Improving MF-DFA model with applications in precious metals market
IImproving MF-DFA model with applications in preciousmetals market
Zhongjun Wang a , Mengye Sun a and A. M. Elsawah b,c, ∗ a Department of Statistics, Science School, Wuhan University of Technology, Wuhan 430070, China b Beijing Normal University-Hong Kong Baptist University United International College, Zhuhai 519085, China c Department of Mathematics, Faculty of Science, Zagazig University, Zagazig 44519, Egypt
Abstract
With the aggravation of the global economic crisis and inflation, the precious metalswith safe-haven function have become more popular. An improved MF-DFA methodis proposed to analyze price fluctuations of the precious metals market. Based onthe widely used multifractal detrended fluctuation analysis method (MF-DFA), wecompare these two methods and find that the Bi-OSW-MF-DFA method possessesbetter efficiency. This article analyzes the degree of multifractality between spot goldmarket and spot silver market as well as their risks. From the numerical results andfigures, it is found that two elements constitute the contributions in the formationof multifractality in time series and the risk of the spot silver market is higher thanthat of the spot gold market. This attempt could lead to a better understanding ofcomplicated precious metals market.
Key words:
The precious metals market; Multifractal analysis; MF-DFA method; Bi-OSW-MF-DFA method.
With the development of capital markets, the types of financial products have been increas-ing rapidly and precious metals trading have been becoming more active in a diversified ∗ Corresponding author. E-mail: a [email protected], [email protected], [email protected] a r X i v : . [ q -f i n . S T ] J un ortfolio. As the main investments in precious metals market, the market price fluctu-ations of gold and silver will inevitably cause widespread concern. Especially when thefinancial crisis comes, the precious metals with safe-haven function are favored by in-vestors. However, a financial market is a complex system including various interior andexterior factors which make it difficult to understand and describe [1].According to the existing researches, the price fluctuations of precious metals market isa process with non-linear characteristics which exhibits complex dynamics features. Notethat, the time series of gold consumer price index of Chinese, Indian and Turkey marketseries are of multifractal nature [2]. The study of multifractality in the financial time seriescould reveal critical information about dynamics and complexity of the financial markets[3]. Mandelbrot, 1988 proposed a loose tentative definition for fractal, a fractal is a shapemade of parts similar to the whole in some way [4]. In recent decades, the fractal theoryhas been developing quickly and has become a new technique for the study of some issuesin financial market which is a very complex system. With the help of nonlinear systemtheory, fractal capital markets theory not only explains the market behavior of manycomplex phenomena, but also provides a way of quantitative analysis. The traditionalrescaled range analysis method (R/S) was first proposed and introduced by Hurst whowas an English hydrologist [5]. It is used to analyze the fractal characteristics and long-term persistence in time series, however this method is only applicable to monofractal. So[6] improved it and used R/S analysis with a q − th order height-height correlarion functionto detect the multifractal nature of KOSPI.It has been acknowledged that the rescaled range analysis (R/S) and the detrendedfluctuation analysis (DFA) are the methods for monofractal.[7-9] .These two method areunable to capture complex dynamics in a time series or to characterize their scaling proper-ties when the processes are governed by more than one scaling exponent [10] .Consideringthis, Kantelhardt et al. (2002) proposed Multifractal Detrended Fluctuation Analysis(MF-DFA) [11] which is an effective method to reliably research long-range correlationsin nonstationary time series and at the same time it can avoid erroneous judgements ofcorrelation. Currently, MF-DFA method has been widely used in the study of multifractalcharacteristics of financial market and has got some attractive results [12-15], it allowsdetermining the correlation properties on large time scale [16]. Zunino et al.(2008) foundthat the multifractal degree for a broad range of stock markets is associated with the stageof their development [12]. Yuan and Zhuang used MF-DFA method to measure multifrac-tailty of stock price index fluctuation [17]. Also, components of empirical multifractalityin financial returns were investigated with MF-DFA method [18]. However, a possible2rawback of the MF-DFA method is the occurrence of abrupt jumps in the detrended pro-file at the boundaries between the segments, since the fitting polynomials in neighboringsegments are not related [16].This paper is organized as follows. In Section 2, we present the brief overview ofexisting research studies with pros and cons, and then improve the MF-DFA method. InSection 3, we use this newly-built Binary Overlapped Sliding Window-Based MF-DFA (Bi-OSW-MF- DFA) to detect the multifractal characteristics of spot gold and silver markets.Furthermore, a comparative study is done for the magnitude of the multifractality betweenthese two markets and analyze their risk. Finally, we close through the conclusions anddiscussions in Section 4. Multifractal detrended fluctuation analysis has been widely used in many fields and it canreliably determine the multifractal scaling behavior of time series [11] and consists of sixsteps as follows.
