---
abstract: 'In a highly interdependent economic world, the nature of relationships between financial entities is becoming an increasingly important area of study. Recently, many studies have shown the usefulness of minimal spanning trees (MST) in extracting interactions between financial entities. Here, we propose a modified MST network whose metric distance is defined in terms of cross-correlation coefficient absolute values, enabling the connections between anticorrelated entities to manifest properly. We investigate 69 daily time series, comprising 3 types of financial assets: 28 stock market indicators, 21 currency futures, and 20 commodity futures. We show that though the resulting MST network evolves over time, the financial assets of similar type tend to have connections which are stable over time. In addition, we find a characteristic time lag between the volatility time series of the stock market indicators and those of the EU CO2 emission allowance (EUA) and crude oil futures (WTI). This time lag is given by the peak of the cross-correlation function of the volatility time series EUA (or WTI) with that of the stock market indicators, and is markedly different ($>20$ days) from 0, showing that the volatility of stock market indicators today can predict the volatility of EU emissions allowances and of crude oil in the near future.'
author:
- Zeyu Zheng
- Kazuko Yamasaki
- Joel Tenenbaum
- 'H. Eugene Stanley'
title: Carbon dioxide emissions trading and hierarchical structure in worldwide finance and commodities markets
---

Introduction and Method
=======================

In the study of complex systems, there has been much work demonstrating the usefulness of extracting underlying structure from the correlations found in statistical data [@Mantegna; @Bonanno2; @Bonanno1; @Onnela1; @Onnela2; @Song; @Micciche; @Tumminello; @ManandStan; @Wang; @laloux; @Plerou1; @Plerou2; @Utsugi]. Any means of selection of statistically reliable information from correlation matrices has been dubbed a “filtering procedure” [@Tumminello]. Useful examples of filtering procedures that use a correlation matrix from return time series are: hierarchical clustering [@Mantegna; @Bonanno2; @Bonanno1; @Onnela1; @Onnela2; @Song; @Micciche], procedures based on the random matrix theory [@Wang; @laloux; @Plerou1; @Plerou2; @Utsugi], and networks from minimum spanning trees [@Mantegna; @Bonanno2; @Bonanno1; @Onnela1; @Onnela2].

Correlation structure studies are not limited only to stock return time series [@Mantegna; @Bonanno2; @Bonanno1], but also extend to quasi-synchronously recorded time series of worldwide stock exchange market indices [@Bonanno1] and stock return volatility increments [@Micciche].

Financial time series can include not only stock price, but also many other types of data, such as commodities price, treasury yield, market index, etc.... Investigation of multi-type quasi-synchronous financial data may yield insights into the interdependent relationships of markets and commodities. Moreover, a relationship map of financial assets can highlight the movement of speculative capital.

We investigate 69 daily financial series from the time period spanning January 2007 to September 2011. The data set includes 21 currency futures, 20 commodity futures which are taken from [http://data.theice.com]{}, and 28 stock market indicators, which are taken from [http://finance.yahoo.com]{} (see Appendix).

Recently, papers have shown the usefulness of the correlation structure described by an ultrametric space and a corresponding hierarchical organization for financial return time series [@Mantegna; @Bonanno2; @Bonanno1]. The approach requires the definition of a metric distance. Because correlation does not fulfill the three [@metric] axioms that define a metric, the aforementioned papers use the Mantegna-Sornette distance defined by

$$d_{ij}=\sqrt{2(1-\rho_{ij})}
  \label{eq5}$$

for each pair of elements $i$ and $j$, where $\rho_{ij}$ is the correlation coefficient of the two time series given by [@feller]

$$\rho_{ij} \equiv \frac{\langle Y_iY_j\rangle-\langle Y_i\rangle \langle Y_j\rangle}{\sqrt{(\langle{Y_i}^2\rangle-\langle Y_i\rangle^2)(\langle{Y_j}^2\rangle-\langle Y_j\rangle^2)}} = \frac{\Big\langle \bigg(Y_i-\langle Y_i \rangle\bigg)\bigg(Y_j-\langle Y_j \rangle\bigg) \Big\rangle}{\sigma_{Y_i}\sigma_{Y_j}},
  \label{eq6.5}$$

where $\langle ... \rangle$ denotes the mean.

