Rare and uncommon risks and the financial meltdown

Abstract

2008–2009 will be remembered as an era where extreme events have come on their own, ex-ante ignored, but factual and painful ex-post. Ex ante we have a tendency to ignore rare events, seeking comfort in numbers we can point out to and ignoring the others (Paul Samuelson). For example, insurance firms concentrate on aggregate risks, avoiding the existence of outliers and non-quantifiable rare events. Yet, risks – true risks, are outliers! Jean Pierre Landau (Vice Governor of the Bank of France), points out that modern finance is based in practice if not in theory on an “implied” ignorance of extreme risks (and implied complexity, my addition) of financial markets. There is thus an overall weakness when confronted with the theoretical implications of rare and extreme events. History and cemeteries are filled with their consequences, however, that have for many reasons and conveniently been forgotten (both due to our inability to confront these events and our own “finiteness” – presuming that it will not happen to us!). Extreme risks are increasingly common and recurring, however. On the one hand climatic changes, population and concentrations growths are causing previously “unthinkable” disasters to be recurrent. Extreme weather is now a TV show while Terror is ever present and everywhere. Extreme and Rare risks, are now at the center stage of our working probability distributions. Yet, they are not always appreciated at their just importance. Financial mathematics for example, presumes both the “predictability” of future prices, interest rates as well as other and related time series emphasizing that “financial uncertainty” is a Martingale, fair and expectedly constant. Such processes have been presumed by Bachelier already in 1900 and underlie the Random Walk Hypothesis (and the Brownian motion) in finance (Cootner, 1964) and in physics (Einstein, 1906). These processes have special characteristics and consist of independent increments, independently and identically distributed Gaussian (thin tail) random variables, contributing to our continued fixation on linearly growing variance (as a measure of uncertainty) over time. This facet of “the growth of uncertainty” has been severely criticized and numerous statistical tests have been based on it to demonstrate that the underlying process need not be Brownian motion. Empirical evidence has shown that financial series are not “well behaved” and cannot be always predicted. They may exhibit unpredictable and “chaotic behavior” which underscores “nonlinear science” approaches to finance. Rather, in many cases, it is observed that data can behave “unpredictably” at time and at others, it may exhibit regular variations. “Bursts” of activity, “feedback volatility” and broadly varying behaviors by stock market agents, “memory” (both long and short, exhibiting persistent behaviors) etc. are characteristics that contribute to the “nonlinearity of uncertainty growth” and thereby to challenging fundamental finance. Further, even aggregation of time series that are mildly auto-regressive can turn out to have long run memory and thereby to serious contentions regarding the assumptions of fundamental finance. By the same token, Vallois and myself [42,43] have indicated that persistence in pure random walks has a short memory and lead to a nonlinear evolution of the process volatility. The study of real time series have motivated a number of approaches falling under a number of themes spanning: fat tails, Leptokurtic distributions, Pareto– Levy stable distributions, long run memory – fractional Brownian models, dependence, persistent processes,