On volume and information

Random agents exploring the state space. 

But are we really satisfied to state that the existence of exchanges, and the fact that information percolates into prices via a series of trades, are facts only “explainable" by human folly, that would be absent in a more perfect (or perfectly-run) world?

John Cochrane

I think an excellent summary of the information equilibrium framework's take on prices and exchange is to answer "yes" to that question.

John Cochrane wrote an nice essay about financial market volume and information (what should really be called knowledge here, because it differs from information theory "information" in the sense that the knowledge people trade on is typically meaningful, and not just measured in bits). That's where I got that quote. The key issue is that, as Cochrane puts it:

Volume is The Great Unsolved Problem of Financial Economics. In our canonical models ... trading volume is essentially zero.

Cochrane then makes a really good analogy:

I gather quantum mechanics is off by 10 to the 120th power in the mass of empty space, which determines the fate of the universe. Volume is a puzzle of the same order, and importance, at least within our little universe.

Correct this to energy density because vacuum energy acts as an outward pressure (accelerating the expansion of the universe, contributing to the cosmological constant) while matter acts gravitationally, but overall this is a correct assessment. However, this is more of a back of the envelope calculation than hard core theory. Because we don't know the fundamental theory of gravity (strings, loops, triangles), we can only guess that it has something to do with the Planck length.

Therefore the scale that we should use to cut off our integrals is the only scale we know, i.e. the Planck length ℓ. And 1/ℓ⁴ [the 4 is for 4-D spacetime] is off by 120 orders of magnitude from the real value of basically zero. The takeaway is not that quantum mechanics is wrong, but that obviously the Planck length does not set the scale of the vacuum energy density. The fundamental theory of quantum gravity should tell us what the real scale we should use is — our naïve guess is wrong.

Getting back to the volume problem in finance, the assumption in finance is that the information (knowledge) content of trades should set the scale of volume in financial markets. Unfortunately, this tells us that volume should be zero if agents are rational.

The obvious conclusion is that the information (knowledge) content of trades does not set the scale of volume in financial markets. Stiglitz and Grossman (1980) set up a new scale — that of noise traders. In that case, the number of noise traders will set the scale of market volume.

If we look at random agents fully exploring the economic state space (maximum entropy), then simply the number of agents participating in the market will set the scale of market volume. Gary Becker wrote a paper about random agents, calling them "irrational" (see here for some slides I put together). I prefer to say they are so complex they appear random. Much like how a sufficiently advanced technology is indistinguishable from magic, sufficiently complex behavior should be indistinguishable from randomness. (Note that this is actually a good definition of algorithmic randomness.)

This is not terribly different from the "noise trader" concept, except it makes no assumptions about individual behavior. I may be irrationally selling Apple stock if you just look at it as a number (observed in the market), but maybe I need to pay medical bills (not observed in the market). But it does make sense of a few of the stylized facts that Cochrane mentions. For example, rising markets are associated with rising volume. This would make sense because more people are drawn into the market when the indices are rising (it doesn't matter the details of the source of the boom). It also makes sense of that great contraindicator: when random people in your neighborhood seem to be talking about markets, it's time to get out.

The thing is that uncorrelated agents fully exploring the state space (maximum entropy, or information equilibrium) does seem to be consistent with informationally efficient markets — at least until there is some kind of correlation (e.g. panic and non-ideal information transfer).

I want to comment on a couple of things Cochrane says towards the end:

Behind the no-trade theorem lies a classic view of information — there are 52 cards in the deck, you have three up and two down, I infer probabilities, and so forth.

Well, this is a classic game theory view of information mixed with information theory information. The three cards face up have zero information entropy, but would contribute to perfect information in game theory (probably a good example to see the difference).

But I did like this:

For a puzzle this big, and this intractable, I think we will end up needing new models of information itself.

Maybe information equilibrium?

...

PS There is also this post on rate distortion and Christopher Sims I wrote last month that is relevant. The limited number of bits of information getting through tell us the vast majority of trades are "entropy trades" (a better term than noise trades) by random agents.

PPS + 3 hrs As a side note, the basic asset pricing can be expressed as a maximum entropy condition.