December 4, 2014 16:25 WSPC/CRC 9.75 x 6.5 1560013

7th Jagna International Workshop (2014)International Journal of Modern Physics: Conference SeriesVol. 36 (2015) 1560013 (8 pages)c© The AuthorDOI: 10.1142/S2010194515600137

Measuring efficiency of international crude oil markets:A multifractality approach

H. M. Niere

Economics Department, Mindanao State University,Marawi City, 9700, Philippines

hmniere@gmail.comwww.msumain.edu.ph

Published 2 January 2015

The three major international crude oil markets are treated as complex systems and theirmultifractal properties are explored. The study covers daily prices of Brent crude, OPECreference basket and West Texas Intermediate (WTI) crude from January 2, 2003 toJanuary 2, 2014. A multifractal detrended fluctuation analysis (MFDFA) is employed toextract the generalized Hurst exponents in each of the time series. The generalized Hurstexponent is used to measure the degree of multifractality which in turn is used to quantifythe efficiency of the three international crude oil markets. To identify whether the sourceof multifractality is long-range correlations or broad fat-tail distributions, shuffled dataand surrogated data corresponding to each of the time series are generated. Shuffled dataare obtained by randomizing the order of the price returns data. This will destroy anylong-range correlation of the time series. Surrogated data is produced using the Fourier-Detrended Fluctuation Analysis (F-DFA). This is done by randomizing the phases ofthe price returns data in Fourier space. This will normalize the distribution of the timeseries. The study found that for the three crude oil markets, there is a strong dependenceof the generalized Hurst exponents with respect to the order of fluctuations. This showsthat the daily price time series of the markets under study have signs of multifractality.Using the degree of multifractality as a measure of efficiency, the results show that WTIis the most efficient while OPEC is the least efficient market. This implies that OPEChas the highest likelihood to be manipulated among the three markets. This reflects thefact that Brent and WTI is a very competitive market hence, it has a higher level ofcomplexity compared against OPEC, which has a large monopoly power. Comparingwith shuffled data and surrogated data, the findings suggest that for all the three crudeoil markets, the multifractality is mainly due to long-range correlations.

Keywords: Multifractality; Hurst exponents; oil markets; efficiency.

This is an Open Access article published by World Scientific Publishing Company. It is distributedunder the terms of the Creative Commons Attribution 3.0 (CC-BY) License. Further distributionof this work is permitted, provided the original work is properly cited.

1560013-1

Int.

J. M

od. P

hys.

Con

f. S

er. 2

015.

36. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by 2

02.9

0.12

9.13

4 on

01/

14/1

5. F

or p

erso

nal u

se o

nly.

December 4, 2014 16:25 WSPC/CRC 9.75 x 6.5 1560013

H. M. Niere

1. Introduction

Fractals as introduced by Mandelbrot1−2 describe geometric patterns with largedegree of self-similarities at all scales. The smaller piece of a pattern can be said tobe a reduced-form image of a larger piece. This characteristic is used to measurefractal dimensions as a fraction rather than an integer. Some examples of fractalshapes are rugged coastlines, mountain heights, cloud outlines, river tributaries, treebranches, blood vessels, cracks, wave turbulences and chaotic motions. However,there are self-similar patterns that involve multiple scaling rules which are notsufficiently described by a single fractal dimension but by a spectrum of fractaldimensions instead. Generalizing this single dimension into multiple dimensionsdifferentiates multifractal from fractals discussed earlier. To distinguish multifractalfrom single fractal, the term monofractal is used for single fractal in this paper.Among the natural systems that have been observed to have a multifractal propertyare earthquakes,3 heart rate variability4 and neural activities.5

Mandelbrot6 introduced multifractal models to study economic and financialtime series in order to address the shortcomings of traditional models such as frac-tional Brownian motion and GARCH processes which are not appropriate with thestylized facts of the said time series such as long-memory and fat-tails in volatil-ity. Further studies confirmed multifractality in stock market indices,7−16 foreignexchange rates17−20 and interest rates,21 to name a few. As a consequence, manystudies have now used the properties of multifractality in forecasting models.22−24

These models are at least as good as, and in some cases, perform better using out-of-sample forecast compared to traditional models. One added advantage of thesemodels is their being parsimonious.