This distance $d_{ij}$ fulfills the three axioms [@metric] of a metric: i) $d_{ij}=0$ if and only if $i=j$, ii) $d_{ij}=d_{ji}$, and iii) $d_{ij} \leq d_{ik}+d_{kj}$  [@Mantegna; @Bonanno1; @Bonanno2].

In this work, we make a modification to the metric above, based on the reasoning that if two time series have a very large [*negative*]{} correlation, they should still be considered close to each other in ultrametric “correlation” space, since this would indicate a strong connection, regardless of the sign. Likewise, weak correlations of either sign indicate a weaker connection. On this basis, strong correlations of either sign should be considered closer than weak ones.

The value of this modification can be readily seen in situations where two entities have strong [ *anti*]{}-correlations, as has been observed in the relationship between bond and stock markets (e.g. between UK gilts and the FTSEMIB), between stocks and currency futures, and between industries and their inputs (e.g. the price of oil and the value of airline stock) [@Norden; @Kwan; @Hammoudeh; @businessweek]. Here, use of the conventional Mantenga-Sornette metric would likely result in any of these two entities manifesting on opposite sides of a tree, even though we know them to be very closely linked. Use of our generalized metric ensures that such entities will be placed nearby when such appropriately strong relations exist.

For this reason, we replace the Pearson correlation coefficient in Eq. \[eq5\] with the absolute value of the Pearson correlation coefficient, defining a new distance as $$d_{ij}=\sqrt{2(1-|\rho_{ij}|)}
 \label{eq6}$$

with the $\rho_{ij}$ defined as before in Eq. \[eq6.5\] as the correlation coefficient of assets $i$ and $j$.

This equation also fulfills three axioms of a metric distance. Because the set of assets considered has no cases such that $\rho_{ij}=-1$, the first axiom is satisfied on this set in that $d_{ij}=0$ if and only if the correlation is total ($\rho=1$, meaning that the stocks perform the same stochastic process). The second axiom (that of symmetry) is trivially satisfied because $\rho_{ij}=\rho_{ji}$ by definition of the Pearson correlation coefficient. For the validity of axiom (iii), consider three time series ${Y_i}$, ${Y_j}$ and ${Y_k}$, which have means equal to 0 and standard deviations equal to 1. All times series have the same length. In order to prove $d_{ij} \leq d_{ik}+d_{kj}$, firstly we define two new time series as

$$Y'_i\equiv \begin{cases}
 Y_i,&\mbox{if }  \rho_{Y_i, Y_k} \geq 0 \\
 -Y_i,&\mbox{if } \rho_{Y_i, Y_k} < 0)
\end{cases}$$

$$Y'_j\equiv \begin{cases}
 Y_j,&\mbox{if }  \rho_{Y_j, Y_k} \geq 0 \\
 -Y_j,&\mbox{if } \rho_{Y_j, Y_k} < 0.
\end{cases}$$

So we can rewrite $d_{ik}+d_{kj}$ of distance Eq. \[eq6\] by using $Y'_i$ and $Y'_j$ as $$\begin{aligned}
d_{ik}+d_{kj} & =\sqrt{2(1-|\rho_{Y_i,Y_k}|)}+\sqrt{2(1-|\rho_{Y_j,Y_k}|)}&\\
&  = \sqrt{2(1-\rho_{Y'_i,Y_k})}+\sqrt{2(1-\rho_{Y'_j,Y_k})}\\
&  \geq \sqrt{2(1-\rho_{Y'_i,Y'_j})}\\
&\begin{cases}
 =\sqrt{2(1-|\rho_{Y'_i,Y'_j}|)}, &\mbox{if } \rho_{Y'_i, Y'_j} \geq 0 \\
 \geq \sqrt{2(1-|\rho_{Y'_i,Y'_j}|)}, &\mbox{if } \rho_{Y'_i, Y'_j} < 0 \\
\end{cases}\\
& =\sqrt{2(1-|\rho_{Y_i,Y_j}|)} \equiv d_{ij}. &\end{aligned}$$

Thus $d_{ik}+d_{kj} \geq d_{ij}$ , and our metric satisfies the three axioms of a metric.