Step 1.
Suppose x k is a series of length N , calculate the ’profile’ and get the series Y ( i ) = i (cid:88) k =1 ( x k − ¯ x ) , i = 1 , , ..., N, ¯ x = 1 N N (cid:88) i =1 x i . Step 2.
Divide the profile Y ( i ) into N s = (cid:2) Ns (cid:3) non-overlapping segments of equal length s .Considered that the length N of the series is often not a multiple of the consideredtime scale s which a short part at the end of the profile may be left. In order not todisregard this part of the series and to ensure data integrity, the same procedure isrepeated starting from the opposite end of the time series. Finally, we can obtained2 N s segments. where [ . ] means integer arithmetic. Step 3.
Calculate the local trend for each of the 2 N s segments by a least-square fit of theseries. Then determine the variance F ( v, s ) = 1 s s (cid:88) j =1 ( Y (( v − s + i ) − y v ( i )) , v = 1 , , ..., N s . for each segment v, v = 1 , ..., N s and F ( v, s ) = 1 s s (cid:88) j =1 ( Y ( N − ( v − N s ) s + i ) − y v ( i )) , v = 1 , , ..., N s . v = 2 N s +1 , ..., N s . Here, y v ( i ) is the fitting polynomial in box v . Because differ-ent order MF-DFAs differ in the capability of eliminating trends in the series, linear(MF-DFA1), quadratic (MF-DFA2), cubic (MF-DFA3), or higher order polynomialscan be considered in the fitting procedure. Step 4.
Generally, we are interested in how the generalized q dependent fluctuation func-tions F q ( s ) depends on the time scale s for different values of q . The fluctuationfunction of q th order is computed by averaging over all the fluctuations of the seg-ments, defined as: F q ( s ) = (cid:32) N s N s (cid:88) v =1 (cid:0) F ( s, v ) (cid:1) q (cid:33) q . The index variable q can take any real non-zero value. For q = 0 , we calculate thefluctuation function as given below: F ( s ) = exp (cid:32) N s N s (cid:88) v =1 ln (cid:0) F ( s, v ) (cid:1)(cid:33) . Step 5.
Finally, the scaling behavior of the fluctuation functions is determined by ana-lyzing log plots of F q ( s ) versus s for each value of q. If the series x k is long-rangepower-law correlated, F q ( s ) increases for large values of s as a power-law, as follows: F q ( s ) ∼ s h ( q ) . Here h ( q ) is known as the generalized Hurst exponent and h (2) is the well-knownHurst exponent H . In general, if the time series is monofractal, h ( q ) is independentof q i.e. ∆ h ( q ) = h ( q min ) − h ( q max ) = 0 and if the time series is multifractal, h ( q )depends on q. It means ∆ h ( q ) > h ( q ). ∆ h ( q )is therefore referred to as the degree of multifractality.For positive values of q , h ( q ) describes the behaviour of segments with large fluc-tuations while for negative values of q, h ( q ) describes the behaviour of segmentswith small fluctuations. In general, h ( q ) is a decreasing monotonic function of q for a stationary time series which means that relatively small fluctuations happenmore often in the series than relatively large ones. If h ( q ) > . , the fluctuationsrelated to q are persistently auto-correlated, if h ( q ) < . , the fluctuations relatedto q are anti-persistently auto-correlated and if h ( q ) = 0 . , the fluctuations relatedto q display a random walk behaviour. See [3].4 tep 6. Multifractality of a time series can also be characterized in terms of the multi-fractal scaling exponent τ ( q ) which is related to the generalized Hurst exponent h ( q )through the relation τ ( q ) = qh ( q ) − . Here τ ( q ) represents the temporal structure of the time series as a function of mo-ments q and it reflects the scaling dependence of small fluctuations for negativevalues of q and large fluctuations for positive values of q. If τ ( q ) is a linear functionof q, the time series can be regarded as monofractal and if τ ( q ) has a nonlineardependence on q, then the series is multifractal.The complexity in a time series can be better captured through the singularityspectrum, f ( α ) .