I was just listening (time-shifted, obviously) to NPR Planet Money's discussion of the recent Nobel Prize in Economics. Aside from the fact that Shiller & Fama won the same prize at the same time for diametrically opposing views (kind of like giving 2 different people prizes for P=NP and P/=NP, simultaneously), the discussion raised some interesting issues about the rationality/irrationality of stock market prices. http://www.npr.org/blogs/thetwo-way/2013/10/14/233934871/americans-win-econo... "Americans Win Economics Nobel For Interpreting Stock Prices" --- Basically, Shiller says that there can be "bubbles" (irrational prices), while Fama says that there can't be, because if there were irrational prices, then some rational entity would step in and make huge profits from this mispricing. I can understand why the Nobel Prize committee gave out both prizes, because I believe that both Shiller and Fama are correct. However, neither the Committee, nor NPR, could explain why this should be possible. I think that it takes an engineer to understand this problem. Consider _waves_. When waves travel through a _linear_ medium, _every wavelength is independent_ (the principle of Superposition), and in particular, the _power_ in one wavelength cannot be converted into power in another wavelength in a perfectly linear system. Exchanging power among wavelengths requires _nonlinearity_. Hence, a water wave can build up (due to wavelength/frequency dispersion even in a linear medium) and subsequently "crash" due to the nonlinearities in water wave equations. But the fundamental property of _waves_ requires _momentum_, without which there would be no waves. This is because the fundamental wave equation is a second order differential equation, and requires the combination of an elastic medium together with a momentum-carrying mass. This wave may quickly die out due to any _first-order_ term in the differential equation, but it is the _second-order_ term that defines the basic property of waves. We now return to the stock market and prices. We can model this system as a system of difference equations which have inevitable _time delays_. With modern high frequency trading (HFT), these time delays can be microseconds, but these time delays are still present. _Any difference equation with time delays may be subject to oscillations_, because any action is predicated on data that is old, and therefore subject to phase shift. When the phase shift reaches 180 or 360 degrees (depending upon the sign) for a given frequency, the system can oscillate and/or potentially diverge (complex & real parts of exp(z), respectively). We now look more carefully at how the stock market works. The stock exchange has an _order book_, which is a current list of bids and asks; each bid gives a price and a size of transaction; each ask gives a price and a size of transaction. When the bid price of one order exceeds the ask price of another order, a _transaction_ takes place, which has the side effect of erasing min(bid_size,ask_size) shares from the order book. The stock exchange then waits for another order (bid or ask), which either gets added to the book, or causes a transaction to happen. The _quoted_ prices on the stock exchange are the highest bid and the lowest ask prices, _regardless of size_. In particular, the quoted prices could depend upon orders worth only $100 on a company whose total market capitalization could be tens of billions of dollars. This is the very definition of "pricing on the margin". What does it take to _change the quoted price_, i.e., "move the market" ? Suppose I wanted to change the quoted price of the ACME company from its current price of $100 to $200. What would I have to do? If I wanted to change the price quickly, I would have to _buy all of the "ask" shares in the order book whose currently asked price was less than $200_, and I would have to issue a bid order whose size was large enough to actually accomplish this. Depending upon how "thinly traded" ACME shares were, this amount could be anywhere between a few hundred dollars to a few hundred million dollars. Basically, any sharp "uptick" would require a buy order large enough to hit an "air pocket" -- a "hole" in the order book price list where there are extremely small sizes of stock available for sale. Causing such a sharp uptick in stock prices would be a stupid thing to do, unless I were afraid that an order even larger than my order would be coming a few microseconds/milliseconds right after me. Thus, I would a) have to believe that the stock would be more than doubling soon, and b) that I had only a split-second jump on the market. If I believed that the stock would be doubling, but that I was the only one in possession of this knowledge, I would buy more judiciously & quietly, hoping that my purchases would drive up the market very slowly -- if at all. In this way, I could purchase my shares at an overall lower aggregate price, prior to the market as a whole becoming aware of the knowledge that I already had. But after my first (small) purchase, I'm now engaged in a _competitive_ market, wherein every transaction is monitored and reacted to by every other player. I.e., we all now go through a series of steps in time where prices are posted, and we all react to these changing prices according to our varying strategies. It is in these _reactions to prices_ that _time delays_ are encountered, which can now produce _oscillations_ due to _momentum_. One player may see a price uptick as an opportunity to sell, while another player may see a price uptick as a signal that someone thinks the stock is undervalued, and uses this presumption as an opportunity to buy. In this case, the seller is _counter-cyclical_ (a "dampening" force), while the buyer is _cyclical_ (an "amplifying" force). A non-cyclical player would have an a priori theory of the _intrinsic_ value of the stock, and would buy/sell accordingly. A cyclical (i.e., "momentum") player would not necessarily have any notion of _intrinsic_ value for the stock, but would attempt to guess whether the stock was headed up or down based upon previous price/volume behavior. We now ask an objective question: what is the ratio of cyclical to non-cyclical players in the market? The answer can be objectively determined: the overwhelming volume of players (measured by dollar volume) in the stock market are _cyclical_ ("momentum" players), and have no notion of intrinsic value (at least over a relatively large range of prices). Whether this fact supports Shiller or Fama requires a bit more analysis. The vast majority of time, the cyclical players beat the non-cyclical players, partially _because_ they know that the vast majority of players are cyclical. When a water wave is headed up, it carries all of the water molecules with it, regardless of how smart they are, how good their computer simulations are, or how well they can see how far above mean sea level they are. Indeed, from the perspective of any water molecule in the wave, no matter how steep the slope is, it appears to getting monotonically steeper, so _until the moment the wave crashes_, there is still as much money to be made (analogizing the water wave with a price bubble) at the last instant before the crash, as at any earlier point on the rising wave. ---

From a purely "rational" point of view, price movements should _never_ be smooth. In a world with "perfect" information and "rational" players, every new bit of information should change everyone's perception of price instantly, and in a discontinuous way.

If I maintain a "discounted cash flow" model for the ACME company, and I hear about some fact that will affect the company seriously in 5 years, I should still "discount" this fact backwards to the present and instantly and discontinuously change my notion of intrinsic value for ACME. That price movements of ACME appear to be smooth at all is due to 2 factors: a) many, many different players have different intrinsic valuations of ACME, and so transactions (and hence price movements) will only occur for those whose intrinsic valuations are close to the current market price; and b) many "market-makers", who have no notion of intrinsic valuation, but are (usually) the quintessential damping mechanism, which attempts to "smooth" price variations about some longer-term "moving average". Market-makers would ordinarily be wiped out, except that they are _fully hedged_ (no net exposure to the market price), and bid/ask _spreads_, which enable them to make small amounts of money even in a completely flat market, so long as there is some volume. The real question is why there are any players whose intrinsic valuations are above the market ? The answer is that these players simply don't have access to enough money to actually move the market. Having the _information_ to move the market isn't the same thing as being _able_ to move the market. These intrinsic players are exactly in the position of the very smart water insect (Maxwell's flea?) who can see the wave is about to break, but isn't in any position to do anything about it (e.g., extract energy). --- Metastability. Metastability in an electronic circuit, or more general "decision-making system", is the name for the indecision that can arise from the time it takes the system to _amplify_ a tiny bit of information into a signal large enough to "move the needle" -- i.e., to affect the larger-scale world. Metastability typically occurs in _arbiters_ and _synchronizers_, where the bit of information is "which signal, A or B, came first?" must be amplified into a large-scale change of some perceptible state. The time to detect and amplify this bit of information may be related in a non-linear way to the time difference between the arrival of the signals, so it could take an unbounded amount of time to amplify this unboundedly small difference to a perceptible level. http://en.wikipedia.org/wiki/Metastability_in_electronics Metastability in electronics --- _The exact same kind of metastability happens in the stock market._ Very small bits of information may take a very long time to become amplified to the point of perceptible changes in stock market prices. This is because the few players who have access to this information simply don't have enough access to capital to _make a big enough splash_, and therefore can't move the market directly. The market is eventually moved -- if at all -- by those _counter-cyclical_ (i.e., "momentum") players who notice these almost imperceptible price signals of the small-but-knowledgeable players and _amplify_ them into larger-scale movements. Thus, the very momentum that enables information transmission in stock market prices also makes the market vulnerable to bubbles and crashes. So Shiller & Fama are both right.