For each of the 22 financial time series, we calculate the return time series, defined as the change of logarithmic price of time series $i$ $$R_i(t) \equiv \ln(Y_i(t+1))-\ln(Y_i(t)).
\label{eq2}$$ Here $Y_i(t)$ is the daily price time series of financial asset $i$. For each of the 22 time series, we also calculate the volatility time series which is defined simply as the absolute value of the return $|R_i|$ $$V_i(t) \equiv |R_i(t)| = |\ln(Y_i(t+1))-\ln(Y_i(t))|.
\label{eq3}$$

Additionally, we define the cross-correlation for our analysis. Consider two time series $\{y_t\}$ and $\{y'_t\}$. The cross-correlation between $\{y_t\}$ and $\{y'_t\}$ is given by $$C_{y,y'}(n)\equiv \overline{(y_t-\mu)(y'_{t+n}-\mu'))}/(\sigma\sigma'),
\label{eq1}$$ where $\mu$ and $\mu'$ are the respective means and $\sigma$ and $\sigma'$ are the respective standard deviations of the series $\{y_i\}$ and $\{y_i'\}$.

The efficient market hypothesis, a basic tenet of modern economics, states that markets are approximately efficient, meaning that one cannot consistently achieve returns better than the market because all information about an asset is already incorporated into that asset’s price [@Samuelson; @ManandStan]. As a result, it is believed that the long-range memory cannot exist in any return time series. Suppose that long-range auto-correlations exist in a return time series: investors may then obtain benefits by using information, which stands in contradiction to the principle of an efficient market. Consider the cross-correlation function (Eq. (\[eq1\])) between return time series of asset $i$ and asset $j$. Any significant cross-correlations $C_{R_i,R_j}(n)$ in $n \neq 0$ of two return time series would also contradict the existence of an efficient market. Therefore we can assume that significant cross-correlations $C_{R_i,R_j}(n)$ will only exist for the case $n=0$.

However, because trading occurs at different times in different cities, some markets are open when others are closed. The effect of non-synchronous trading in time series analysis has been well stated [@Lo; @Lin]. In fact, the highest degree of correlation between different markets may be detected at a one-day time lag because of the time difference.

Therefore, significant cross-correlations $C_{R_i,R_j}(n)$ may also exist for $n=-1$ or $n=1$. Additionally, we only care about the magnitude and not the sign of cross-correlation. Thus, we define the absolute correlation coefficient as $$\rho_{ij}=\max \left (\left \vert C_{R_i,R_j}(n)\right \vert \right )
\label{eq4}$$ for $n=-1, 0, 1$.

In volatility time series, long-range correlations $C_{V_i,V_j}(n)$ have been shown to exist \[10-13\]. It follows that significant cross-correlations $C_{V_i,V_j}(n)$ may exist for $n\gg0$ or $n\ll0$. Additionally, the existence of long-range negative correlations between past returns and future volatility [@Bouchaud; @Perello], known as the leverage effect, has also been reported. This correlation is moderate and decays exponentially over the long term. However, while both of these types of correlations may help predict the financial risks on a long-range time interval, we point out that neither the negative correlation between returns and volatilities nor long-range autocorrelation of volatility can be used to obtain benefits. This is because the price volatility does not include the direction of price changes, and so neither contradicts the efficient market hypothesis.

As a sample, the correlation functions of volatility $C_{V_i, V_j}(n)$ and return $C_{R_i, R_j}(n)$ between the FTSE100 and DJIA are shown in Fig. \[1\]. The peaks (highest correlations) of both correlation functions are near n=0; however, the correlation function for returns is fast decaying, quickly approaching 0 for $n \neq 0$, while the volatility correlation function is slow decaying with $C_{V_i, V_j}(n) > 0$ for $n > -50$ and $n < 50$.