This paper investigates the presence and compares the degree of multifractalityof the daily prices of crude oil of the three major international crude oil mar-kets namely the Brent crude, OPEC reference basket and West Texas Intermediate(WTI) crude from January 2, 2003 to January 2, 2014. The Brent crude is sourcedfrom the North Sea and is the main European oil market; OPEC is mainly sourcedfrom the Middle East; and WTI is the benchmark used in Chicago and New Yorkmercantile exchange. Furthermore, since multifractality can be due to long-rangecorrelations or due to broad fat-tail distributions, this paper identifies which of thetwo factors dominates the multifractality of the daily crude prices time series ofthe said markets. The paper is arranged as follows. Methodology is discussed inSection 2. Data are described in Section 3. Presentation of results is in Section 4.Finally, the paper concludes in Section 5.

2. Methodology

In measuring multifractality, the paper uses the method of Multifractal DetrendedFluctuation Analysis (MFDFA) as outlined in Kantelhardt et al.25 Matlab codesused are based in Ihlen.26 The procedure is summarized in the following steps.

1560013-2

Int.

J. M

od. P

hys.

Con

f. S

er. 2

015.

36. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by 2

02.9

0.12

9.13

4 on

01/

14/1

5. F

or p

erso

nal u

se o

nly.

December 4, 2014 16:25 WSPC/CRC 9.75 x 6.5 1560013

Measuring efficiency of international crude oil markets: A multifractality approach

(1) Given a time series ui, i = 1, . . . , N , where N is the length, create a profileY (k) =

∑ki=1 ui − u, k = 1, . . . , N , where u is the mean of u.

(2) Divide the profile Y (k) into Ns = N/s non-overlapping segment of length s.Since N is not generally a multiple of s, in order for the remainder part ofthe series to be included, this step is repeated starting at the end of the seriesmoving backwards. Thus, a total of 2Ns segments are produced.

(3) Generate Ys (i) = Ys [(v − 1) s + i] for each segment v = 1, . . . , Ns, and Ys (i) =Ys [N − (v − Ns) s + i] for each segment v = Ns + 1, . . . , 2Ns.

(4) Compute the variance of Ys (i) as F 2s (v) = 1

s

∑si=1 [Ys (i) − Yv (i)]2, where Yv (i)

is the mth order fitting polynomial in the vth segment.(5) Obtain the qth order fluctuation function by

Fq (s) =

{1

2Ns

2Ns∑v=1

[F 2

s (v)]q/2

}1/q

.

If the time series are long-range correlated then Fq (s) is distributed as powerlaws, Fq (s) ∼ sh(q). The exponent h (q) is called as the generalized Hurst exponent.When h (q) = 0.5, this implies that the fluctuations are just random walks.

For monofractals, the Hurst exponent is a constant equal to h (2). The closerthe value of h (2) to 0.5, the more closely the time series mimics random walk.Hence, market efficiency can be measured by the distance of h (2) from 0.5. Formultifractals however, h (q) varies with q. Thus, a spectrum of h (q) values impliesthe presence of multifractality.

The degree of multifractality can be quantified as |∆h| = h (qmin) − h (qmin).Moreover, the higher the degree of multifractality, the lower the market efficiency.23

To identify whether the multifractality is due to long-range correlations or is dueto broad fat-tail distributions, shuffled data and surrogated data are generated. Inthe spirit of Zunino et al.,11 100 different shuffled time series and surrogated timeseries are produced to reduce statistical errors. Shuffling the data will remove thelong-range correlation in the time series. It is done by randomizing the order of theoriginal data. The multifractality due to long-range correlation can be computed ashc = ∆h − ∆hf where the index f refers to shuffled data.

Surrogated data is produced by randomizing the phases of original data inFourier space. This will make the data to have normal distribution. The multifrac-tality due to broad fat-tail distributions can be measured as hd = ∆h−∆hr wherethe index r refers to surrogated data.

3. Data

The daily crude oil prices of the Brent crude, OPEC reference basket and WTIcrude from January 2, 2003 to January 2, 2014 are used for a total of 2788,2839 and 2765 observations respectively. The number of observations differs forthe three markets because the number of business trading days also differs due

1560013-3

Int.