α and f ( α ) can be obtained through a Legendre transform of q and τ ( q ). α = dτ ( q ) dq = h ( q ) + qh (cid:48) ( q ) .f ( α ) = qα ( q ) − τ ( q ) = q h (cid:48) ( q ) + 1 . Both the exponent α and spectrum f ( α ) express the singularity of the price series.The width of singularity spectrum f ( α ) is always denoted as ∆ α, where ∆ α = α max − α min . The ∆ α is used to denote the uniform degree of the distribution ofnormalized prices for the whole fractal structure. The bigger of ∆ α, is the smallereven distribution probability measure is, and the more violent price fluctuationswill usually be expected. And ∆ α = 0 corresponded to the situation of completelyuniform distribution. The plot of singular exponent α and singularity spectrum f ( α ) can reflect the properties of probability distribution. If the series exhibits asimple monofractal scaling behavior, the value of singularity spectrum f ( α ) will bea constant; if the series exhibits a simple multifractal scaling behavior, the value ofsingularity spectrum f ( α ) will change dependently on the singular exponent α. Thereader can refer to [13, 19, 20].
In our paper, we use different detrending procedures to improve MF-DFA in two aspectsand named it as Bi-OSW-MF-DFA.
Firstly.
We want to extract data just for one time, so we divide the profile Y ( i ) into twosections equally, the length n of these two sections both equal (cid:4) N (cid:5) , where (cid:98) x (cid:99) means thelargest integer not greater than x, then we get two sub-series. Secondly.
We divide both the two sub-series into N ∗ s = (cid:108) N − ss − l (cid:109) , where (cid:100) x (cid:101) means the5mallest integer not less than x, overlapping segments of equal length s, l, (cid:0) < l < s (cid:1) isthe overlapped length of neighboring segments. Since the length of the sub-series is oftennot a multiple of the considered time scale, the last segment of the first profile will includea short part in the beginning of the second sub-series. In order to ensure data integrity,the same procedure is repeated starting from the opposite end of the second sub-series.Thereby, 2 N ∗ s segments are obtained altogether. In order to explain why our method performs better and illustrate specifically, we caneasily see that the number of segments in Bi-OSW-MF-DFA is half of that in MF-DFAwhich denotes as 2 N ∗ s = N s . Therefore, F q ( s ) = N ∗ s N ∗ s (cid:88) v =1 (cid:0) F ( s, v ) (cid:1) q q = (cid:32) N s N s (cid:88) v =1 (cid:0) F ( s, v ) (cid:1) q (cid:33) q . As well as, we use three figures to show the different ways of dividing the intervals ofprofile Y ( i ) between MF-DFA method and Bi-OSW-MF-DFA method. From the followingFigures 1, 2 and 3, it can be easily seen the different detrending procedures.On one hand, to ensure data integrity, MF-DFA processes data both in sequential orderand in reversed order. However, there will produce some repeated information which mayincrease error and increase the time of computation. Bi-OSW-MF-DFA method onlyextracts data for once which saves the computer running time. On the other hand, [16]pointed out that the unrelated fitting polynomials in neighboring segments may causeabrupt jumps in the detrended profile at the boundaries between the segments. So asimple way to avoid these jumps would be the calculation of F q ( s ) based on polynomialfits in overlapping windows. In Section 3.2, we will use numerical simulations to prove ourmethod: Bi-OSW-MF-DFA method possesses better efficiency in detail.6igure 1: Way of dividing the intervals Y ( i ) in sequential order of MF-DFA methodFigure 2: Way of dividing the intervals Y ( i ) in reversed order of MF-DFA methodFigure 3: Way of dividing the intervals of profile Y ( i ) of Bi-OSW-MF-DFA method7 Multifractal analysis and risk of gold and silver markets
The main objective of this study is to investigate the characteristics of precious metalsmarket, using some multifractal measures to judge the market fluctuations. Daily closingprice of spot gold (from February 9, 2004 to March 23, 2015) and spot silver (from February13, 2004 to March 23, 2015) are used as our samples. Eliminating weekends and holidays2851 and 2880 different data were obtained respectively. All these data have taken fromthe JIJIN silver and gold analysis software. Figure 4 and Figure 5 show the daily closingprice trends of spot gold and spot silver.As a first step, the unit root test is used to detect the stationarity of these two timeseries. Table 1 and Table 2 show the results by using ADF test (defined below).
