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Conceived and designed the experiments: DYK MR TL EBJ. Performed the experiments: DYK MR TL EBJ. Analyzed the data: DYK MR TL EBJ. Contributed reagents/materials/analysis tools: DYK MR TL EBJ. Wrote the paper: DYK MR TL EBJ.

Eshel Ben-Jacob is an academic editor in PLoS ONE. This does not alter the authors' adherence to all the PLoS ONE policies on sharing data and materials.

In the current era of strong worldwide market couplings the global financial village became highly prone to systemic collapses, events that can rapidly sweep throughout the entire village.

We present a new methodology to assess and quantify inter-market relations. The approach is based on the correlations between the market index, the index volatility, the market Index Cohesive Force and the meta-correlations (correlations between the intra-correlations.) We investigated the relations between six important world markets—U.S., U.K., Germany, Japan, China and India—from January 2000 until December 2010. We found that while the developed “western” markets (U.S., U.K., Germany) are highly correlated, the interdependencies between these markets and the developing “eastern” markets (India and China) are volatile and with noticeable maxima at times of global world events. The Japanese market switches “identity”—it switches between periods of high meta-correlations with the “western” markets and periods when it behaves more similarly to the “eastern” markets.

The methodological framework presented here provides a way to quantify the evolvement of interdependencies in the global market, evaluate a world financial network and quantify changes in the world inter market relations. Such changes can be used as precursors to the agitation of the global financial village. Hence, the new approach can help to develop a sensitive “financial seismograph” to detect early signs of global financial crises so they can be treated before they develop into worldwide events.

Has the world become one small financial global village? Coupling between the world's different markets has become stronger and stronger over the past years, as is evidenced by the financial difficulties, which are affecting many markets around the globe, especially since late 2008. The growing financial integration allows capital to flow rather freely between countries and markets. Investments in stocks can be diversified into global portfolios, consisting of multiple assets from a large number of markets. As a result, stock markets have turned into an extended and strongly coupled complex system, in which large movements in price and volatility are likely to be transferred from one market to the other due to portfolio readjustments. Engle et al.

When it comes to the analysis of individual markets, a wealth of different measures have been devised and used to analyze similarity between financial time series. These include Pearson's correlations

Recently, Kenett et al. investigated the dynamics of correlations between stocks belonging to the S&P 500 index, and the residual (partial) correlations after removing the influence of the index

The similarity between stock price changes is commonly calculated via the Pearson's correlation coefficient. The raw stock correlations

Partial correlation is a powerful tool to investigate how the correlation between two stocks depends on the correlation of each of the stocks with a third mediating stock or with the index as is considered here. The residual, or partial, correlation between stocks

Note that according to this definition,

To investigate the dynamics of correlations in capital markets, we make use of a running window analysis. We use a short time window, of 22-trading days, which is equivalent to one work month, with a full overlap. Thus, for example the first window will be days 1–22, the second window day 2–23, etc. At each window we calculate stock correlation and partial correlation matrices, and average them. This results in a value of correlation (partial correlation) for each stock, representing its average correlation (partial correlation) to all other stocks. This is defined as

Finally, we calculate the average of average correlations (partial correlations), which represents the total average correlation (partial correlation) in the market,

We denote this variable as the intra-correlation (intra partial correlation), as it represents the average correlation of stocks belonging to one given market.

Next, we investigate the synchronization of two given markets. To this end, we calculate correlation and lagged cross correlation between the intra correlations of each market. The correlation of market correlations is denoted as market meta-correlation (MC), given by

Recently, it was shown that the market index has a cohesive effect on the dynamics of the stock correlations

For the analysis reported in this paper we use data of the daily adjusted closing price from stocks in six different markets, all downloaded from Thomson Reuters Datastream. The markets investigated are the U.S., U.K., Germany, Japan, India and China. These include the four main stock markets as well as two less developed markets for comparison of the results. For each market we aimed for a sample as broad as possible, without any ex ante selection of branches. See

Market | Stocks used | Index used | # before | # filtered |

U.S. | S&P 500 | S&P 500 | 500 | 403 |

U.K. | FTSE 350 | FTSE 350 | 356 | 116 |

Germany | DAX Composite | DAX 30 Performance | 605 | 89 |

Japan | Nikkei 500 | Nikkei 500 | 500 | 315 |

India | BSE 200 | BSE 100 | 193 | 126 |

China | SSE Composite | SSE Composite | 1204 | 69 |

It should be noted that the correlations measured have some explanatory limitations, which are mainly due to structural differences of the markets and to selection issues of the stocks. First of all, the dataset is by construction biased towards long-lived stocks. Secondly, the intra-market correlations have been calculated on the basis of a market index, which composition is undergoing changes over time. However, we are pretty certain that the exact composition of the index used for the normalization does not have significant influence on the results.

