The anatomy of a bubble

Summary of collapse points in markets

Ref also to the work of Didier Sornette

Informally, we can think of a bubble as an advance in an asset's price to levels that are "detached from fundamentals" - essentially, the primary motive for investing ceases to be the expectation of future cash flows or consumption, and instead centers on the expectation of further increases in price.

Formally, a bubble can be defined as a "non-fundamental" component of price which grows exponentially.

Mathematically, the bubble value is B_t, in the model Price_t=V_t + B_t, where V_t is discounted value of future dividends and B_t is an arbitrary constant s.t. B_(t+1)=(1+k)B_t where K is the long-term return.

Major parabolic bubbles also tend to include short-term fluctuations which include a log-periodic component-- corrections become shallower and more frequent within the parabolic trend which leads to a crash at a finite-time singularity.
"When you have to fit a sixth-order polynomial to capture price history because exponential growth is too conservative, you're probably close to a peak."