
Parameters for analysis of jumps in the stock dynamics

where ( )
Y t is the value of Ukrainian Stock Exchange Index at moment t ; is fractional Brownian motion with a Hurst coefficient H ; μ and σ are coefficients of drift and volatility respectively.First, we consider the problem of estimation of the Hurst coefficient H by discrete observations of process H B t

Proposition 2 .
Let the process ( ) is a fractional Brownian motion with Hurst coefficient H , μ and σ are constant, and this H B t . be a fractional Brownian motion with Hurst coefficient H . Then the centered H B twhere ( )

. Statistical test of correspondence to the analysed model
particular t M is a martingale.Thus, the fractional Brownian motion can be transformed into Wiener process through simple integral transformation.Note that this property is characteristic for fractional Brownian motion.Thus, if the process t is fractional Brownian motion.This feature allows to build a test that verifies whether a time series can be described by the analysed lognormal model.Our task is to build a test based on the observations of time series is a fractional Brownian motion with Hurst coefficient H , and μ and σ are the coefficients of drift and volatility, respectively.
H B t H B t