Measuring efficiency of international crude oil markets: A multifractality approach

Transcription

1 7 th Jagna International Workshop (2014) International Journal of Modern Physics: Conference Series Vol. 36 (2015) (8 pages) c The Author DOI: /S Measuring efficiency of international crude oil markets: A multifractality approach H. M. Niere Economics Department, Mindanao State University, Marawi City, 9700, Philippines hmniere@gmail.com Published 2 January 2015 The three major international crude oil markets are treated as complex systems and their multifractal properties are explored. The study covers daily prices of Brent crude, OPEC reference basket and West Texas Intermediate (WTI) crude from January 2, 2003 to January 2, A multifractal detrended fluctuation analysis (MFDFA) is employed to extract the generalized Hurst exponents in each of the time series. The generalized Hurst exponent is used to measure the degree of multifractality which in turn is used to quantify the efficiency of the three international crude oil markets. To identify whether the source of multifractality is long-range correlations or broad fat-tail distributions, shuffled data and surrogated data corresponding to each of the time series are generated. Shuffled data are obtained by randomizing the order of the price returns data. This will destroy any long-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 of the price returns data in Fourier space. This will normalize the distribution of the time series. The study found that for the three crude oil markets, there is a strong dependence of the generalized Hurst exponents with respect to the order of fluctuations. This shows that 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 WTI is the most efficient while OPEC is the least efficient market. This implies that OPEC has the highest likelihood to be manipulated among the three markets. This reflects the fact that Brent and WTI is a very competitive market hence, it has a higher level of complexity compared against OPEC, which has a large monopoly power. Comparing with shuffled data and surrogated data, the findings suggest that for all the three crude oil 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 distributed under the terms of the Creative Commons Attribution 3.0 (CC-BY) License. Further distribution of this work is permitted, provided the original work is properly cited

2 H. M. Niere 1. Introduction Fractals as introduced by Mandelbrot 1 2 describe geometric patterns with large degree of self-similarities at all scales. The smaller piece of a pattern can be said to be a reduced-form image of a larger piece. This characteristic is used to measure fractal dimensions as a fraction rather than an integer. Some examples of fractal shapes are rugged coastlines, mountain heights, cloud outlines, river tributaries, tree branches, blood vessels, cracks, wave turbulences and chaotic motions. However, there are self-similar patterns that involve multiple scaling rules which are not sufficiently described by a single fractal dimension but by a spectrum of fractal dimensions instead. Generalizing this single dimension into multiple dimensions differentiates multifractal from fractals discussed earlier. To distinguish multifractal from 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 property are earthquakes, 3 heart rate variability 4 and neural activities. 5 Mandelbrot 6 introduced multifractal models to study economic and financial time series in order to address the shortcomings of traditional models such as fractional Brownian motion and GARCH processes which are not appropriate with the stylized facts of the said time series such as long-memory and fat-tails in volatility. Further studies confirmed multifractality in stock market indices, 7 16 foreign exchange rates and interest rates, 21 to name a few. As a consequence, many studies have now used the properties of multifractality in forecasting models These models are at least as good as, and in some cases, perform better using outof-sample forecast compared to traditional models. One added advantage of these models is their being parsimonious. This paper investigates the presence and compares the degree of multifractality of the daily prices of crude oil of the three major international crude oil markets namely the Brent crude, OPEC reference basket and West Texas Intermediate (WTI) crude from January 2, 2003 to January 2, The Brent crude is sourced from the North Sea and is the main European oil market; OPEC is mainly sourced from the Middle East; and WTI is the benchmark used in Chicago and New York mercantile exchange. Furthermore, since multifractality can be due to long-range correlations or due to broad fat-tail distributions, this paper identifies which of the two factors dominates the multifractality of the daily crude prices time series of the said markets. The paper is arranged as follows. Methodology is discussed in Section 2. Data are described in Section 3. Presentation of results is in Section 4. Finally, the paper concludes in Section Methodology In measuring multifractality, the paper uses the method of Multifractal Detrended Fluctuation Analysis (MFDFA) as outlined in Kantelhardt et al. 25 Matlab codes used are based in Ihlen. 26 The procedure is summarized in the following steps