ADF test:
In statistics and econometrics, an Augmented Dickey-Fuller test (ADF) is atest for a unit root in a time series sample. Using unit root test could detect the stationarityof a time series [21]. According to data in Tables 1 and 2 under the certain confidenceTable 1: ADF test of gold daily closing price return series.
Variable Test Type ( c, t, p ) ADF Statistic Threshold of 1% Threshold of 5% Threshold of 10% Prob. Conclusion Y ( c, ,
0) 1 . − . − . − .
617 0 .
958 Nonstationary∆ Y ( c, , − . − . − . − .
617 0 .
000 Stationary
Table 2: ADF test of silver daily closing price return series.
Variable Test Type ( c, t, p ) ADF Statistic Threshold of 1% Threshold of 5% Threshold of 10% Prob. Conclusion Y ( c, ,
0) 0 . − . − . − .
616 0 .
839 Nonstationary∆ Y ( c, , − . − . − . − .
616 0 .
000 Stationary level, statistic ADF > critical value, the spot price of gold and silver are not stable andtheir first differences estimators are stationary. From the result of ADF test, we can learnthat gold and silver spot price time series are non-stationary which also with a certaintrend. Since logarithmic return rate can eliminate the dependence of price fluctuation onprice level and return equals one order difference of logarithmic price series, we deal withthe data in this way. Based on the original time series, let I t be the closing price of time t and I t − be the closing price of last time. The rate of return can be calculated as followingformula r t = ln I t − ln I t − . Finally, we finally get 2850 and 2879 gold and silver daily return data respectively.8n order to obtain some further analysis of the data and get more accurate information,we use SPSS to do descriptive statistics analysis of the return series as follows.Table 3: Descriptive statistics of return series.Statistics N Mean Minimum Maximum Skewness Kurtosisgold 2879 0.0003 -0.20 0.13 -1.074 8.358silver 2850 0.0004 -0.11 0.10 -0.416 6.706From the Table 3, a positively large skew is seen. The skewness of spot gold and silverreturn time series are both less than 0 and the peakedness are both greater than 3. Theresults also show that these two series have a high degree of peakedness and fat tails andindicate that the return series of spot gold and silver do not follow a normal distribution.On the other hand, the following figures (see, Figures 6 and 7) describe the trends ofspot gold and spot silver daily rate of return.Figure 4: Daily closing price history of spot gold market (2004-2015)9igure 5: Daily closing price history of spot silver market (2004-2015)Figure 6: Daily return series history of spot gold market (2004-2015)Figure 7: Daily return series history of spot silver market (2004-2015)10 .2 Comparison of MF-DFA and Bi-OSW-MF-DFA
Compared with some mature markets such as the stock market, the future market and thebond market, precious metals market is still in its development stage. The precious metalsmarket is a kind of complex non-liner dynamic system where the market information isasymmetrical. Presently the fluctuation of precious metals market seem quite confusingeven to the regular traders, and it becomes almost impossible to predict its accurate riseand fall [2]. Although many natural phenomena seem rather complicated at the first sight,they do share some consistent and simple features [5].On the other hand, Step 5 of MF-DFA has pointed out that if the series are long-rangepower-law correlated, the fluctuation function F q ( s ) increases, for large value of s , as apower-law, F q ( s ) ∼ s h ( q ) . Note that, fluctuation function F q ( s ) can be defined only for s ≥ m + 2, where m is theorder of the detrending polynomial. Moreover, F q ( s ) is statistically unstable for large scale s ( ≥ N ) . The generalized Hurst exponent h ( q ) is extracted from a straight line fit to thelog-log plot of F q ( s ) versus s. See Figure 8 and 9.Figures 8 and 9 give the results, we use moments with -20, -10, -2, + 2, +10 and +20and the scale ranging from 