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Multifractal Properties in China’s
Agricultural Futures MarketsFictions or Facts?
Shu-Peng CHEN, Ling-Yun HE
China Agricultural University
Aug. 13, 2010
1
Contents
 Background
•
•
Economic Markets
Agricultural Futures Markets
 Data Source
•
•
China Agricultural Futures Markets
Distribution of Returns
 Multifractals
•
•
Partition Function
Multifractal Detrended Fluctuation Analysis (MF-DFA)
 Sources of Multifractals
•
•
Shuffling Procedures
Phase Randomization Procedures
 Conclusion
2
Background
Long-range correlation
Fat-tails
Fundamental
Hypothesis
of traditional
theories
Normal
Distribution
Random Walk
Volatility clustering
Fractals/multifractals
Chaos
Rejected
Failed to explain
the market
phenomena
Economic Markets
B. B. Mandelbrot, (1967), The variation of the prices of cotton, wheat, and
railroad stocks, and of some financial rates, Journal of Business 40 393413.
E. E. Peters, (John Wiley & Sons, Inc., 1991), Chaos and Order in Capital
Markets: A New View of Cycles, Prices and Market Volatility.
J. Alvarez-Ramirez, M. Cisneros, C. Ibarra-Valdez, and A. Soriano, (2002),
Multifractal Hurst analysis of crude oil prices, Physica A: Statistical
Mechanics and its Applications 313, 651-670.
K. Matia, Y. Ashkenazy, and H. E. Stanley, (2003), Multifractal properties
of price fluctuations of stocks and commodities, Europhysics Letters 61,
422-428.
4
Economic Markets
G. Lim, S. Kim, H. Lee, K. Kim, and D.-I. Lee, (2007), Multifractal
detrended fluctuation analysis of derivative and spot markets, Physica A:
Statistical Mechanics and its Applications 386, 259-266.
M. Corazza, A. G. Malliaris, and C. Nardelli, (1997), Searching for fractal
structure in agricultural futures markets, Journal of Futures Markets 17,
433 - 473.
A. Chatrath, B. Adrangi, and K. K. Dhanda, (2002), Are commodity prices
chaotic? , Agricultural Economics 27, 123–137.
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Agricultural Futures Markets
US Agricultural Futures Markets:
Corazza et al. (1997) investigated six representative US agricultural
futures markets and found the existence of fractals.
Chatrath et al. (2002) studied four futures markets as the representative
ones in US and found low-dimensional chaotic structures in those markets
China’s Agricultural Futures Markets:
1.
Are China’s Agricultural Futures Markets fractal?
2.
If fractals do exists, is it mon-fractal or multifractal?
3.
What are the nonlinear dynamical sources of multifractals/mon-fractals?
6
Contributions
1. we gave a piece of empirical evidence of the esistence
of fractal features;
2. we were the earliest scholars who investigated the
sources of the multifractals in China’s Agricultural
Futures Markets
Data Sources
Futures
Commodity
Period(mm-dd-yy) Observations
Markets
Exchange
Dec. 28th, 1993 to
Wheat
3280
ZCE
th
Mar. 12 , 2010
Jul. 17th, 2000 to
Soy Meal
2313
DCE
th
Mar. 12 , 2010
Mar. 15th, 2002 to
Soybean
1924
DCE
th
Mar. 12 , 2010
Sep. 22th, 2004 to
Corn
1325
DCE
th
Mar. 12 , 2010
8
All of our data are taken from Reuter© database
Data Sources
Summary Statistics
Mean
Std. Dev.
Skewness
Kurtosis
Jarque-Bera
Wheat
1491.6
239.94
0.1347
2.2074
95.776*
Soy Meal
2454.2
603.66
0.5884
2.7749
138.33*
Soybean
3192.3
756.04
0.8958
3.1522
259.18*
Corn
1530.0
213.93
-0.2941
1.8017
98.375*
Note: *means reject the null hypothesis that the sample comes from a normal distribution at the significance of 0.01.