When we compare time series from different markets, some adjustments must be made, mostly due to differences in trading days. To this end, we either only used data from days in which trading was done in both markets, or we replaced missing data with that of the observation of the last trading day. These two methods yield very similar results for the correlation analysis. When comparing all markets together, we used the joint trading period of the London (U.K.) and Frankfurt (Germany) stock exchange (the bilateral pair which has the most overlap with all other markets) and again replaced missing observations for all other markets with last day observation. For comparisons of the U.S. and Japan one should be aware that it makes sense to consider observations of day

A first proxy to the dynamics of the different world's economies is the dynamics of their leading market indices. Here we focus on six of the world's largest economies, representing western markets – U.S., U.K., and Germany – and eastern markets - Japan, India and China. The stock price indices of these countries are presented in

All indices have been normalized by their mean. The indices of the U.S., U.K. and Germany (blue, green and red line) appear almost as if they are shifted parallel, which is a sign of their high correlation. Note that all price indices are based on stock prices in local currency.

Investigating the index volatility, rather than the index price reveals meaningful hidden information. Studying

The price indices data was standardized for the 10-year interval (the mean is zero and the variance is 1 for each complete time series). The volatility peaks for the U.S, U.K. and Germany mostly coincide while there is less similarity with Japan. India and China show a very different behavior of volatility, especially until 2007.

To understand the dynamics of capital markets, much research has focused on the analysis of correlations

For each market, we use a 22-day window, and in each window calculate the intra-correlation. This results in the dynamics of the intra-correlation for the period of 2000–2010, for each market separately. Each horizontal line represents the average correlation of one stock (the left y-axis displays the number of the stock). The western markets and Japan show a similar behavior, visualized through vertical stripes at the same time, showing synchronized waves of strong correlations. The black line shows the average of all correlations at a given 22-day window (corresponding to the right y-axis). The trend is increasing for all countries except for China.

For each market, a bursting behavior for the intra-correlations is observed. This is consistent with previous findings

Next, we calculate for each of the markets the Index Cohesive Force (ICF,

The dynamics of the ICF for each market is plotted, for the period of 2000–2010.

Markets featuring similar values of the ICF will have a similar dependency on the market index. Thus, if the indices of these markets are highly correlated, these markets should be strongly coupled. As such, the ICF provides new important information on these couplings.

The observed similarities of indices and correlation patters leads to the question of how synchronized stock markets are with respect to changes in these measures. Thus, we computed the meta-correlations – the correlations between the intra-correlations, using a 66-day window. The index correlations, the index volatility correlations and the ICF correlations were calculated using the same window size.

According to the index correlations the three “western” markets - U.S., U.K. and Germany - are highly correlated. The index correlations between Japan and India and all other markets are significantly weaker (the difference between these two groups is even more visible for the index volatility). China finally seems rather uncorrelated with the rest of the world, although some upward trend is visible (see a year by year breakdown for the market pair correlations in

The introduction of the Index Cohesive Force makes things easier when one is considering the dynamics of a particular market and especially if one is interested in its stability, and provides a valuable measure to assess the state of each individual market. Previous work has shown that in the case of the U.S. market, low values of the ICF (lower than 10) correspond to a relatively healthy state of the market

Much better results are obtained using the meta-correlations. Using this measure, we found that the three “western” markets have a high level of uniformity. The Japanese market appears to be significantly more influenced by the “west” than the Indian market, the Chinese has the lowest correlations. The latter is in line with what was expected for example given the capital controls and regulations in China and limitations for foreign investors

Both calculated using a 66-day window. The U.S. and Germany show a higher similarity for both measures than the U.S. and Japan. While both measures fluctuate over time, we observe that high correlations do not necessarily show jointly in the top and the bottom figure. We can thus differentiate between times of identical price movements (high index correlation) and global stress (high index correlation and high meta-correlation).

The coupling of markets, as quantified by the meta-correlation, changes over time. The Japanese market switches between following the “western” and following the “eastern” worlds: for some time intervals it behaves very similar to the U.S. market (which is also similar to the U.K. and Germany markets), and at other times, the intra-correlations of Japan behave more similar to that of the Asian countries. Similar observations can be made for U.K. and Germany and their similarity to the U.S. vs. Asia. The interdependencies between India and China and the more developed markets are very volatile over time and show maxima in years with important global events (2001: 9/11-attacks, 2003: Iraq war, SARS, etc.). To illustrate the general development, we show the differences in coupling between markets during 2001 and 2010 (see

The width of the edges of the graph is proportional to the meta-correlation between the markets it connects (right legend). The node size is proportional to the inter-correlations (left legend). For 2000 we observe markets with low intra-correlations and inter-correlations of similar magnitude, excluding China. For 2010 we observe much higher intra-correlations in all markets and a denser network of interdependencies. (The nodes for the U.K. and Germany are further away from each other than their geographical position).

This paper presents a new framework for quantitative assessments of the coupling and interdependences between different markets in the global financial village. The new approach also provides the means to study feedback between the micro (intra market) and the macro (inter markets) levels. More specifically, the stock-stock correlations in the individual markets represent local market dynamics, whereas the meta-correlations represent global market dynamics. Thus, the methodology presented here of intra and meta correlation analysis provides the means to study the bottom-top and top-down feedback mechanisms which take place in the world's economies.