100 to 500. Obviously, there exists some crossovers betweendifferent scaling regimes in the fluctuation functions . In this case a multitude of scalingexponents is required for a full description of the scaling behavior in the same range oftime scales, and a multifractal analysis must be applied [11]. The following part we willuse MF-DFA and Bi-OSW-MF-DFA to do multifractal analysis of spot gold and silvermarkets as well as the comparison between MF-DFA and Bi-OSW-MF-DFA.To conform that the newly-built Bi-OSW-MF-DFA method is more robust than MF-DFA, we use both methods to analyze the same gold and silver return series. In order toavoid errors caused by differences of other parameters in multiple analysis, we select thesame sub-interval length and order where the time scale ranges from 10 to 50. Table 4gives the Hurst exponent h ( q ) calculated from these two methods. For convenience, wedenote h ( q ) for gold return series and h ( q ) for silver return series in Table 4.Because of F q ( s ) ∼ s h ( q ) , it can be easily found that if pseudo fluctuations and somerepeated information are reducing, the errors will become less which leads to the decreaseof fluctuation functions F q ( s ) with different values of time scale s . So the slope of log-logplot of the fluctuation functions F q ( s ) which equals Hurst exponent h ( q ) will become moreconcentrated and slighter. Comparing the results from MF-DFA and Bi-OSW-MF-DFA,we find that to the gold return time series: ∆ h ∗ g ( q ) = 0 . < ∆ h g ( q ) = 0 . h ∗ s ( q ) = 0 . < ∆ h s ( q ) = 0 . . These comparisons couldmanifest the (Bi-OSW-MF-DFA) method possesses better robustness.Table 4: Hurst exponent h ( q ) calculated from MF-DFA and Bi-OSW-MF-DFA.Order MF-DFA B i-OSW-MF-DFAq h g ( q ) h s ( q ) h ∗ g ( q ) h ∗ s ( q )-20 0.8704 1.0400 0.7217 0.7849-16 0.8569 1.0237 0.7129 0.7757-12 0.8331 0.9942 0.7005 0.7619-8 0.7832 0.9287 0.6807 0.7398-4 0.6887 0.7729 0.6387 0.68560 0.5908 0.6207 0.5546 0.58062 0.5316 0.5480 0.5129 0.5164 0.4634 0.4326 0.4719 0.43028 0.3629 0.2833 0.4034 0.322112 0.3139 0.2294 0.3674 0.274816 0.2885 0.2042 0.3477 0.249220 0.2739 0.1899 0.3357 0.2334∆ h h ( q ) and order q. Bycomparing the results of Figures 10 and 11, we can suggest that, with Bi-OSW-MF-DFAmethod the Hurst exponent is more concentrated and the oscillation of Hurst exponent isslighter. From the Figures 10 and 11, when assign the order q from -20 to 20, the Hurstexponent h ( q ) is changing dependently. It indicates that, both in gold and silver markets,local trends of these two series are not uniform, they do not exhibit a simple monofractalscaling behavior but with multifractal characteristics. Meanwhile, the volatility of thegeneralized Hurst exponent can measure the degree of multifractality in a financial marketthe more obvious, the greater its risk is. Compared with silver, the range of gold returnseries generalized Hurst exponent ∆ h ∗ g ( q ) = 0 . h ∗ s ( q ) = 0 . , so we can get, the multifractality of silver return series is stronger and12he risk of silver market is higher than gold market.Figure 8: F q ( s ) obtained from Bi-OSW-MF-DFA for gold return series for some q -orders.Figure 9: F q ( s ) obtained from Bi-OSW-MF-DFA for silver return series for some q -orders.13igure 10: h ( q ) versus q of the gold return series based on the two methods.Figure 11: h ( q ) versus q of the silver return series based on the two methods.14 .3 Factors of multifractal structure in precious metals market According to some existing literatures, we have found that there are two different typesof multifractality in time series. (i) Multifractality due to a broad probability densityfunction for the values of the time series. (ii) Multifractality due to different long-rangecorrelations of small and large fluctuations [10]. In order to get further investigation ofprecious metals market multifractality, we will respectively process in shuffling procedureand phase randomization procedure.