ADF test
Varibles
Exogenous
Wheat
Soy Meal
Soybean
Corn
(c,t,0)1
(c,t,0)
(c,t,0)
(c,t,1)
ADF test
statistics
-2.5225
-2.3545
-2.1381
-2.0850
Critical values at 5%
level of significant
-3.4112
-3.4118
-3.4121
-3.4132
Conclusion
Non-stationary
Non-stationary
Non-stationary
Non-stationary
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Note: c stands for constant, t stands for linear trend, the number 0 stands for the estimated optimal lag length in ADF test.
Distribution of Returns
100
50
a Wheat
40
Probability density
80
Probability density
b
returns distribution
fitted normal distribution
Levy distribution
60
40
20
30
20
b Soy Meal
10
0
-.1
0.0
.1
.2
0
-.10
.3
-.05
returns
c
d
60
50
.05
.10
100
c Soybean
80
Probability density
Probability density
0.00
returns
40
30
20
60
40
d Corn
20
10
0
-.2
-.1
0.0
.1
returns
.2
0
-.10
-.05
0.00
.05
.10
returns
The estimated parameters of Levy distribution
Agricultural
Futures
Wheat
Soy Meal
Soybean
Corn
α
β
γ
μ
1.1898
1.3704
1.3956
1.3552
0.0462
-0.1120
0.0445
0.0770
0.00373392
0.00695781
0.00612666
0.00366281
-0.00006111
0.00069307
0.00037866
-0.00008803
Note: Shu-Peng Chen and Ling-Yun He, Physica A, 389 (7) (2010), 1434-1444.
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Partition Function
Linear, which is a piece of evidence
of power-law relationship or fractal
b
40
40
30
30
20
20
lnSq()
lnSq( )
a
10
10
0
0
-10
-10
-20
-20
-6
-5
-4
-3
-2
-1
a Wheat
b Soy Meal
-6
0
-5
-4 2D Graph
-3 1 -2
0
c Soybean
ln
ln
c
-1
d
40
40
d Corn
30
30
20
lnSq( )
lnSq( )
20
10
10
0
0
-10
-10
-20
-20
-30
-6
-5
-4
-3
ln
-2
-1
0
-6
-5
-4
-3
-2
-1
0
ln
Note: Shu-Peng Chen and Ling-Yun He, Physica A, 389 (7) (2010), 1434-1444.
11
Partition Function
a6
tk=1
4
tk=5
tk=63
tk=125
-2
tk=250
tk=5
2
tk=21
0
tk=125
-2
tk=250
-4
-4
-6
-6
-8
-8
a Wheat
b Soy Meal
-10
-10
-8
-6
-4
6
-2
q
0
2
4
6
d
tk=1
4
tk=5
2
tk=21
-10
tk=125
-2
tk=250
(q)
0
-8
-6
-4
-2
q
6
tk=1
4
tk=5
tk=125
-2
tk=250
-4
-6
-6
-8
-8
2
4
6
c Soybean
d Corn
tk=63
0
-4
0
tk=21
2
tk=63
(q)
tk=1
4
tk=63
(q)
0
-10
c
6
tk=21
2
(q)
b
-10
-10
-10
-8
-6
-4
-2
q
0
2
4
6
-10
-8
-6
-4
-2
0
2
4
6
q
12
Note: Shu-Peng Chen and Ling-Yun He, Physica A, 389 (7) (2010), 1434-1444.
Partition Function
a 1.2
b 1.2
tk=1
tk=5
1.0
tk=250
tk=125
f()
tk=125
f()
a Wheat
tk=63
.8
tk=250
.6
.4
b Soy Meal
.4
.2
.2
.6
.8
1.0

1.2
1.4
1.6
.6
.8
1.0
1.2
1.4
1.6
1.8

c 1.2
d 1.2
tk=1
tk=5
1.0
.8
tk=250
tk=63
.8
tk=125
f()
tk=125
.6
d Corn
tk=5
tk=21
tk=63
c Soybean
tk=1
1.0
tk=21
f()
tk=5
tk=21
tk=63
.6
   max   min
tk=1
1.0
tk=21
.8
Width, which is an estimate
of multifractal strength
tk=250
.6
.4
.4
.2
.2
0.0
.6
.8
1.0
1.2

1.4
1.6
1.8
.6
.8
1.0
1.2
1.4
1.6
1.8

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Note: Shu-Peng Chen and Ling-Yun He, Physica A, 389 (7) (2010), 1434-1444.
