What is a black swan? In Taleb (2007) a black swan is an event that cannot be predicted. These dark creatures of probability are what are more esoterically referred to as unknown unknowns. Unknown unknowns are events that are not possible of knowing what and when such events take place. In chaos theory this can be described as a situation where the non-deterministic chaotic system goes through a full range of attraction points—when a certain threshold is met.
In financial trading, many market participants think that their models or trading strategy will reveal the inner workings of the market. But since the financial markets have many degrees-of-freedom this complex chaotic system will exhibit black swan behavior. For example, the actual risk associated with the CDS market per–Lehman was not fully understood—with all its complex counterparty interactions—leading to institutions taking more risk than might be prudent. This was mainly due to the way the risk was measured. This risk was measured using a normal distribution for counterparty defaults within the copula functions that risk management departs used. Unfortunately this risk should have been measured using a default distribution that was more skewed with a high kurtosis, e.g. t-copula methodology.
Dragon kings are an entirely different beast. The mythical creatures are not so benign after all. In Sorrentte (2003) a dragon king is when endogenous risk is building up in a complex system. This complex system could be a deterministic or non-deterministic chaotic system. The main point that Sorrentte is conveying is that in many complex systems there are signals within the system that can be filtered out to determine a regime switching or system failure. Sorrettes’ work in rupture dynamics helped with his modeling of market crashes using a log periodic function method.
The periodic wave function has higher frequency before a mean reversion. In theory, one can parameterize a log periodic function and project the critical time that an asset will mean revert. Sorrentte (2003) suggests time scales of 8 years to be able to determine critical times of mean reversion.