J. M

od. P

hys.

Con

f. S

er. 2

015.

36. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by 2

02.9

0.12

9.13

4 on

01/

14/1

5. F

or p

erso

nal u

se o

nly.

December 4, 2014 16:25 WSPC/CRC 9.75 x 6.5 1560013

H. M. Niere

to national holidays and other idiosyncracies. Daily price data for OPEC ref-erence basket has been downloaded from the OPEC online database website:http://www.opec.org/opec web/en/data graphs/40.htm. The daily price data forBrent and WTI crude was downloaded from the website of the U.S. Energy Infor-mation Administration: http://www.eia.gov/dnav/pet/pet pri spt s1 d.htm.

4. Results

Figures 1 to 3 show the plots of the daily crude prices, the daily returns, andthe associated shuffled and surrogated time series of daily returns for Brent crude,OPEC reference basket and WTI crude respectively. The original daily returns andthe shuffled time series show some extreme fluctuations which is a sign of havingfat-tail distribution. The surrogated time series do not have extreme fluctuation, acharacteristic of a normal distribution.

In doing the MFDFA procedure, m = 3 is used as the order of polynomial fitin Step 3. The length s varies from 20 to N/4 with a step of 4 as suggested inKantelhardt et al.25 Finally, q runs from −10 to 10 with a step of 0.5. Figure 2presents the generalized Hurst exponents for the original returns, shuffled returnsand surrogated returns. For monofractals, the Hurst exponent is independent of q

which is also equal to the generalized Hurst exponents of multifractals atq = 2, that

Fig. 1. (Color online) Plots of the (a) daily Brent crude oil price, (b) its daily returns, (c) shuffledtime series, and (d) surrogated time series.

1560013-4

Int.

J. M

od. P

hys.

Con

f. S

er. 2

015.

36. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by 2

02.9

0.12

9.13

4 on

01/

14/1

5. F

or p

erso

nal u

se o

nly.

December 4, 2014 16:25 WSPC/CRC 9.75 x 6.5 1560013

Measuring efficiency of international crude oil markets: A multifractality approach

Fig. 2. (Color online) Plots of the (a) daily OPEC crude oil price, (b) its daily returns, (c) shuffledtime series, and (d) surrogated time series.

Fig. 3. (Color online) Plots of the (a) daily WTI crude oil price, (b) its daily returns, (c) shuffledtime series, and (d) surrogated time series.

1560013-5

Int.

J. M

od. P

hys.

Con

f. S

er. 2

015.

36. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by 2

02.9

0.12

9.13

4 on

01/

14/1

5. F

or p

erso

nal u

se o

nly.

December 4, 2014 16:25 WSPC/CRC 9.75 x 6.5 1560013

H. M. Niere

Fig. 4. (Color online) Generalized Hurst exponent, h(q), as a function of q for the original, shuffledand surrogated daily returns for (a) Brent crude, (b) OPEC reference basket, and (c) WTI crude.

is, h (2). In other words, monofractals have only one single Hurst exponent whichis h (2) regardless of the value of q. In contrast, multifractals have a spectrum ofgeneralized Hurst exponents which vary depending upon the value of q. It is notedin Figure 2 that for the daily returns time series, h (q) is dependent upon q. Asq increases, h (q) decreases. This is a confirmation that the daily crude price timeseries of the three international crude oil markets are indeed multifractals. Thissuggests that monofractal models are not appropriate for this time series.

Table 1 presents the generalized Hurst exponents, h (q)with values of q rangingfrom –10 to 10 for the original return time series, shuffled and surrogated timeseries. Since for all the three markets, we have |hc| > |hd|. This means that themultifractality is mainly due to long-range correlations.

Using |∆h| as a measure of efficiency, we can conclude that WTI is the mostefficient while OPEC is the least efficient market. This implies that OPEC has thehighest likelihood to be manipulated among the three markets. This reflects thefact that Brent and WTI is a very competitive market hence, it has a higher levelof complexity compared against OPEC, which has a large monopoly power.