Our results provide new information about the uniformity preset in the world's economies. We find significant uniformity for the three western markets, whereas Japan and India display a greater extent of multiformity; however, this multiformity is time dependent, and periods of significant uniformity with the western markets are observed. Unlike these, the case of China is significantly different. For all our measures it shows an amount of segmentation that is not in line with China's important role in the world economy, especially the huge trade flows we observe.

Earlier studies hint that the sole legal possibility to invest in emerging markets is not sufficient for their full integration with other markets. Investment funds and other institutional investors need accompanying financial products (i.e. country funds, depository receipts, and other derivatives), which are only gradually becoming available in emerging markets. Country specific risks, taxes and holding time requirements can further dampen cross-border investments

Finally, some interesting observations can be made about the general development of financial markets. It has been much debated that markets have become more coupled over the last years, and that we are observing the downside of this development right now during the debt crisis within the Eurozone and the U.S., expressed in pronounced synchronized movements of stock markets. From our analysis it becomes evident that this uniformity does not only stem from an increase of correlation between markets, but that there has also been an ongoing simultaneous shift towards uniformity in each single market.

In conclusion, using new specially devised analysis methods, we provide the means to investigate and quantify uniformity and multiformity in the global market, and changes in these measures. In the current era, when the global financial village is highly prone to systemic collapses which can sweep the entire village, our approach can provide a sensitive “financial seismograph” to detect early signs of global crises.

Dynamics of the intra partial correlation. For each market, we use a 22-day window, and in each window calculate the intra partial correlation, removing the effect of the index. Each horizontal line represents the average correlation of one stock (the left ordinate displays the number of the stock).

(TIF)

Average correlation of price indices, on a year-by-year basis. The year-by-year correlations were calculated as the average over all 66 day windows of each year. The pairs are sorted in descending order by total average correlation. Averaging by years allows us to judge on the general – medium to long run – interdependence between markets.

(EPS)

Average correlation of ICF, on a year-by-year basis. The year-by-year correlations were calculated as the average over all 66 day windows of each year. The pairs are sorted in descending order by total average correlation. Averaging by years allows us to judge on the general – medium to long run – interdependence between markets.

(EPS)

Average meta-correlation, on a year-by-year basis. The year-by-year correlations were calculated as the average over all 66 day windows of each year. The pairs are sorted in descending order by total average correlation. Averaging by years allows us to judge on the general – medium to long run – interdependence between markets.

(EPS)

Average index volatility correlation, on a year-by-year basis. The year-by-year correlations were calculated as the average over all 66 day windows of each year. The pairs are sorted in descending order by total average correlation. Averaging by years allows us to judge on the general – medium to long run – interdependence between markets.

(EPS)

Distributions of the Index Cohesive Force (ICF) values for the Japaneses market in different periods - 2000–2003 (blue), 2004–2007 (orange), and 2008–2010 (red). It is observable that the distributions are different for the studied periods, and that the ICF values are higher with a fat tail distribution for periods marked by strong economic fluctuations.

(TIFF)

Index correlations (top) and meta-correlations (bottom) for US-Germany and US-Japan, Lag with maximum correlation (blue cross) and correlation at lag 0 for the cross-correlation of the indices. Both performed for a 66-day window.

(EPS)

Index correlations (top) and meta-correlations (bottom) for US-UK and US-India, Lag with maximum correlation (blue cross) and correlation at lag 0 for the cross-correlation of the indices. Both performed for a 66-day window.

(EPS)

Index correlations (top) and meta-correlations (bottom) for US-China and UK-Germany, Lag with maximum correlation (blue cross) and correlation at lag 0 for the cross-correlation of the indices. Both performed for a 66-day window.

(EPS)

Index correlations (top) and meta-correlations (bottom) for UK-Japan and UK-India, Lag with maximum correlation (blue cross) and correlation at lag 0 for the cross-correlation of the indices. Both performed for a 66-day window.

(EPS)

Index correlations (top) and meta-correlations (bottom) for UK-China and Germany-Japan, Lag with maximum correlation (blue cross) and correlation at lag 0 for the cross-correlation of the indices. Both performed for a 66-day window.

(EPS)

Index correlations (top) and meta-correlations (bottom) for Germany-India and Germany-China, Lag with maximum correlation (blue cross) and correlation at lag 0 for the cross-correlation of the indices. Both performed for a 66-day window.

(EPS)

Index correlations (top) and meta-correlations (bottom) for Japan-India and Japan-China, Lag with maximum correlation (blue cross) and correlation at lag 0 for the cross-correlation of the indices. Both performed for a 66-day window.

(EPS)

Index correlations (top) and meta-correlations (bottom) for India-China, Lag with maximum correlation (blue cross) and correlation at lag 0 for the cross-correlation of the indices. Both performed for a 66-day window.

(EPS)

Cross-correlation plot of meta-correlations. Black marker on maximum if the correlation is higher than .7, U.S. against all other markets. Performed for a 66-day window.

(EPS)

Cross-correlation plot of index correlations. Black marker on maximum if the correlation is higher than .7, U.S. against all other markets. Performed for a 66 day window.

(EPS)

MR and DYK wish to thank Friedrich Wagner for fruitful conversations and comments. DYK and EBJ wish to thank Tobias Preis and Yoash Shapira for all of their comments and suggestions on this work.