We now turn our attention to the stability and structure of MSTs, which are made using the distance based defined in Eq. \[eq6\]. Following our discussion on MSTs, we will show the correlation function graphs of volatility that relate the correlation $C(n)$ to the time-lag $n$, specifically focusing on the value of $n$ that gives the LOWESS (the smoother which uses locally weighted polynomial regression) equal to its maximum value. Here, the LOWESS is defined by a complex algorithm, proposed by W. S Cleveland [@Cleveland] in 1981. For each value, we define 10th nearest neighbor values as the local region, which is used to calculate the LOWESS value.

An MST is defined as the set of $n-1$ links that connect a set of $n$ elements in the smallest possible total distance [@Graham]. MSTs have been used in prior papers [@Mantegna; @Bonanno2; @Bonanno1; @Onnela1; @Onnela2] to connect financial data, illustrating the MST’s usefulness in highlighting the interactions between a number of financial time series.

In Fig. \[2\], we find that, although MSTs show significantly different structures in different calendar years, the same type of financial assets tend to group together consistently over time.

We also find that the stock market indicators (blue in Fig. \[2\]) and currency futures (green) groups show stronger interconnections than the commodities (red) group. For stock market indicators and currency futures, financial factors are the predominant reason for price changes. On the other hand, commodity futures may be just as much affected by investment as they are by actual supply and demand. Speculation in commodity futures may alter pricing in a way that contradicts the law of supply and demand. The existence of such contradictions depends on commodity type and the calendar year and therefore serve to decrease the stability of the MST. If this reasoning is correct, increasing the time span of the time series for cross-correlation should make the observed connections more stable. In Fig. \[3\], we show the MST using the time series from January 2007 to September 2011. We note that only two coal futures are not connected to the commodity group and the stability is greater than for single-year time series.

In Fig.\[4\], we describe the cross-correlation functions of the volatility time series. We show the cross-correlation function of main stock market indicators with EUA in (a), and with WTI in (b). A systemic time shift between EUA or WTI, and stock market indicators can clearly be seen. Since the maximum cross-correlation coefficient in most functions is not much greater than 0.2, the connections are not so strong, but certainly they are significant. Further tests of Granger causality also show that the volatility of stock indices is useful in forecasting the volatility of EUA and WTI approximately 20 to 120 days in advance.

As mentioned before, the correlation function of volatility is a slow-decaying function. It is much more slowly decaying than the correlation function of return time series (see Fig. \[1\]), meaning that a long-range cross-correlation relationship exists. If we consider significant cross-correlations between different volatility time series to be an information transfer between different assets, the time lag corresponding to the highest values of correlation gives the time lag of that information transfer. It is worth pointing out that the time lag between each pair of stock markets is approximately 0, such as the time lag between DJIA and other 27 stocks indicators are shown in Fig. \[4\] (d) . We show a simple summary of such time lag in Fig. \[4\] (c) and (d) If there is information affecting stock markets at a time of 0 days, the EUA and WTI crude oil futures will be affected by this information roughly 30 and 90 days later respectively. Because both crude oil and the ability to emit carbon are major inputs in the world economy, the existence of this time lag can have strong implications in terms of potential economic feedback loops.

Discussion
==========

The drawn MSTs are reflective of a number of easily reasoned underlying economic relationships, both through the stability and specificity of the links.

As expected, we find a stable tendency for like financial assets to cluster. Even during the otherwise anomalous 2007 subprime lending crisis and 2008 global financial crisis, this clustering tendency is preserved. This indicates the existence of strong stable connections, which come out of the strong cross-correlations, reflective of basic economic features and interactions. These connections are stable over time and not affected by market conditions.

On the other hand, certain portions of the MSTs are consistently unstable, like those relating to coal. This also may be reflective of economic relationships. Unlike other commodities like oil, speculation in coal is limited, so the movement of the coal futures may be simple supply and demand, as opposed to driven by speculation. Coal’s lack of strong connections to other commodities may be a result of investor’s low speculation in stock when building their commodities portfolios.

