Friday, September 4, 2009

Hedging Gaia

Indulge me as I take a stab at an idea that has been wandering around in my head, lost in the desert for some time now.

There are all kinds of patterns in market prices data, on all kinds of scales. How do we think about those patterns? We know that they are not random fluctuations around a mean value -- the market is not a classic random walk binomial distribution. It has fat tails and black swans. Or maybe it's got a fractal time structure like Mandelbrot claims. At any rate, we have yet to come up with a good statistical description of most market time series data. This is what confounds risk management, fools the physicists, and ends in the ritual sacrifice of Vikram Pandit and other former high priests of VaR.

So what kind of statistical beast is the market then? Well, let's think about how it gets built up. The first markets (conceptually, but also historically to some extent) are just instantaneous spot markets. Think of the medieval fair. Pigs and salt and harlots. Supply and demand meet casually in the street, talk shop over a tall cool glass of mead. These markets aren't very random at all. They pretty quickly find their way to equilibrium, and then they mostly stay there. These markets also do experience fluctuations, but these aren't random either. The price of strawberries goes up when they are out of season. If you want more grain at harvest time, it will cost you dearly. The fluctuations are largely as predictable as the seasons from which they derive. The only thing that shakes them loose from these patterns is a change in one of the two terms, typically the supply one -- a drought, a plague, a war, etc ...

But fluctuations that everyone can predict in advance are clearly not sustainable in a market system. Chinese pig farmers will hoard aluminum and washing machines when they are cheap and sell them when they are expensive. This damps out the natural cycle. In addition, other clever monkeys will invent futures markets and other derivatives that institutionalize these mechanisms and make them much more powerful and systematic. Under these conditions (iff storable surplus production, speculative capital accumulation, and information propagation, as far as I can see) markets become forward looking. Eventually, the idea is that this would lead to conditions of perfect arbitrage and perfect equilibrium. I believe there's even a famous theorem about how instruments that allow the market to be more forward looking serve, mathematically speaking, to "complete" it, and, at the limit (you need not just futures, but futures on futures, etc ...) to guarantee that is machine for producing perfect equilibrium. Lacking this perfection in practice, the tiny remaining fluctuations around equilibrium would presumably be distributed along a bell curve.

If the market did produce perfect equilibrium, and the world were completely predictable, then everything that could happen would already be reflected in the market, and it would never change. All the predictable variations of the world, all the seasonal patterns, etc ... would be incorporated in one fixed and unchanging price in advance. It would be a sort of perfect crystal ball that told us in advance of all sorts of thing that we might think improbable a priori, but that are still predictable. The question of what kind of patterns we find in this market would then be identical to asking what kind of patterns we don't find in the world. "How random is the stock market" would become "how random is the part of the world we don't know about"? And this is in fact exactly what the efficient markets hypothesis guys have in mind when they say that the market reflects all the known information at a given time. It's a strange sort of equilibrium, as the price of securities would continue to jump around as new shit came to light, but it's a sort of equilibrium nevertheless, in that there is no force within the system that would cause its state to evolve.

Well, but so then we can wonder just how random is the part of the world we don't know about. Do we believe that the distribution of unknown outcomes for our world falls along a bell curve? That seems remarkably unlikely. The world is truly Random. It's not little random-walk model random, which is actually just domesticated randomness with its tiny predictable variations. For true Randomness, not even statistics works. Everything is sui generis. Or not even, because even that would be a form of pattern. True randomness would be utterly patternless, if such a thing can be conceived ...

And this, ultimately is my point. The market is the closest thing we have to a true random number generator, just not in the sense it is usually claimed.


Okay, so that was originally ultimately my point, but after a few days of thinking about it I now have a new and improved point that supercedes that original ultimate point and you certainly wouldn't want to miss that, now would you? My new and improved point picks up where the last one left off -- the market is truly Random. It's as random, to quote a great philosopher of yesteryear, as the unknown unknowns of our world. And yet the market fluctuations that track this deep randomness are not merely reflections of it; the market is not a passive, objective observer of the world, but actually a part of it (and what isn't these days?). When the market incorporates a new fact, it not only has to change the price of securities to reflect this new information, but it has to incorporate the new real reactions people have to the fact that this information is now available. In other words, the new information has an impact not only on the market, but also on the world. But this means that the market's original incorporation of the predictable aspects of the world was actually wrong (though in an unpredictable way -- the known unknowns were, by hypothesis, already incorporated in the market, albeit probabalistically). And what's worse, if people were earlier using the market to make real decisions (which the market of course incorporated, and which was of course a good idea, because the market is always right honorable) those decisions have now been obsoleted.

The upshot is that the market can't be random. It can't be as Random as the world (because we can predict some of that world, and the market incorporates those predictions, as well as the real actions taken to make those predictions come true). It can't even be as RRandom as the things we don't know about the world, because the things we don't know about the world change the things we do know about the world, when, in fact, this new shit comes to light. So the market has to be RRRandom or Random2 or maybe even RandomRandom -- the randomness of the market feeds back on itself like infinity, so it's hard to know whether twice as Random is more Random than half as Random.

Having reached this level of randomness, you should clearly spend the rest of the day slumped on the couch with your feet propped up on a copy of Gesammelte Abhandlungen mathematischen und philosophischen inhalts and a cold compress over your eyes. Drink exotic cocktails through a Twizzler, and, whatever you do, do not trade any more stocks -- you'll just make things worse.

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