Instantaneous Frequency to Stock Returns in a Financial Crisis
Tsair-chuan Lin
Department of Statistics, National Taipei University
Amanda S. M. Chang
Center of General Education, Chang Gung University
Abstract
Most of the classical time series analyses require the time series to be stationary and/or linear. However, financial time series are usually nonlinear and nonstationary. In this study,
we use the Hilbert-Huang Transformation (HHT) to investigate the dairy return of several international stock markets from 2005 to 2010. The HHT approach mainly consists of application of
the empirical mode decomposition (EMD) and the instantaneous frequency analysis. We find that there is an obvious change in the behavior of the trading activity among these stock markets
since the U.S. sub-prime mortgage credit crunch in 2008.
Keywords: instantaneous frequency, empirical mode decomposition, HHT.
INTRODUCTION
4
3.5
x 10
twi
sti
sp
nasdq
hsi
ftse
cac
dax
3
2.5
stock price
Stock markets are complex dynamical systems whose elements are investors and traders with
varying capital sizes using different investment and trading strategies. For the last two decades
the international finance has higher correlation because of the internationalization. For example:
Exchange rate and interest rate, economic policy adjustment, international main stock market's
change, petroleum price fluctuation, financial crisis, war or other big events. This study tries to
address the question: “Is there a high similarity in the trading activities of different markets
when encounter a financial crisis?” To resolve these problems, we use the instantaneous angle
and instantaneous frequency, developed by Huang et al. [2], to examine whether there is any
similarity in time evolution of prices in the two segments and any change in similarity between
two indexes before and after a financial crisis. This method was proposed based on nonlinear or
nonstationary concern and has been designed to extract dynamic information from time series.
FIGURES
2
1.5
Fig. 1. The dairy series of the TWI,
STI, SP, NASDQ, HSI, FTSE, CAC,
and DAX indexes from 2005 to 2010.
1
0.5
0
2005/01
HILBERT-HUANG TRANSFOMTION METHOD
2005/11
2006/09
2007/07
date
2008/06
2009/05
2010/03
0.2
Rt
The algorithm to create IMFs in the EMD has two main steps. After repeated the sifting
process up to k times until some acceptable tolerance is reached:
C1
0.1
C3
0
h1k (t)  h1(k-1) (t)- m1k (t)
If the resulting time series is the first IMF, it is designated as c1  h1k (t). Finally, we get
x(t)  i 1 c i (t)  rn (t)
-0.1
ri (t)  ri-1 (t)- c i (t)
n
Fig. 2 The application of EMD for
the NASDAQ index. The return
series is decomposed into 9 IMFs
and 1 residue; here, only the first 3
components are displayed.
-0.2
For the k-th IMF, this can be done by first calculating the conjugate pair of c k (t), i.e.,
-0.3

c (t')
y k (t)  P  k
dt',
 - t - t'
1
C2
where P indicates the Cauchy principal value. We define z k (t)  c k (t) iyk (t)  A k (t)exp(i k (t)).
Then, we can calculate the instantaneous phase according to the above equations.
-0.4
-0.5
2005/01
2005/11
2006/09
2007/07
2008/06
2009/05
2010/03
(a)
STOCK INDEXES
0.2
C1
0.15
P
In this study, we use several major world stock indexes from 2005 to 2010 for analysis. These
indexes include the TWI, HSI, FTSE, CAC, DAX, S&P, STI and NASDAQ. The data was
retrieved from the Yahoo database [3]. The international main stock prices in the last decade
were changing enormously. In the years 1999-2000, the NASDAQ vigorously exploded and
jumped up around 50% and bring the positively impact to the other international stock markets.
This rally was based on expected earnings of brand new internet and tech companies with no
track record. Promptly, the stock price declined even more powerful since the internet bubble.
The years 2007-2008 were another period of extremes. Financial companies found a way to
make money even easier - trading and selling financial derivatives. This finincial crises of subprime mortgage credit, causes the American housing market murky dispirited. The association
attacks the American economy, and has the negative chain effect to the global economic and
stocks.