Certain connections in our MSTs reveal underlying relationships that are created by regulation. For example, we find that EUA futures mostly connect with the base electricity and natural gas futures, which show stable correlations among them. EU allowance permits, as a part of the European Union Emission Trading Scheme, are either allocated or auctioned and allow a firm to emit a designated amount of carbon dioxide [@Stavins]. Since power generation accounts for about one-quarter of total emissions of carbon dioxide, and natural gas is the most resource of electricity generation in UK, the stable connections of EUA to UK base electricity futures and UK natural gas futures in our MST graph is reasonable.

Also intuitive is the location specific clustering of stock market indices. AORD, an Australian index, consistently appears closely linked to those of nearby countries like New Zealand, Japan, and China. Similarly, the HSI not only keeps connections with most of the Asian stock indices like the JK11, 000001.SS, BSESN, and TWII, but also keeps connections with or otherwise stays closely connected to indices from America and Australia. Thus, the MST created reflects the common knowledge that Hong Kong is the financial center of Asia.

One benefit of the novelty of our approach is that it connects indices of dissimilar type, yielding new insights. The index that connects most to coal is OSEAX, that of Norway. Norway is rich in oil and natural gas, which explains why the Norway stock index appears as the most “coal-like” of the national indices. Similarly, the most “currency-like” of the national stock indices seems to be AEX, the Dutch securities index.

We also note the relationship between the centrality in the network and real world geographical knowledge. SOK/SEK is a currency future that is among the furthest from the center. This is intuitive, since currency trading between Norway and Sweden has little to do with financial activity in the rest of the world. The same principle applies to the trading of Euros with the British pound, shown as EUR/GBP. The Australian dollar, on the other hand, plays a central role in its exchanges with far away currencies like the US dollar, Euro, Japanese yen, and Canadian dollar.

Conclusion
==========

In this paper, we have analyzed the correlation function of return and volatility time series, constructed MSTs based on return time series, and found consistent time lags in the correlation functions of the volatilities. From these analysis, we have two main conclusions. (i) The stability of MST structure clustering between like commodities reflects a basic rule of economic activity, that the interaction between economic actors is not easily affected by capital movement. The method of absolute cross-correlation coefficient based MSTs has strong implications in the ongoing debate about the relationships of different financial commodity time series. (ii) We find that a time lag of correlation functions of volatility appears between stock markets and EUA and WTI. From this finding, we hypothesize that there may be systemic differences in the spread of financial risk, most often quantified as volatility. In other words, as concerns risk, different types of markets may have different sensitivities to economic information and other influencing factors. It would be interesting, from theoretical point of view, to generalize this time lag to predict the financial risks on a much longer time interval. However, much more would need to be understood first, such as the properties and mechanisms for this time lag. Hence, further empirical study is needed first. We endeavor to address this question more in future work.

Appendix: Set of Data
=====================

The data under investigation includes 28 stock market indicators, 21 currency futures, and 20 commodity futures.

The stock market indicators investigated are:

    Symbol    Meaning                        Notes                Symbol  Meaning                         Notes
  ----------- ------------------------------ ------------------- -------- ------------------------------- -------------------
   000001.SS  SSE Composite Index            Shanghai stocks       ISEQ   Irish Stock Exchange Quotient   
      AEX     Amsterdam Exchange Index       Dutch securities      JKII   Jakarta Islamic index           
     AORD     All Ordinaries                 Australian stocks     KLSE   Kuala Lumpur Stock Exchange     
      ATX     Austrian Traded Index                                KS11   Seoul Composite (South Korea)   
     BSESN    SENSEX                         Bombay stocks         MXX    Mexican Stock Exchange IPC      
     BVSP     Bovespa Index                  São Paulo stocks      N225   NIKKEI 225                      Tokyo stocks
      CAC     CAC 40                         French stocks         NZ50   NZX 50 Index                    New Zealand index
     DJIA     Dow Jones Industrial Average   American index        OMX    OMX Stockholm 30                
     FTSE     FTSE 100                       London stocks        OMXC20  OMX Copenhagen 20               
    FTSEMIB   FTSE MIB                       Italian stocks       OSEAX   Oslo Børs All Share Index       
     GDAXI    Deutscher Aktien Index         German blue chips     STI    Straits Times Index             Singapore stocks
    GSPTSE    S&P/TSX Composite Index        Toronto stocks        SSMI   Swiss Market Index              
      HSI     Hang Seng Index                Hong Kong stocks     TA100   Tel Aviv 100                    
     IBEX     IBEX 35                        Spanish stocks        TWII   TSEC weighted index             Taiwanese stocks