Multifractal-Detrended Fluctuatuion
Analysis(MF-DFA)
-2
b
-2
-3
-3
-4
-4
lnFq(s)
lnFq(s)
a
-5
a Wheat
-5
-6
-6
q=-5
-7
-7
q=-3
b Soy Meal
-8
-8
2
3
4
5
6
2
7
3
4
5
6
7
q=-1
lns
lns
c Soybean
q=1
c
d
-2
q=3
-4
-3
d Corn
q=5
lnFq(s)
-4
lnFq(s)
-3
-5
-5
-6
-6
-7
-7
-8
-8
2.0
2.5
3.0
3.5
4.0
lns
4.5
5.0
5.5
6.0
2.0
2.5
3.0
3.5
4.0
4.5
5.0
5.5
lns
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Note: Ling-Yun He and Shu-Peng Chen, Physica A, 389 (18) (2010), 3828-3836.
MF-DFA
1.2
Wheat
Soy Meal
Soybean
Corn
1.1
1.0
h(q)
.9
.8
.7
.6
.5
.4
-6
-4
-2
0
2
4
6
q
15
Note: Ling-Yun He and Shu-Peng Chen, Physica A, 389 (18) (2010), 3828-3836.
MF-DFA
4
Wheat
Soy Meal
Soybean
Corn
2
(q)
0
-2
-4
-6
-8
-6
-4
-2
0
2
4
6
q
16
Note: Ling-Yun He and Shu-Peng Chen, Physica A, 389 (18) (2010), 3828-3836.
MF-DFA
1.2
Wheat
Soy Meal
Soybean
Corn
1.0
f()
.8
.6
.4
.2
.2
.4
.6
.8
1.0
1.2
1.4

17
Note: Ling-Yun He and Shu-Peng Chen, Physica A, 389 (18) (2010), 3828-3836.
Sources of Multifractals
 Two major sources of multifractals
• Long-range temporal correlation for small and large fluctuations
(K. Matia, et. al., Europhysics Letters, 61 (3) (2003), 422-428.)
• Non-Gaussian probability distribution of increments
(J. Kwapien et. al., Physica A, 350 (2-4) (2005), 466-474.)
 Two procedures to identify
•
•
Shuffling procedures
(K. Matia, et. al., Europhysics Letters, 61 (3) (2003), 422-428.)
• Phase Randomization Procedures
(M. Small and C. K. Tse, IEEE Transactions on Circuits and Systems. 1,
Fundamental Theory and Applications, 50 (5) (2003).)
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Sources of Multifractals
 Shuffling Procedures
•
•
•
Step 1: generating pairs (m, n) of random integer numbers, which
satisfies m, n≤N, where N is the length of the time series to be
shuffled ;
Step 2: interchanging entries m and n of the time series;
Step 3: repeating the first and second steps for 20N times. It is critical
to ensure that ordering of entries in the time series is fully shuffled,
thus the long-range or short-range memories, if any, will be destroyed.
The shuffling is repeated with different random seeds to avoid the
systematic errors caused by random number generators.
 Phase Randomization Procedures
•
•
•
Step 1: Taking the discrete Fourier transform of the time series;
Step 2: Shuffling the phases of the complex conjugate pairs;
Step 3: Taking the inverse Fourier transformation.