Shuffling procedure consists of three steps [14]:Step1:
Generate pairs ( m, n ) of random integer numbers (with m, n ≤ N ) where N isthe total length of the time series to be shuffled. Step2:
Interchange entries m and n. Step3:
Repeat steps 1 and 2 for 20 N steps. (This step ensures that ordering of entriesin the time series is fully shuffled.) Phase randomization procedure consists of three steps:Step1:
Process the original series with discrete Fourier transform.
Step2:
Rotate the phase with one phase angle.
Step3:
Then process the data with inverse Fourier transform.Compared with the original series, the shuffled time series destroys correlations but canretain its volatility, while the surrogate time series weakens the non-Gaussian distributionof time series. Using the more efficient Bi-OSW-MF-DFA method to analyze the originalseries, the shuffled series and the surrogate series of spot gold and silver markets. Theresults are as in Figures 14 and 15 and Table 5.It is can be seen from Figures 14 and 15 that the Hurst exponent of gold and silvershuffled series are both decreasing sharply. The changes indicate that market price fluctu-ations dominantly affect the multifractality of gold and silver market. Comparing the twomarkets: the Hurst exponent of gold shuffled series is farther from 0.5. After destroyingthe correlations, gold market shows a stronger anti-persistence and this result proves thatgold return series has a higher degree of correlation.Additionally, in view of Table 5, we have the ∆ h = 0 . h = 0 . h ( q ) versus q of original, shuffled and surrogate series in gold market basedon Bi-OSW-MF-DFA method.Figure 13: h ( q ) versus q of original, shuffled and surrogate series in silver market basedon Bi-OSW-MF-DFA method. With the relation between h ( q ) and τ ( q ), we can get the figure that τ ( q ) is changing withdifferent q. Since h ( q ) is a constant in monofractal analysis, the τ ( q ) ∼ q chart is linear.16able 5: h ( q ) of the original series, the shuffled series and the surrogate series. Order h ( q ) of gold return series h ( q ) of silver return seriesq original series shuffled series surrogate series original series shuffled series surrogate series-20 0.7217 0.3489 0.5862 0.7849 0.3818 0.5848-16 0.7129 0.3144 0.5743 0.7757 0.3526 0.5766-12 0.7005 0.2619 0.5577 0.7619 0.3052 0.5667-8 0.6807 0.1887 0.5379 0.7398 0.2289 0.5546-4 0.6387 0.1147 0.5243 0.6856 0.1322 0.53850 0.5546 0.0672 0.5195 0.5806 0.0708 0.52052 0.5129 0.0514 0.5176 0.5160 0.0512 0.51214 0.4719 0.0386 0.5142 0.4302 0.0349 0.50408 0.4034 0.0190 0.5015 0.3221 0.0076 0.485312 0.3674 0.0044 0.4852 0.2748 -0.0167 0.461116 0.3477 -0.0068 0.4706 0.2492 -0.0397 0.439520 0.3357 -0.0156 0.4595 0.2334 -0.0605 0.4244∆ h And h ( q ) will change with different q in multifractal analysis, so the curve is nonlinear.We provide the τ ( q ) ∼ q chart of these three different return series (see, Figures 16 and17). A strong nonlinearity between τ ( q ) and q can be easily seen from Figures 16 and 17.At the same time, both in gold and silver return series, the nonlinearity is more obvious inthe shuffled series which makes a more significant contribution to multiple scale. Becauseof a stronger nonlinearity existing in silver return series, the price of silver market is morelikely to vibrate suddenly and unexpectedly. We can obtain the plot of singular exponent α and multifractal spectrum f ( α ) with thesetwo formula α = h ( q ) + qh (cid:48) ( q ) and f ( α ) = q h (cid:48) ( q ) + 1 . The width of multifractal spectrum could reflect the risk of financial market. We can seefrom the Figures 18 and 19 and Table 6 that the width of the original silver return seriesmultifractal spectrum ∆ α equals 0.6160 is greater than it of the gold market which equals0.4625. In silver market, there is a stronger price fluctuation which manifests the higherrisk. Although the risk is more significant, but it also represents that we can get more17igure 14: τ ( q ) versus the order q of original, shuffled and surrogate series in gold marketbased on Bi-OSW-MF-DFA method.Figure 15: τ ( q ) versus the order q of original, shuffled and surrogate series in silver marketbased on Bi-OSW-MF-DFA method. 