The commodity and currency futures investigated are all traded in the markets of intercontinental exchange. The commodity futures are:

    Symbol   Meaning                             Notes                    Symbol   Meaning                               Notes
  ---------- ----------------------------------- ---------------------- ---------- ------------------------------------- ----------------------
    Barley   Western Barley Futures                                      Gasoline  RBOB Gasoline Futures                 
   BrCrude   Brent Crude Futures                 North Sea crude oil     HeatOil   Heating Oil Futures                   
    Canola   Canola Futures                                               NatGas   UK Natural Gas Futures                
     CCI     Consumer Confidence Index Futures                           PeakElec  UK Peak Electricity Futures           
    Cocoa    Cocoa Futures                                                RBCoal   Richards Bay Coal Futures             
    Cotton   Cotton No. 2 Futures                                         RCoal    Rotterdam Coal Futures                
    FCOJA    FCOJ-A Futures                      Florida orange juice     RJ/CBR   Thomson Reuters/Jefferies CRB Index   assorted commodities
   Electric  UK Base Electricity Futures                                  Sugar    Sugar No. 11 Futures                  
    Coffee   Coffee “C” Futures                                            WTI     West Texas Intermediate               Texas crude oil
    GasOil   Gas Oil Futures                                               EUA     EU emission allowance                 

Additionally, the currency futures in the form A/B refers to the value of currency A in units of currency B. For example, USD/EUR would be the value of US dollars in units of Euros. The currency futures investigated are:

   Symbol  Currency             Traded in units of
  -------- -------------------- -----------------------------------
    AUD    Australian dollar    CAD, JPY, NZD, USD
    CAD    Canadian dollar      JPY
    EUR    Euro                 AUD, CAD, GBP, JPY
    GDP    British pound        AUD, CAD, JPY, NOK, NZD, ZAR, SEK
    JPY    Japanese yen         
    NOK    Norwegian krone      JPY, SEK
    NZD    New Zealand dollar   JPY, USD
    SEK    Swedish krona        JPY
    ZAR    South African rand   

[99]{}

R. N. Mantegna, Eur. Phys. J. B [**11**]{}, 193-197 (1999). G. Bonanno, N. Vandewalle and R.N. Mantegna, Phys. Rev. E [**62**]{}, R7615-R7618 (2000). G. Bonanno, F. Lillo, and R.N. Mantegna, Quantitative Finance [**1**]{}, 96-104 (2001). J.-P. Onnela, K. Kaski and J. Kertesz, Eur. Phys. J. B [**38**]{}, 353-362 (2004). J.-P. Onnela, A. Chakraborti, K. Kaski, J. Kertesz, and A. Kanto, Phys. Rev. E [**68**]{}, 056110 (2003) D. Song, M. Tumminello, W. Zhou, and R. N. Mantegna, Phys. Rev. E [**84**]{}, 026108 (2011) S. Miccichea, G. Bonannob, F. Lilloa, and R. N. Mantegna, Physica A [**324**]{}, 66-73, (2003) M. Tumminello, F. Lillo, and R. N. Mantegna, J. Econ. Behav. Org. [**75**]{} 40 (2010); Phys. Rev. E, [**76**]{} 031123 (2007).

