People like to go to excesses. I think that what started out with a tulip, maybe four-hundred years ago, and continued through the South sea bubble and all of those sorts of things ( Tronic bubble in 60’s, new economy bubble in 90’s). I’m not saying here that that all human choices are orthogonal to rational ones but in human nature is to heard. Herding can result from a variety of mechanisms, such as anticipation by rational investors of noise traders strategies, agency costs and monetary incentives given to competing fund managers, sometimes leading to the extreme Ponzi schemes, rational imitation in the presence of uncertainty, and social imitation. Many financial economists recognize that positive feedbacks and in particular herding is a key factor that can push prices upward (downward) faster-than-exponentially which if unchecked can lead to bubbles.
Roughly 10 years ago Johansen and Sornette proposed model that attempts to incorporate those ingredients into a traditional rational expectations model of bubbles proposed by Blanchard.
The Jochansen-Sornette model assumes the financial market is composed of two types of investors: perfectly rational investors who have rational expectations and irrational traders who are prone to exhibit herding behavior. The noise traders drive the crash hazard rate according to their collective herding behavior, leading its critical behavior. Due to the no-arbitrage condition, this is translated into a price dynamics exhibiting super-exponential acceleration, with possible additional so-called “log-periodic” oscillations associated with a hierarchical organization and dynamics of noise traders.
A+B*(t-tc)^C*(1+D*COS(w*LN(t-tc)+O)
This so-called log-periodic power law (LPPL) dynamics given by has been previously proposed in different forms in various papers. The power law A−B(t−tc)_ expresses the super-exponential acceleration of prices due to positive feedback mechanisms. The term proportional to cos(w ln(t-tc)+o) describes a correction to this super-exponential behavior, which has the symmetry of discrete scale invariance. This formulation results from analogies with critical phase transitions (or bifurcations) occurring in complex adaptive systems with many interacting agents. The key insight is that spontaneous patterns of organization between investors emerge from repetitive interactions at the micro-level, possibly catalyzed by top-down feedbacks provided for instance by the media and macro-economic readings, which are translated into observable bubble regimes and crashes.
In previous posts I fit the LPPL formula into several financial time series. This time I fit LPPL into EUR/PLN exchange rate. As it is seen on the chart the formula fits remarkably well into the data. Even more surprisingly the LPPL model explains not only the periods when the positive feedbacks let to the zloty overvaluation but also it fits well to the period when zloty was depreciating (2001-2003). This confirms the universality of the process described by LPPL formula which describes well not only bubbles but also antibubbles.