C2
C3
0.1
0.05
0
-5
-4
-3
-2
-1
0

2
3
4
5
1
(b)
150
C1
C2
100
Figure 3. The distribution of (a)
phases and (b) amplitudes of the first
three IMFs of the NASDAQ index
return.
P
C3
50
0
0
0.005
0.01
0.015
0.02
0.025
A1
ANALYSIS OF STOCK INDEXES
In the stock time series analysis, the researchers usually use the logarithmic return scale for
application. The logarithmic return is simply defined by R(t)  ln(yt ) - ln(yt-1 ) ,where y t stands for
the stock price at time t. Fig. 2 gives the logarithmic return of NASDAQ index and the first 3
IMFs obtained through the EMD method result in decomposition of 9 components and 1 residue.
Therefore, the further application of the phase statistics approach is based on analysis on the
first IMF. Consequently, we take an illustration with the NASDAQ index, the probability
density functions of the amplitude for its first-IMF is roughly Boltzmann distributed. Beside this,
we find from Fig. 3 that except for the first IMF, the phases of the other IMFs have almost equal
probability for -      .
To investigate the correlative behaviors between any two different stock markets, we pair-wisely
calculate the instantaneous phases between two indexes. Here, the phase differences 1 (i, j) is
defined as 1 (i, j)  1 (Stocki ) - 1 (Stockj ).
Fig. 4 gives the probability distributions of phase differences between the first-IMFs of returns
for two indexes for 2005–2010. We summarize some finding based various market regions and
certain periods. (1) The third row plots provide the angle difference distribution of S&P against
the other indices. It can be seen, when subtracting against those of TWI, STI, or HSI, each
distribution all have positive mod. Surprisingly, similar results are displayed in the fourth row
pots. That indicates that the stock activity of American market have the leading impact to the
Asia stock indexes. (2) The phase difference distribution between the S&P and the NASDAQ has
high kurtosis and narrow range. Hence these two markets have high correlated activities in the
whole period. More specifically, the (3,4)-th and the (4,3)-th subplots show that the more
correlative activities occurred after the sub-prime mortgage credit crunch in 2008. (3) Such
correlative activities can also be found among the FTSE, the CAC, and the DAX. (4) From the
first, the second and the fifth rows, one can see that each of the Asia indexes verse the European
indexes, all have low Kurtosis, and hence a relatively less correlated activities between these two
Continents.
From above observation, the universal existence of regional correlation among international
stock markets, hence we take only TWI and NASDAQ as benchmarks for the following event
investigation. We summarize the main result as follows: (1) The first row of Fig. 4 shows the
TWI and the American stock markets have higher correlative after the sub-prime mortgage
credit crunch in 2008. However, the TWI verse the other Asia stock markets or the TWI verse
the European markets have higher correlative during 2007b-2008a rather than the period
2008b-2009a. (2) For the NASDAQ index verse the other international indexes, the correlations
have remarkable change during 2007b-2008a and elevate continually after 2010.
1
TWI
STI
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CAC
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CAC
FTSE
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HSI
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FTSE
Nasdaq
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S&P
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S&P
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2005*-2006a
1
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2006b-2007a
2007b*-2008a
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2008b*-2009a
2009b-2010
Fig. 4. The array plot of the probability distribution for. The indexes i and j are labeled by the
order: 1(TWI), 2(STI), 3(SP), 4(NASDQ), 5(HSI), 6(FTSE), 7(CAC), and 8(DAX).
REFERENCE
1.
2.
3.
M.-C. Wu and C.-K. Hu, Phys. Rev. E 73, 051917 (2006).
N. E. Huang, Z. Shen, S. R. Long, M. C. Wu, H. H. Shih, Q. Zheng, N.-C.
Yen, C.-C. Tung and H. H. Liu, Proc. R. Soc. Lond. A 454, 903 (1998).
Data sets are available from http://finance.yahoo.com/