R. Mantegna and H. E. Stanley, An Introduction to Econophysics Correlations and Complexity in Finance, Cambridge University Press (2000). L. Laloux, P. Cizeau, J.-P. Bouchaud and M. Potters Phys. Rev. Lett. [**83**]{} 1467 (1999). V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Amaral, and H. E. Stanley, Phys. Rev. Lett. [**83**]{} 1471 (1999). V. Plerou, P. Gopikrishnan, B. Rosenow, L. A. N. Amaral, T. Guhr, and H. E. Stanley, Phys. Rev. E [**65**]{} 066126 (2002). A. Utsugi, K. Ino, and M. Oshikawa Phys. Rev. E [**70**]{} 026110 (2004).

D. Wang, B. Podobnik, D. Horvatic, I. Grosse, and H. E. Stanley, Phys. Rev. E [**83**]{} 046121 (2011). The strict mathematical definition of a metric actually requires that the metric satisfy a fourth axiom, that of positive definiteness (non-negativity), which both the Mantegna-Sornette metric and ours satisfy trivially. See: A. V. Arkhangel’skii, and L. S. Pontryagin, [*General Topology I: Basic Concepts and Constructions Dimension Theory, Encyclopaedia of Mathematical Sciences*]{}, (Springer, 1990). L.Norden and M.Weber European Financial Management [**15**]{} 529 (2009). S. H. Kwan Journal of Financial Economics [**40**]{} 63 (1996). S. Hammoudeh and H. Li Journal of Economics and Business [**57**]{} 1 (2005). http://www.businessweek.com/ap/2012-05/D9V3TCDG0.htm W. Feller, An Introduction to Probability Theory and Its Applications, Wiley 1997 . P. Samuelson (1965). Industrial Management Review [**6**]{}: 41. W.-L. Lin, R. F. Engle, and T. Ito, The Review of Financial Studies [**7**]{} 507 (1994) A. W. Lo, and A. C. Mackinlay, Journal or Econometrics [**45**]{} 181 (1990) J-P Bouchaud, A. Matacz, and M. Potters, Phys. Rev. Lett. [**87**]{}, 228701 (2001) J. Perello, and J. Masoliver Phys. Rev. E [**67**]{}, 037102 (2003) W. S. Cleveland, The American Statistician [**35**]{} 54 (1981) R. L. Graham,and P. Hell, Annals of the History of Computing [**7**]{} 43 (1985) S. R. Stavins, Handbook of Environmental Economics, Elsevier, Edition 1 Volume 1 (2001)

![ The cross-correlation function $C(n)$ of volatility (a) and return (b) time series between the DJIA and FTSE100. Both show statistically significant correlation coefficients at their maxima near time lag=0 (dotted curve). Solid lines show the LOWESS (locally weighted scatter plot smoothing) values of $C(n)$, smoothed over a span of 30 days. []{data-label="1"}](fig1.eps "fig:"){width="90.00000%"}\

![The MST obtained from the absolute correlation coefficients $|\rho_{ij}|$ of the set of 68 return time series during in individual calendar years (a) 2007 (b) 2008 (c) 2009 (d) 2010. Red indicates commodity futures, blue indicates currency futures, and green represents stock market indicators. See Appendix for a listing of symbols and their meanings. []{data-label="2"}](fig2.eps "fig:"){width="99.00000%"}\

![The minimal-spanning tree (MST), similar to Fig. \[2\], but for the longer time span January 2007 to September 2011. See Appendix for symbol definitions.[]{data-label="3"}](fig3.eps "fig:"){width="90.00000%"}\

![Cross-correlation function $C(n)$ of volatility daily time series between 5 main stock market indices and (a) EUA (European carbon emissions permits), or (b) WTI (light sweet crude oil). Red lines indicate the locally weighted scatter plot smoothing values (LOWESS) of $C(n)$. Graphs show systemic time shift for the highest cross-correlation value. This time shift is observed in most stock markets. (c) indicates the average time-lag between EUA, WTI, and 28 stock market indicators, with error bars showing the standard deviations. (d) indicates the average time-lag between the DJIA and other 27 stock market indicators, with error bars showing the standard deviations. The time lags (in days) are calculated from the time lag $n$ of highest LOWESS values.[]{data-label="4"}](fig4.eps "fig:"){width="90.00000%"}\
