ISSN 1991-3087     НА ГЛАВНУЮ Стохастический процесс, основанный на эффекте «Северо-Восточного ветра волатильности»   Пучков Андрей Александрович, магистр экономики, докторант Рижского технического университета.   Stochastic process, based on North-East Volatility Wind Effect   Andrejs Puchkovs, Economist, Mg. oec., PhD student of Riga Technical University.   This article describes volatility evolution of a stochastic process. Evolution of volatility is affected by 'North-East Volatility Wind Effect' described in previous papers. 'North-East Volatility Wind Effect' is describing behavior of stock market indices. An exposure of this allows to predict stock indices volatility bursts [7, 8]. Analysis of 'North-East Volatility Wind Effect' is based on signal decomposition by using the wavelet filtering. Wavelet filtering is applied by using Direct and Inverse CWT for each scaling parameter [2, 3]. Thus for each scaling parameter the signal component (which is part of the original signal) is obtained. For subsequent research volatility indicator is analyzed by using 20-days time window, which is shifted on the time axis. Volatility analysis is done for each signal component. As a result volatility evolution in time is obtained for each signal component. According to research, a slight increase in volatility in the low-frequency components of the signal leads to significant disturbances in high-frequency components destine entire signal volatility growth. In current paper ’North-East Volatility Wind Effect' is formalized in a form of stochastic process, which volatility is evolving in time. Volatility is decomposed in parts or layers. Complicated volatility evolution is happening in each volatility layer. Volatility layers have complicated interrelationships and volatility is transmitted from layer to layer as it is happening in stock markets. The main object to be described in paper is wave nature of volatility.             1. Stochastic process A stochastic process is defined in time is started from zero , has evolving variance in time, which can be scripted according (1) equation. (1) where is a logarithmic volatility indicator, which is decomposed in parts, or so called volatility layers, see (2) equation. (2) (3) where is a volatility layer or a volatility component. In case of integer volatility layers, the formula could be accepted by (3) equation. Volatility transmission from volatility layer in time (or simply ) to volatility layer in time (or simply, additional parameter denotes where volatility was transmitted from) is defined in (4) equation. (4) where is a interlayer volatility gradient, which indicates how much energy or volatility was transmitted from volatility layer (in time ) to volatility layer (in time ). Consequently volatility from all layers transmitted to is calculated by (5) equation. (5) For illustration see 1. Figure. Fig. 1. Transmission of volatility between layers.   Volatility layers have complicated relationships, taking and transmitting volatility from one layer to another. The nature of volatility is determined by interlayer volatility gradient.   2. Interlayer volatility gradient Suppose, Interlayer Volatility Gradient depends on one parameter, which is called interlayer radius. Interlayer radius between layer and is calculated by (6) equation: [4,5] (6) Interlayer Volatility Gradient is time dependent, and has a wave nature. For example it can be calculated by (7) equation. (7) where: - Volatility Wave; - is a constant, ; - Magnitude (amplitude) of Volatility Wave; - frequency of Volatility Wave; - is (angle) shifting parameter; - is (infinite-) small number (usually ); - running index of harmonics; - argument of power fuction (usually );   Harmonics should approximation volatility wave in time . Volatility wave defines Interlayer Volatility Gradient, which defines behavior of stochastic process. According (7) equation, Volatility wave is evolving in time and, evolution is made by harmonics argument.   3. The nature of volatility wave Consider the simplest case of volatility wave , which has only one evolving parameter: . In current example, other parameters are fixed in time: The number of harmonics is equal to one . In such case (7) equation can be rewritten. (8) In order to understand the nature of last expression, it is considered by parts. For example, consider sine wave with following parameters: The main part of expression is evolving wave , see the next figure 2. Fig. 2. Evolution of sinusoid in time.   