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Sources of Multifractals
The multifractal properties after the procedures
Procedures
Shuffling
Procedures
Phase
Randomization
Procedures
After Procedures
Conclusion
Remain the same
Not come from long-range
correlation
Disappear
Complete come from longrange correlation
Exist, but strength
become weaker
Mainly come from long-range
correlation, but also be
influenced by other factors
Remain the same
Not come from non-Gaussian
distribution
Disappear
Complete come from nonGaussian distribution
Exist, but strength
become weaker
Mainly come from nonGaussian distribution, but also
be influenced by other factors
Sources of Multifractals
Original
Wheat
Shuffled
Soy Meal Soybean
Corn
Surrogate
Wheat Soy Meal Soybean Corn Wheat Soy Meal Soybean
Corn
m=1
h(2)
0.7919
0.7978
0.8049 0.7641
0.4859
0.5197
0.4671 0.4439 0.7550
0.7392
0.7198
0.7177
h(-5)
1.1777
0.8881
1.0293
0.9614
0.7121
0.6242
0.6857 0.6526 0.7640
0.7106
0.7531
0.7734
h(5)
0.6396
0.7430
0.6598
0.6383
0.2783
0.5064
0.3422 0.3646 0.7552
0.7430
0.7389
0.7117
⊿α
m=2
0.8121
0.2425
0.5989
0.5526
0.6721
0.2195
0.5660 0.4994 0.0156
0.0702
0.0277
0.1183
h(2)
0.7632
0.7011
0.6721
0.7148
0.5065
0.5150
0.4853 0.5025 0.6999
0.6621
0.6590
0.6899
H(-5)
1.1424
0.8592
0.9258
0.9450
0.7273
0.6160
0.6947 0.7139 0.7260
0.6890
0.6903
0.7566
h(5)
0.6287
0.6658
0.4943
0.5948
0.2786
0.4944
0.3591 0.4224 0.6860
0.6544
0.6635
0.6799
⊿α
0.7855
0.3320
0.6824
0.5884
0.6897
0.2288
0.5507 0.5064 0.0787
0.0677
0.0487
0.1525
h(2)
0.7306
0.6733
0.6390
0.6948
0.5303
0.5024
0.5004 0.5404 0.6862
0.6451
0.6250
0.6675
h(-5)
1.1137
0.8672
0.9199
0.9358
0.7486
0.6118
0.7163 0.7861 0.7059
0.6945
0.6819
0.7455
h(5)
0.5777
0.6299
0.4603
0.6018
0.3111
0.4807
0.3596 0.4627 0.6821
0.6378
0.6198
0.6539
⊿α
0.8138
0.4050
0.7121
0.5650
0.6736
0.2458
0.5775 0.5408 0.0459
0.1129
0.1219
0.1829
21
m=3
Note: Ling-Yun He and Shu-Peng Chen, Physica A, 389 (18) (2010), 3828-3836.
Conclusion
1. We found power-law auto-correlation in Agricultural Futures
Markets, which means a large increment of price change is more
likely to be followed by a large increment
2. All of the Hurst Exponents are greater than 0.5, which means
comparing with random walk, the price change show persistent
properties
3. We found nontrivial multifractal spectra, which is a piece of
empirical evidence of the existence of multifractal features;
4. Non-Gaussian distribution constitutes the major contribution in
multifractals and long-range correlation also have an effect on the
formation in the multifractals of the markets.
22
Important References
B. B. Mandelbrot, The variation of the prices of cotton, wheat, and railroad stocks, and of
some financial rates, Journal of Business 40 (1967), 393-413.
C. K. Peng, S. V. Buldyrev, S. Havlin, M. Simons, H. E. Stanley and A. L. Goldberger,
Mosaic organization of DNA nucleotides, Physical Review E 49 (2) (1994), 1685.
Edgar E. Peters, Chaos and Order in Capital Markets: A New View of Cycles, Prices and
Market Volatility, John Wiley & Sons, Inc. 1991.
Jan W. Kantelhardt, Stephan A. Zschiegner, Eva Koscielny-Bunde, Shlomo Havlin, Armin
Bunde and H. Eugene Stanley, Multifractal detrended fluctuation analysis of nonstationary
time series, Physica A 316 (1-4) (2002), 87-114.
Zhi-Qiang Jiang and Wei-Xing Zhou, Multifractal analysis of Chinese stock volatilities
based on the partition function approach, Physica A 387 (19-20) (2008), 4881-4888.
Acknowledgement
Thank you for your kindly attention!
Your valuable comments and
suggestions are greatly appreciated!
Email: [email protected]
[email protected]
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