It has been a while that I haven’t done a book review. I want to review this book to lay the foundation for my future discussions on several important topics. This post is a bit mathematical, and if you are not into it, you can skip the paragraphs forward to the RED sentence, starting after the equations. The post is less of a review, but more of a discussion on the results from the book.

Didier Sornette is a UCLA (University of California at Los Angeles) geophysics professor. For him to write such a book with his arcane mathematical model is almost outright weird. Such is the inter-disciplinary nature in the advanced studies of scientific frontier. The scientific frontier here is essentially the studies related to **complex systems**, not an informative name but descriptive nevertheless. Mathematicians have not been able to sort out many of the complex systems in nature. The mathematical study of complex systems is called **chaos theory**. Here is a pointer to the introduction of chaos theory. Many of nonlinear dynamic complex systems have tremendous amount of random inputs and numerous known or unknown rules, and yet exhibiting some simple order or behavior. Some systems has very few simple rules, and yet exhibiting much more complex results. One of the better known studies in chaos theory is Fractals. Within the complex systems, sometimes there exists one or many **strange attractors**. I can’t find any good webpages on strange attractors. Here is from wikipedia. Essentially, the attractor is (one of) the convergence of the most initial states evolved through time. There are many books on **Chaos Theory** if you’re interested. Lots of math involved, but extremely interesting. In any case, let me regress to this particular study of the complex system: Stock Market.

Through Sornette’s study on predicting earthquake and phase changes (from liquid to solid, etc), he applied his knowledge to this critical phenomenon of a stock market crash event. Here, “*critical”* carries more of the mathematical meaning of having some orders of derivative undefined (usually the 1st order, which is the slope of the curve) or discontinous. After such critical event, things no longer behave the same way. You can pretty much concatenate a different function after that point. By observing the information network among traders which gives rise to the inherent fractal nature in the stock market price, Sornette was able to come up with a simple model for the behaviors of financial bubbles (equation 18, pg.335), where P(T) is the price of the financial product in time=T:

log( P(T) ) = A + B ((Tc – T)^m) (1 + C cos(w log((Tc-T) + beta) )) for bubbles, and

log( P(T) ) = A + B ((T – Tc)^m) (1 + C cos(w log((T-Tc) + beta) )) for anti-bubbles or deflation of the bubble.

where A, B, C, m, beta, and Tc are modelling constants. In the special case of C=0, no price oscillation, the equation simplifies to

**log( P(T) ) = A + B ((Tc – T)^m)** for bubbles

For the following discussion, I will only use the simplified equation. Obviously, for the swing traders, oscillation in the price is extremely important to identify the local peak and valley in the prices for sell & buy points.

Noting from the boldfaced equation, **Tc is the Crash Time**. At T=Tc, the value of 0^m becomes undefined. If you take the derivative respect to T for both sides, you get

d (P(T)) / dP = 1 / P(T) = – B m (Tc – T)^(m-1) * (dT / dP) or

dP / P(T) = – B m (Tc – T)^(m-1) dT or

**dP / dT = -B m (Tc – T)^(m-1) * P(T)** or

**dP / dT = b / ((Tc – T)^n) * P(T),**

**where both b and n are positive, T < Tc, for a bubble**

Essentially, this is a **super-exponential** function. An exponential function has its increase in price proportional to the price (dP/dT is proportional to P(T) ). A super-exponential function increases even faster than exponential function. As time increases towards Tc, the rate of increase or dP/dT increases even faster. The term that is multiplied to P(T) races towards infinity as T approaches Tc. For obvious reason, no physical processes can have a rate of increase infinitely large. At such rate of increase, the price P(T) is bound to be **busted**.

Let’s plug in some numbers into the dP/dT equation. Assuming that b=1, and n=2, when Tc-T =1, dP/dT = P(T). When Tc-T=0.5, closer to crash point, dP/dT=4P(T), or dP=4P(T)dT. When Tc-T=0.25, even closer to crash point, dP=16P(T)dT. Putting into discrete terms, **the time it takes for the price to double (again and again) exponentially shrinks shorter**, or can be expressed as

**dP / P(T) = b / ((Tc – T)^n) * dT**

Alright, I’m very sorry if I have lost all of you. Just remember this: **a Bubble-like behavior is such that the time it takes for the price to double (again and again) exponentially shrinks shorter.** By the way, it doesn’t need to do a double to qualify. You can substitute double by any percentage amount of increase, more or less, as long as the time to achieve that multiple shrinks exponentially. In some way, a bubble closely resembles a money-making pyramid scheme. In a pyramid scheme, the late comers funnels their money to the early comers. Since the growth of the pyramid is exponential, that is the growth rate is porportional to the current size of the participants, very soon the pyramid runs out of new participants, stops growing and is unable to bring money to the late participants.

In the book, Sornette is not that crazy yet to ask readers to go through all of these derivations. The only equation that you will find are the very first 2 or 3 equations. The rest is my contribution (or dis-contribution for more confusion). Now let’s go back and answer this question of *Why Stock Market Crash*. In the evolution of the bubble, as described by the so-called **log-periodic power law** equation, the bubble experiences an **unsustainable** exponential growth. The bubble may collapse earlier due to a help from external factors or events, due to its inherent instability going towards the crashing peak. However, bubble is destined to collapse because of its own weight. Nothing in nature can have a growth rate reaching infinity as dictated by the equation. And nothing can grow exponentially indefinitely when the resource or money to participate in the market is finite. Actually, another term in the technical analysis of stocks is that the price has gone **parabolic**. I believe parabolic, a 2nd order function, is simply used as an approximation to this super-exponential model. The key observation again is that price increases faster even with less time. Using NASDAQ 2000 bubble as an example, the price for NASDAQ took more than 3 years to double to about 2500, and then it only took about 6 months for another double to complete. Such growth is indicative of the existence of a bubble.

So don’t blame anyone on any stock market crash. What goes up must come down. That is simply the law of mathematics and physics. That’s the way it works. It is so much better to have a steady growth than an unsustainable growth. However, bubbles are repeated throughout the history, and it is probably inherent in the human nature of greed and fear in the fight of grabbing ever increasing returns.

Want more predictions from this book. In fact, I will be posting more on the unsustainable human population growth and its derived consequences. In the last chapter of the book, Sornette has applied his model to various other sets of data, and based on the data fitting, the growth era of human may stop at around 2050. What does that mean for the human race as a whole? Read my future posts. By the way, his prediction on USA real estate market bubble is set to peak around summer 2006. I guess that is turning out to be quite close so far. Some may date the peak of this bubble at November 2005. It depends. I personally believe that the current cycle of real estate has peaked already.