3 Measuring efficiency of international crude oil markets: A multifractality approach (1) Given a time series u i, i =1,...,N,whereN is the length, create a profile Y (k) = k i=1 u i ū, k =1,...,N,whereū is the mean of u. (2) Divide the profile Y (k) inton s = N/s non-overlapping segment of length s. Since N is not generally a multiple of s, in order for the remainder part of the series to be included, this step is repeated starting at the end of the series moving backwards. Thus, a total of 2N s segments are produced. (3) Generate Y s (i) =Y s [(v 1) s + i] for each segment v =1,...,N s,andy s (i) = Y s [N (v N s ) s + i] for each segment v = N s +1,...,2N s. (4) Compute the variance of Y s (i)asfs 2 (v) = 1 s s i=1 [Y s (i) Y v (i)] 2,whereY v (i) is the m th order fitting polynomial in the v th segment. (5) Obtain the q th order fluctuation function by { 2N 1 s [ F q (s) = F 2 2N s (v) ] } 1/q q/2. s v=1 If the time series are long-range correlated then F q (s) is distributed as power laws, F q (s) s h(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 closer the 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. For multifractals however, h (q) varies with q. Thus, a spectrum of h (q) values implies the presence of multifractality. The degree of multifractality can be quantified as h = h (q min ) h (q min ). 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 due to broad fat-tail distributions, shuffled data and surrogated data are generated. In the spirit of Zunino et al., different shuffled time series and surrogated time series are produced to reduce statistical errors. Shuffling the data will remove the long-range correlation in the time series. It is done by randomizing the order of the original data. The multifractality due to long-range correlation can be computed as h c = h h f where the index f refers to shuffled data. Surrogated data is produced by randomizing the phases of original data in Fourier space. This will make the data to have normal distribution. The multifractality due to broad fat-tail distributions can be measured as h d = h h r where the index r refers to surrogated data. 3. Data The daily crude oil prices of the Brent crude, OPEC reference basket and WTI crude 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 for the three markets because the number of business trading days also differs due

4 H. M. Niere to national holidays and other idiosyncracies. Daily price data for OPEC reference basket has been downloaded from the OPEC online database website: web/en/data graphs/40.htm. The daily price data for Brent and WTI crude was downloaded from the website of the U.S. Energy Information Administration: pri spt s1 d.htm. 4. Results Figures 1 to 3 show the plots of the daily crude prices, the daily returns, and the associated shuffled and surrogated time series of daily returns for Brent crude, OPEC reference basket and WTI crude respectively. The original daily returns and the shuffled time series show some extreme fluctuations which is a sign of having fat-tail distribution. The surrogated time series do not have extreme fluctuation, a characteristic of a normal distribution. In doing the MFDFA procedure, m = 3 is used as the order of polynomial fit in Step 3. The length s varies from 20 to N/4 with a step of 4 as suggested in Kantelhardt et al. 25 Finally, q runs from 10 to 10 with a step of 0.5. Figure 2 presents the generalized Hurst exponents for the original returns, shuffled returns and 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) shuffled time series, and (d) surrogated time series

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

6 H. M. Niere Fig. 4. (Color online) Generalized Hurst exponent, h(q), as a function of q for the original, shuffled and 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 which is h (2) regardless of the value of q. In contrast, multifractals have a spectrum of generalized Hurst exponents which vary depending upon the value of q. Itisnoted in Figure 2 that for the daily returns time series, h (q) is dependent upon q. As q increases, h (q) decreases. This is a confirmation that the daily crude price time series of the three international crude oil markets are indeed multifractals. This suggests that monofractal models are not appropriate for this time series. Table 1 presents the generalized Hurst exponents, h (q)with values of q ranging from 10 to 10 for the original return time series, shuffled and surrogated time series. Since for all the three markets, we have h c > h d. This means that the multifractality is mainly due to long-range correlations. Using h as a measure of efficiency, we can conclude that WTI is the most efficient while OPEC is the least efficient market. This implies that OPEC has the highest likelihood to be manipulated among the three markets. This reflects the fact that Brent and WTI is a very competitive market hence, it has a higher level of complexity compared against OPEC, which has a large monopoly power

7 Measuring efficiency of international crude oil markets: A multifractality approach Table 1. Generalized Hurst exponents, h (q) with q = 10 to 10. q Brent OPEC WTI Original Shuffled Surrogated Original Shuffled Surrogated Original Shuffled Surrogated h hc = hd = hc = hd = hc = hd =

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