Introduction

A system consisting of coupled units can self-organize into a critical transition if a majority of the units suddenly and synchronously change state1,2,3. For example, in sociology, the actions of a few can induce a collective tip** point of behavior of the larger society4,5,6,7,8,9,46. The intuition is that if an unstable system is perturbed it returns more slowly to its natural state compared to a stable system. The more stable the system, the stronger the tendency to return to its natural state, so the more quickly it responds to transient perturbations.

We compute the first-order autoregression coefficient of the fluctuations of each maturity IRS time series for all possible window sizes and show a representative set of results in Figure 2; see the Methods section for details. We find indeed signs of critical slowing down around the Lehman Brothers bankruptcy for certain window sizes. However, it is difficult to find parameter values that provide a sustained advance warning, that is, where the indicator crosses the warning threshold for more than a few days before the bankruptcy. Only in the EUR data for the 1-year maturity and a sliding window size of around 1000 trade days we find a significant early warning, which disappears for a sliding window larger than 1250 trade days (see Section S5.1 in the SI).

Figure 2
figure 2

The solid blue line is the coefficient of the first-order autoregression of the detrended time series, which is a measure of critical slowing down.

The dashed red line is the warning threshold of two standard deviations above the mean of a sliding window of 750 trade days, as in Figure 1. The coefficient is computed of a sliding window of 500 (left) and 1000 (right) trade days which is detrended using a Gaussian smoothing kernel with a standard deviation of 5 trade days. We show the critical slowing down indicator for the first, second, fifth and tenth maturity in the USD and EUR markets.

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Another type of generic leading indicators used in the literature are the spatial correlation and spatial variance of the signals of the units of a system3,47,48,49,50,51,52. See Figure 3. In our data, the dimension of maturities can be taken as the ‘spatial’ dimension. The traditional correlation function used is the linear Pearson correlation, shown in the top panels of Figure 3. We also compute the correlations using the mutual information function, shown in the middle panels, since this function can capture non-linear relationships as well. In finance, correlations are often calculated using the relative differences (returns) of time series instead of the absolute values (19), so we repeat the calculations on the returns shown in the right half of Figure 3.

Figure 3
figure 3

Alternative leading indicators for the IRS time series in both markets, in levels and in returns.

We computed the average cross-maturity Pearson correlations for sliding window sizes of 100 days (blue line), 300 days (green line) and 500 days (red line) between the 1-year IRS and all other maturities. The variance at time t is computed of the levels (returns) of all maturities of the single trade day t. Time point 0 on the horizontal axis corresponds to the day of the Lehman Brothers bankruptcy.

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We find that in our time series these indicators do not show a distinctive change of behavior around the time of the bankruptcy in both markets. One possible explanation is that all IRS prices correlate strongly with external financial indices (such as the home-price index), which may dominate the observed correlations in the IRS prices across the maturities. In this scenario the IDL is still capable to be a leading indicator because it ignores the correlation that is shared among all IRS prices. That is, the information in the IRS prices of different maturities decays as a + b·(ft)1−i where a is the information (or correlation) shared among all IRS prices and the estimated rate of decay ft is independent of a.

More traditional indicators used for financial time series are the magnitude or spread of interest rates53. However, Figure 1 as well as the variances in the level data in Figure 3 show that neither measure provide a clear warning: a high (USD) and low-spread period (EUR) occurred more than a year before the bankruptcy and was returning to normal at the time of the bankruptcy.

Lastly, the same swap with a different variable payment frequency (e.g., monthly, quarterly, semi-annually) were quoted at the same price in the market before 2007. During the recent crisis, a significant price difference across frequencies emerged54. Although this has a major impact on the valuation and risk management of derivatives, this so-called ‘basis’ does not provide a clear early warning (see Section S5.2 in the SI).

