The term *Black Swan* has gained popularity of late, largely due to the enormous success of Hollywood’s recent thriller of the same name. The Black Swans we discuss in this article, however, are more unpopular, involving neither ballet, nor popcorn, nor – most importantly – any synopsis of what we are about to experience. These are rare events with dramatic consequences, and extremly difficult to predict. In this post, we will look at several key indicators of looming financial disaster to determine whether we can predict a would-be Black Swan Event.

Students rarely remember “skewness” after a Statistics course. The measure captures how asymmetric the probability distribution of a random variable is. A *left skewed* distribution indicates that the tail on the left side is longer than the tail on the right side; even if most outcomes are above the mean, low outcomes can be much larger than high outcomes.

After the stock market crash of October 1987, the Chicago Board of Options Exchange (CBOE) introduced a Skew Index on the S&P 500 that allows investors to gauge tail risk in the stock market, with returns lower than two standard deviations below the mean indicating high lower tail risk. The Skew is calculated from the prices of out-of-the-money put options on the S&P 500.[1] Traders pay attention to the volatility smile of options, which relates the option-implied volatility to the degree of the option’s moneyness. Out-of-the-money options tend to have higher implied volatilities than at-the-money options. As lower tail risk increases, the slope of the left part of the smile steepens, turning the smile into a grimace of apprehension at the prospect of a higher probability of extreme downward moves.

In recent weeks, the menace of an impending crash has agitated financial reporters and analysts. Rupture appears imminent: high sovereign debt risk in Europe and the US, increasing fragility of the US economy, a slowdown in China, rising emerging markets inflation, and a dangerous ineptitude on the part of policymakers to tackle the problems. The economic fault lines are so wide and unstable that they create the ideal conditions for an economic earthquake of severe proportions. The result, if not properly addressed, is a Black Swan event that the Skew Index should serve to predict.

So, has the Skew been hinting at something big?

Not really. The Figure above plots the volatility (VIX) and the skewness (SKEW) from the beginning of May 2010 to last Friday, June 3, 2011. The measures are normalized, with a starting value of 100. The VIX fluctuates much more than the SKEW (as expected), and one must remember that the CBOE SKEW typically ranges from 100 to 150. As the CBOE explains: “A SKEW value of 100 means that the perceived distribution of S&P 500 log-returns is normal, and the probability of outlier returns is therefore negligible. As SKEW rises above 100, the left tail of the S&P 500 distribution acquires more weight, and the probabilities of outlier returns become more significant.” In other words, the higher the SKEW, the higher lower end tail volatility becomes. The average value of the SKEW over the last 13 months is 122.5. Since the beginning of May 2011 the average value of the SKEW is 123.8. Although the values have generally been higher this year than last year, they have not been much higher in recent weeks, and are still significantly below the maximum value of 136.4 reached in January 25, 2011.

Another way to answer the question is to determine how many times the S&P 500 has fallen by at least 2 standard deviations from the mean (previous 12-months mean). The Figure below indicates that this has occurred very few times since last September, never more than 2 times in a period of 20 days. Last week, it happened once. That may have shaken some people, but it is impossible to conclude that the risk of a large sigma-event has now been heightened.

Another method to determine if we might face a crash pertains to cross-market correlations. A common view is that cross-market correlations increase in times of trouble, as discussed in a *Financial Times* article by Aline van Duyn last Thursday. High cross-market correlations can drastically reduce the benefits of portfolio diversification and are particularly problematic if they occur in bad economic periods, because it is precisely during these periods that diversification is more valuable. Cross-market correlations appear uncontrollable: there are no good models that explain why they increase. While higher levels of risk aversion, tighter financial constraints and contagion certainly play a role, none of these factors provides an endogenous mechanism, coherently formulated. We will return to this topic in a later post.

For now, we want to check whether cross-market correlations have gone up recently. The following Figure shows the average (20-day rolling) correlation (blue line and left axis) among various commodities (agriculture, energy, softs, precious metals, industrial metals). It also tracks the DJ-UBS index of commodities (normalized to 100 on May 24, 2010). The figure shows that correlations across commodities have risen in May 2011 to levels last seen in November. But while the rise in correlation was associated with rapid falls in prices during the first week of May, prices and correlations have both risen since then.

The correlation between commodities and the US stock market, in the next Figure, has also been on the rise since February, but it has recently plateaued at last year’s levels, suggesting nothing extraordinary.

As our figures show, the indicators do not suggest a high level of imminent danger in the system. Neither the volatility (VIX), nor the cross market correlations, nor the frequency and magnitude of price falls, nor tail risk (SKEW). Nothing! It is always possible that an unforeseen storm of unknown proportions might hit – a true Black Swan event. But speculating about a growing financial crisis is, for now, simply speculation.

Sentiment among retail investors appears to be changing, with continued outflows from mutual funds since May. The strange calm in the presence of huge macroeconomic problems has left a feeling of uneasiness among many financial analysts. Option traders, on the other hand, appear to be remaining cool, at least with regard to the near-term outlook. With each group pursuing a different strategy, only the future will be able to tell whether it was the analysts’ bias or the traders’ shortsightedness and overconfidence that got it wrong this time. The indicators tell us not to be alarmed, but if the Black Swan turns out to exist after all, we can only hope that we are prepared and deal with its aftermath.

[1] Also the excess of demand for put options over call options.

## One Comment

I believe “S&P making a larger than 2sigma move” is somewhat analogous to “realized volatility”. One can potentially calculate the “break even vol” for an interval of days & a strike assuming we hedge according to Black-Scholes. VIX on the other hand would give us the vol that is being traded.

Now one can view a call option as a variance swap – the one that is dynamically hedging is swapping realized variance with traded (VIX) variance. Looks like for the time frame you mention the realized vols are smaller than traded vols. This might be because the non-dynamic hedgers (retail option buyers who do it to seek protection) are willing to pay a higher premium based on their views. The hedgers (banks mostly) are willing to receive that “higher” VIX premium based on their view of lower “realized” vol.