from Dr. John Hussman's market commentary earlier this week:

This first link is to a paper on the website of the National Bureau of Economic Research. The NBER is the group that OFFICIALLY declares when a recession begins and ends. In their paper, they explain the phenomenon that as a bubble matures, it ACCELERATES higher, and why that contributes to the ensuing crash. Here is a quote that summarizes the paper:

“The probability that the bubble ends may well be a function of how long the bubble has lasted, or of how far the price is from market fundamentals. If the probability of a crash increases for example, the price, in the event the crash does not take place, will have to increase faster, not only to compensate for the increased probability of a fall, but also to compensate for the large risk involved in holding the asset.”

And here is the paper (pdf format). The paper is very academic and arcane, with lots of complex math, so unless you are a math whiz, don't expect to be able to understand all of it:

http://www.nber.org/papers/w09 45.pdf

This second link is to Dr. John Hussman's market commentary from earlier this week:

http://www.hussmanfunds.com/wm c/wmc131230.htm

Here is a quick summary of this rising risk of a crash, which I've taken from Hussman's most recent market commentary:

“As the price variation speeds up, the no-arbitrage condition, together with rational expectations, then implies that there must be an underlying risk, not yet revealed in the price dynamics, which justifies this apparent free ride and free lunch. The fundamental logic here is that the no-arbitrage condition, together with rational expectations, automatically implies a dramatic increase of a risk looming ahead each time the price appreciates significantly, such as in a speculative frenzy or in a bubble. This is the conclusion that rational traders will reach. This phenomenon can be summarized by the following proverb applied to an accelerating bullish market: ‘It’s too good to be true.’” Didier Sornette, Why Stock Markets Crash, 2003

“Our positions are always built on observable evidence rather than scenarios. We already have sufficient evidence to be fully defensive. Only later will we read in the headlines exactly why this defensive position was warranted.” John Hussman, PhD 12/23/2013

"My guess is that the present speculative advance may have a few percent to run – I’ll be particularly concerned if the market does so in a rapid, uncorrected manner in the next couple of weeks, which could suggest crash probabilities approaching 100% based on the sort of analysis above." John Hussman, PhD, 12/23/2013