Discussion

From an optimistic viewpoint, the IDL indicator may improve the stability of the financial derivatives market. Our observation that previously introduced leading indicators did not provide an early warning for the Lehman Brothers bankruptcy and the crisis that followed, is consistent with the hypothesis that leading indicators lose their predictive power in financial markets55. A plausible explanation is that an increase of a known leading indicator could be directly followed by preemptive policy by central banks56, a change of behavior of the market participants, or both, until the indicator returns to its normal level. This would imply that the financial system is capable of avoiding the type of critical transitions for which it has leading indicators: it changes behavior as it approaches such a transition, while it remains vulnerable to other critical transitions for which it has no indicators. The fact that the IDL indicator provides an early warning signal suggests that it is capable of detecting a type of transition for which the financial system had no indicators at the time. Therefore, from this viewpoint the IDL indicator potentially makes the financial system more resilient because it improves its capability of avoiding catastrophic changes.

From a pessimistic viewpoint, on the other hand, the IDL indicator may actually decrease the stability of the financial system. Upon an increase of IDL, participants may respond in a manner that increases the IDL further, reinforcing the participants' response and so on, propelling the financial system towards a crisis. This is a general dichotomy for all early warning indicators in finance57. In the absence of a mechanistic model of the financial derivatives market it is difficult to predict the effect of a warning indicator.

Our results are a marked step forward in the analysis of complex dynamical systems. The IDL is a generic indicator that may apply to any self-organizing system of coupled units. For many such systems we lack the mechanistic insight necessary to build models with sufficient predictive power. Remarkably, we find evidence that the percolation of information can provide a tell-tale of self-organized critical phenomena even in the absence of a descriptive model. Although we study the financial derivatives market here, it seems reasonable to expect that it is true for a wide range of systems such as the forming of opinions in social networks5,6,7,8,9,60,61, phase transitions and spontaneous magnetization in physics47,62,63,64, robustness in biological systems65,66 and self-organization of populations of cells67 and even software components68.

Methods

Calculating the IDL in the IRS time series

Because the IRS price levels are not stationary within the sliding window sizes we use relative differences (returns) instead. Let denote the return of an IRS with maturity i = 1,…,15 at time t, where denotes the corresponding price level. We fit the exponential decay to the measured Shannon information as function of i, where a is the mutual information that all IRS rates have in common, b is the normalizing factor and f(t) is the rate of decay of the mutual information between the IRS rates across maturities. We define the IDL as the corresponding halftime. The mutual information is estimated by constructing an adaptive69 contingency table of the two vectors and , which are the most recently observed returns in the market at time t. To construct this table we divide the range of values of each vector into h bins of variable size such that each bin contains about the same number of samples. Two observed pairs of returns are considered equal if they fall into the same bin. Our results are robust against choosing the parameters and h; see Section S4 in the SI for more details. The results in Figure 1 were produced with a window of trade days and binning the return values into h = 10 bins.

Calculating the first-order autoregression coefficient of fluctuations

Calculating this coefficient of a given time series requires two parameters: the standard deviation of the Gaussian smoothing kernel g, which de-trends the signal and the number of most recent IRS prices which are used to compute the autoregression. The procedure is identical for each maturity. First we use the smoothing kernel to compute a running weighted average of the time series, where each IRS price level becomes the weighted average of its neighbors. Then we subtract it from the original time series to obtain the de-trended signal, i.e., the short-term fluctuations. Of these fluctuations we calculate the first-order autoregression coefficient at time t using the preceding prices. The autoregressive model used is the Yule-Walker model70. The results in Figure 2 were produced with a kernel of size g = 5 and a sliding window of price levels. This procedure can be calculated for the price levels regardless of non-stationarity since it contains a de-trending step. See Sections S5.1 and S5.4 in the SI for more details as well as results for different values of g and .

Calculating the spatial correlation and variance

At each time point we calculate the spatial correlation coefficient at time t as , using the preceding IRS rates of maturity 1 and maturities i, i = 1,2,…,15. Here, Fcorr is either the standard Pearson correlation for the upper plots in Figure 3, or the mutual information function for the middle plots; denotes the arithmetic average of the correlation values for the different maturities and denotes the price of an IRS of maturity i at time t. The results in Figure 3 were produced using sliding windows of sizes . The spatial variance is computed at each time point t as . We repeat the calculations after replacing each original level by its relative difference (returns) .