1560013-6

Int.

J. M

od. P

hys.

Con

f. S

er. 2

015.

36. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by 2

02.9

0.12

9.13

4 on

01/

14/1

5. F

or p

erso

nal u

se o

nly.

December 4, 2014 16:25 WSPC/CRC 9.75 x 6.5 1560013

Measuring efficiency of international crude oil markets: A multifractality approach

Table

1.

Gen

eralize

dH

urs

tex

ponen

ts,

h(q

)w

ith

q=

−10

to10.

qBre

nt

OPEC

WTI

Ori

gin

al

Shuffl

edSurr

ogate

dO

rigin

al

Shuffl

edSurr

ogate

dO

rigin

al

Shuffl

edSurr

ogate

d

−10

0.4

991

0.5

427

0.5

750

0.5

496

0.5

337

0.6

397

0.4

928

0.5

410

0.5

380

−90.4

974

0.5

412

0.5

742

0.5

477

0.5

322

0.6

387

0.4

913

0.5

393

0.5

375

−80.4

957

0.5

397

0.5

734

0.5

458

0.5

307

0.6

377

0.4

897

0.5

374

0.5

369

−70.4

940

0.5

381

0.5

727

0.5

439

0.5

291

0.6

367

0.4

882

0.5

355

0.5

365

−60.4

923

0.5

365

0.5

720

0.5

420

0.5

276

0.6

358

0.4

867

0.5

336

0.5

361

−50.4

907

0.5

349

0.5

713

0.5

401

0.5

260

0.6

350

0.4

852

0.5

316

0.5

357

−40.4

892

0.5

333

0.5

707

0.5

382

0.5

244

0.6

342

0.4

837

0.5

296

0.5

354

−30.4

878

0.5

316

0.5

702

0.5

365

0.5

228

0.6

334

0.4

823

0.5

276

0.5

353

−20.4

865

0.5

300

0.5

697

0.5

349

0.5

212

0.6

328

0.4

810

0.5

255

0.5

352

−10.4

854

0.5

284

0.5

692

0.5

334

0.5

197

0.6

322

0.4

798

0.5

234

0.5

352

00.4

846

0.5

268

0.5

689

0.5

322

0.5

181

0.6

318

0.4

788

0.5

213

0.5

353

10.4

841

0.5

251

0.5

686

0.5

313

0.5

165

0.6

315

0.4

781

0.5

191

0.5

356

20.4

841

0.5

235

0.5

684

0.5

309

0.5

150

0.6

312

0.4

778

0.5

170

0.5

360

30.4

846

0.5

219

0.5

682

0.5

312

0.5

135

0.6

311

0.4

779

0.5

148

0.5

365

40.4

860

0.5

203

0.5

681

0.5

322

0.5

120

0.6

312

0.4

786

0.5

127

0.5

371

50.4

883

0.5

187

0.5

681

0.5

343

0.5

105

0.6

313

0.4

801

0.5

105

0.5

378

60.4

917

0.5

171

0.5

682

0.5

379

0.5

091

0.6

316

0.4

826

0.5

083

0.5

387

70.4

967

0.5

156

0.5

683

0.5

432

0.5

076

0.6

320

0.4

864

0.5

061

0.5

396

80.5

034

0.5

140

0.5

685

0.5

508

0.5

062

0.6

324

0.4

919

0.5

038

0.5

406

90.5

122

0.5

123

0.5

687

0.5

611

0.5

047

0.6

329

0.4

993

0.5

016

0.5

416

10

0.5

231

0.5

107

0.5

689

0.5

743

0.5

032

0.6

335

0.5

087

0.4

993

0.5

427

∆h

0.5

360

0.5

090

0.5

691

0.5

904

0.5

017

0.6

341

0.5

200

0.4

970

0.5

437

hc

=−0

.1787

hd

=−0

.1179

hc

=−0

.2053

hd

=−0

.1485

hc

=−0

.1195

hd

=−0

.0226

1560013-7

Int.

J. M

od. P

hys.

Con

f. S

er. 2

015.

36. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by 2

02.9

0.12

9.13

4 on

01/

14/1

5. F

or p

erso

nal u

se o

nly.