18rofits. Generally speaking, silver market is suitable for speculators to get into. In bothof gold and silver markets, there exist obvious differences of the multifractal spectrumwidth between the shuffled series and the surrogate series which confirms once again themultifractality in precious metals market is determined by two factors. In addition, thewidth of multifractal spectrum becomes narrower after phase randomization procedureindicates that extreme non-Gaussian events also have affected the multifractal propertiesin time series.Table 6: Multifractality degree of three different series in both gold and silver markets.spot gold ∆ h ∆ α spot silver ∆ h ∆ α original series 0.3860 0.4625 original series 0.5515 0.6160shuffled series 0.3645 0.3629 shuffled series 0.4423 0.3947surrogate series 0.1267 0.3178 surrogate series 0.1604 0.3835Figure 16: Multifractal spectrum f ( α ) of original, shuffled and surrogate series in goldmarket based on Bi-OSW-MF-DFA method.19igure 17: Multifractal spectrum f ( α ) of original, shuffled and surrogate series in silvermarket based on Bi-OSW-MF-DFA method. Based on fractal theory, this paper improves the MF-DFA method which is widely used inmany fields and analyzes the precious metals market. We select the spot gold and silvermarket to do the research and obtain the following conclusions.1. We improve Multifractal Detrended Fluctuation Analysis (MF-DFA method) and ob-tain a better model–Bi-OSW-MF-DFA method. Binary Overlapped Sliding Window-based Multifractal Detrended Fluctuation Analysis possesses stronger robustness.This model not only reduces the extraction of repeated information which decreaseserror, but also speeds up operation in simulation.2. There are some crossovers both in the fluctuation function F q ( s ) of spot gold marketand spot silver market. Precious metals market does not exhibit a simple monofractalscaling behavior but with significant multifractal properties. The statistics suggestthat the multifractality in gold market is less obvious than silver market which ofmore profitts and higher risk for investors.3. After shuffling procedure and phase randomization procedure, we can find that themultifractal properties of the precious metals market is jointly caused by small andlarge fluctuations and volatility of the different long-range correlation and volatilityfat tail probability distribution, and the volatility of time series is the main factor.20. Multifractal spectrum analysis of gold and silver markets shows that the multifractalspectrum width of silver series is wider than gold series, so the potential risk is higherin silver market. The wider the multifractal spectrum width is, the more significantthe oscillation of market and the higher the rate of return; on the contrary, the lesssignificant the oscillation of market is, the less risk exists. All the evidences suggestthat there is a higher degree of security in gold market which makes it more suitablefor investors to hedge.In real transaction, the smaller volume and the more speculative tendencies of investorslead to more serious price oscillations in silver market, and we have got the same conclusionby analyzing empirical data. Moreover, dispersion, high costs and other factors also makesilver market price fluctuate strongly. But comparing with the stock market, preciousmetals market has a shorter history, is still an immature developing financial market whichneeds to be researched more deeply. Thus, more valuable and comprehensive informationcould be provided for investors to investigate the precious metals market. Acknowledgments
This work was partially supported by the UIC Grants (Nos. R201810, R201912 andR202010), the Zhuhai Premier Discipline Grant. and the National Natural Science Foun-dation of China (No. 61304181).