Since is dependent from and it is positive , is rising in time and evolution is moving the sinusoid towards zero. Such effect is called 'North-East Volatility Wind' effect, which is explained by volatility transmission from low-frequency components to high-frequency components. Opposite effect is called 'South-West Volatility Wind' effect. Logarithmic sine wave , has a form presented in next figure. Fig. 3. Evolution of logarithmic sine wave in time.   Transformation of logarithmic sine wave in form of is presented in the next figure. An addition of in equation is explained by effect, which take place in a stock markets. In example parameters are selected. Fig. 4. Evolution of transformed logarithmic sine wave in time.   In a complex plane, where Volatility Wave plot has a form of radar, since . Fig. 5. Volatility Wave in a complex plain.   4. Modifications of process The process with volatility evolution may have different modifications, for example interlayer radius, which is calculated in (6) equation may have following modifications: (9) This modification is called 'diagonal case', while in this case only central diagonal x+y=1 affects V(t) (logarithmic volatility indicator), other arguments are transmitting volatility between layers. 'Two pole' modification is calculated by (11) equation. (10) where: - the 'gravity' of poles 1 and two. Two pole model is based on following property: by increasing interlayer radius , Volatility Wave W(r, t) decreases. Both modifications ('two pole' and 'diagonal') are used to make mathematical model more realistic and applicable for stock markets.   5. Computational part In current part of paper, computational algorithm for volatility wind calculation is provided. Computations are realized in MATLAB environment. See the code provided:   clc; clear all; close all;   x = 0:0.01:1; n = length(x); y = x';   X = repmat(x',1,n); Y = repmat(y',n,1);   B =((1-X)+Y<=1);   Z1 = exp((X + Y*1i)/2*pi); Z2 = exp((X*1i +Y)/2*pi);   Z = B.*Z1 + ~B.*Z2;   Xm = real(Z); Ym = imag(Y); R = sqrt(Xm.^2 + Ym.^2); %R = R/max(max(R));   %Define values of Volatility Wave M = 1; c = 2; omega = 2*pi*1; k = 1; d = 1.5; ep = 0.1;   [n,m] = size(R); T = 100; VWM = zeros(n,m,T);   %Compute the dynamics for t = 1:T  VWM(:,:,t) =-c*log(M*sin(omega*R+k*t)+1)+(R+ep).^-d;    figure(1)  mesh(X,Y,VWM(:,:,t))  axis tight  title('Volatility Wave - transmission of volatility by layers')  xlabel('Layer x');  ylabel('Layer y');  pause(0.1)    %Volatility wave for each layer  VW.a(:,t) = sum(VWM(:,:,t));    figure(2)  area(VW.a(:,t) )  title('Volatility Wave by layers')  xlabel('Volatility Layers a');  axis tight  pause(0.1)    VW.t(t) = sum(VW.a(:,t));   end   figure plot(((VW.t-std(VW.t))/mean(VW.t))) title('Volatility Wave in time') xlabel('Time t'); axis tight               Computational algorithm provides the following figures in output. Fig. 6. Transmission of Volatility by layers. Fig. 7. Volatility Wave dynamics by layers.               Proposed mathematical model brings the light on wave nature of volatility which is observed in stock markets. Proposed algorithms can be modified in order to provide very realistic models of stock price behavior.   Bibliography   1.                   Кроновер: Фракталы и хаос в динамических системах. Основы. теории. М: Постмаркет, 2000. — 352 с. – 293. стр. 2.                   Смоленцев: Основы теории вейвлетов в MATLAB. М: ДМК Пресс, 2003. - 304. стр. 3.                   Яковлев: Введение в вейвлет преобразования. Новосибирск, 2003. – 104 стр. 4.                   http://en.wikipedia.org: Complex number. [http://en.wikipedia.org/wiki/Complex_number] (Accessed 1 January. 2014.). 5.                   www.youtube.com Complex Number. [http://www.youtube.com/watch?v=b3adw5igSzI] (Accessed 1 January. 2014.). 6.                   www.youtube.com Complex Number Theory. [http://www.youtube.com/watch?v=b3adw5igSzI] (Accessed 1 January. 2014.). 7.                   PUCKOVS, A., MATVEJEVS, A.: Equity Indexes Analysis and Synthesis by using Wavelet Transforms. CFE 2013 7-th International Conference on Computational and Financial Econometrics, London: Birkbeck University of London, 2013, ISBN 978-84-937822-3-8, p. 98. 8.                   PUCKOVS, A., MATVEJEVS, A.:’North-East Volatility Wind’ Effect.No: 13th Conference on Applied Mathematics (APLIMAT 2014): Book of Abstracts: 13th Conference on Applied Mathematics (APLIMAT 2014), Slovakia, Bratislava, 4.-6. Feb., 2014. Bratislava: 2014, 67.-67.pp. ISBN 9788022741392.   Поступила в редакцию 15.05.2014 г.
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