December 4, 2014 16:25 WSPC/CRC 9.75 x 6.5 1560013

H. M. Niere

References

1. B.B. Mandelbrot, Fractals: Form, Chance and Dimension (W. H. Freeman and Co.,San Francisco, 1977).

2. B.B. Mandelbrot, The Fractal Geometry of Nature (W. H. Freeman and Co., NewYork, 1982).

3. G. Parisi and U. Frisch, Turbulence and Predictability in Geophysical Fluid Dynamicsand Climate Dynamics, in Proc. of the International School “Enrico Fermi”, (North-Holland, Amsterdam, Netherlands, 1985).

4. A.L. Goldberger, L.A. Amaral, J.M. Hausdorff, P.C. Ivanov, C. K. Peng and H.E.Stanley, Fractal dynamics in physiology: Alterations with disease and aging, in Proc.Natl. Acad. Sci. (2002), p. 2466.

5. Y. Zheng, J.B. Gao, J.C. Sanchez, J.C. Principe and M.S. Okun, Phys. Lett. A 344,253 (2005).

6. B.B. Mandelbrot, Fractals and Scaling in Finance (Springer, New York, 1997).7. H. Katsuragi, Phys. A 278, 275 (2000).8. Z.-Q. Jiang and W.-X. Zhou, Phys. A 387, 4881 (2008).9. X. Sun, H. Chen, Z. Wu and Y. Yuan, Phys. A 291, 553 (2001).

10. P. Oswiecimka, J. Kwapien, S. Drozdz, A. Z. Gorski and R. Rak, Act. Phys. Polo. B37, 3083 (2006).

11. L. Zunino, A. Figliola, B.M. Tabak, D.G. Perez, M. Garavaglia and O.A. Rosso, Chaos,Solitons & Fractals 41, 2331 (2009).

12. L. Zunino, B.M. Tabak, A. Figliola, D.G. Perez, M. Garavaglia and O.A. Rosso, Phys.A 387, 6558 (2008).

13. C.-T. Lye and C.-W. Hooy, Int. J. of Econ. and Mgt. 6(2), pp 278–294, 2012.14. X. Lu, J. Tian, Y. Zhou and Z. Li, Working Papers 2012-08 (Department of Economics,

Auckland University of Technology, 2012).15. Y. Yuan, X.-T. Zhuang and X. Jin, Phys. A 388, 2189 (2009).16. W. Hui, Z. Zongfang and X. Luojie, Mgt. Sci. and Engg. 6, 21 (2012).17. N. Vandewalle and M. Ausloos, Eur. Phys. J. B 4, 257 (1998).18. P. Norouzzadeh and B. Rahmani, Phys. A 367, 328 (2006).19. G. Oh, C. Eom, S. Havlin, W.-S. Jung, F. Wang, H.E. Stanley and S. Kim, Eur. Phys.

J. B 85, 214 (2012).20. T. Ioan, P. Anita and C. Razvan, Ann. of Fac. of Econ. 1, 784 (Faculty of Economics,

University of Oradea, 2012).21. D.O. Cajueiro and B.M. Tabak, Phys. A 373, 603 (2007).22. T. Lux, Economics Working Paper No. 2003, 13 (Department of Economics, Christian-

Albrechts-Universitat Kiel, 2003).23. T. Lux, Economics Working Paper No. 2006, 17 (Department of Economics, Christian-

Albrechts-Universitat Kiel, 2006).24. T. Lux, L. Morales-Arias and C. Sattarhoff, Kiel Working Paper 1737, (Kiel Institute

for the World Economy, 2011).25. J.W. Kantelhardt, S.A. Zschiegner, E. Koscielny-Bunde, S. Havlin, A. Bunde and H.E.

Stanley, Phys. A 316, 87 (2002).26. E.A.F. Ihlen, Front Physiol 3, 141 (2012).

1560013-8

Int.

J. M

od. P

hys.

Con

f. S

er. 2

015.

36. D

ownl

oade

d fr

om w

ww

.wor

ldsc

ient

ific

.com

by 2

02.9

0.12

9.13

4 on

01/

14/1

5. F

or p

erso

nal u

se o

nly.