Альтернативная мера оценки рисков для финансовых рядов с помощью эффекта'Северо-Восточного ветра волатильности'
Пучков Андрей Александрович,
магистр экономики, докторант Рижского технического университета.
Economist, Mg. oec., PhD student of Riga Technical University.
This paper illustrates alternative indicator for risk estimation. This indicator is beneficial for financial market instability prediction on early stages. Alternative risk measure is based on North-East Volatility Wind Effect and its helical (spiral) structure investigation in frequency domain. North-East Volatility Wind effect is described in [6-10]. but helical structure exploration is made in article “North-East Volatility Wind Effect in Traveling Wave Perspective. Helical Structure of Volatility wave Fourier Coefficients” which is expected publication in current journal.
Fig. 1. Volatility pie function representation in frequency domain. Evolution in time.
In this article “North-East Volatility Wind Effect in Traveling Wave Perspective. Helical Structure of Volatility wave Fourier Coefficients” evolution of 'Volatility pie' function in a frequency domain is considered. The same picture is obtained for function. Results are represented next.
Fig. 2. Volatility pie function representation in frequency domain. Evolution in time.
According to results volatility pie in frequency domain anddid not have clear structure. The same conclusions are done about function.But in time (during financial crisis 2007 peak) the picture has been changing and functions aandacquired clear helical (spiral) structure. This effect is explored by less number of explicit harmonics (harmonics with higher magnitude).  After market crisis 2007, clear clear helical (spiral) structure was lost.
Since Fourier transform is done for investigated volatility wave which is fixed in time , obtaining (while ), helical structure 'regularity' can be measured as a minimal total distance between Fourier coefficients (from spiral), since is a three dimensional function.
Minimal total distance is calculated as a total distance, which includes (spiral) chain of all Fourier coefficients, which belongs to . Next Fourier coefficient is included in chain, if it is located in minimal distance to previous Fourier coefficient. Procedure is repeated until the last Fourier coefficient is included in spiral chain.
This procedure could be simplified by (2) equation.
The smaller distance , more regularity in helical (spiral) structure. Acquired measure is calculated for all time variables , obtaining function which is used as alternative risk measure.
In the next figure calculation results of alternative risk measure are shown comparing with volatility measure (logarithmic variance indicator). Results are illustrated on the Dow Jones Industrial Index data.
Fig. 3 Volatility indicator and alternative risk indicator.
The dynamics of analyzed indicators are visually very similar, but correlation in fact is not high (30%).
The main question to be answered how alternative risk indicator could be used for volatility forecasting. In order to answer this question consider early forecasting task by using alternative risk indicator . Future volatility indicator is forecaster in 100 time units (about~100 weeks) by using Alternative risk indicator in past. For forecasting NARX (Nonlinear AutoRegressive with External Input) neural network with following architecture.
Fig. 4. NARX Neural Network architecture.
This network is trained on 70% of data, validated on 15% and tested on 15% of data. After 7-th epoch, the best best result is obtained on validation set. Trained Network provides output with following dynamics.
Fig. 5. NARX Neural Network output dynamics.
Output provided by network is highly correlated with Target. But results could be improved. implementing input data decomposition. Overall forecasting results are not bad. Output-Target regression data are shown in next figure.
Fig. 6. NARX Neural Network output - target regression.
The following R/MSE results are acquired:
Fig. 7. NARX Neural Network result estimation.
For estimated Neural Network, Target and Output correlation on Test, Validation and Testing observations are about 90%, MSE indicator is small. According research Alternative risk indicator obtained from distance between 'helical' Fourier coefficients could be used for Volatility indicator forecasting for about 100 weeks, but algorithm modification could be used for better forecast.
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Поступила в редакцию 11.07.2014 г.