Paul Samuelson’s proof that properly anticipated prices fluctuate randomly

You'l find the paper here and it is one of the best introductions to Samuelson's method and writing style.  The caveats at the very end, as to what an efficient markets hypothesis might mean, and what the probability distributions might mean, remain valuable reading to this day.

Comments

Benoit Mandelbrot went beyond Samuelson'd a priori theorizing, and made an empirical study of markets in the early 1980s. He discovered that prices were not random, but exhibited fat-tailed distributions. He published a book about his findings, The (Mis)Behavior of Markets.
But we have it on Samuelson's authority that the Austrians were unscientific because of their a priori approach to economics.
Maybe Samuelson was not quite as much of a Scientist as he wanted the world to think.

Bill,

If you are going to argue about this, at least get the terms correct. Fat tailed and random are not in conflict-in fact, fat tailed distributions are random.

anon,

My reading of Mandelbrot is that he unequivocally thinks that fat-tailed distributions are not random.
Here is a quote from pp. 247-48 of his book:

"The classic Random Walk model makes three essential claims. First is the so-called martingale condition:
that your best guess of tomorrow's price is today's price. Second is a declaration of independence: that tomorrow's price is independent of past prices. Third is a statement of normality: that all the price changes taken together, from small to large, vary in accordance with the mild, bell-curve distribution. In my view, that is two claims too many. The first, thought not proven by the data, is at least not (much) contradicted by it; and it certainly helps, in an intuitive way, to explain why we so often guess the market wrong. But the others are simply false. The data overwhelmingly show that the magnitude of price changes depends on those of the past, and that the bell curve is nonsense [N.B.: except for stuff like IQ and height]. Speaking mathematically, markets can exhibit dependence without correlation.... Large price changes tend to be followed by more large price changes, positive or negative. Small changes tend to be followed by more small changes. Volatility clusters."

Samuelson was wrong. And any stock trader with a mathematical bent knew it all along.

Norman,

I don't have a summary at hand, but you might try Wikipedia, "Benoit Mandelbrot," or maybe "fractals."

I'll copy and past my comment on Samuelson and the Le Chatlier principle as soon as it is moderated at the Think Markets blog.

Bill Stepp: I don't know if you characterize Mandelbrot correctly, but you certainly don't understand or don't know what Samuelson claimed. From page 4 in the linked paper (page 42 in the journal):

"If P(X, x0, x1...) (note: the probability distribution of prices) had the Markov property, then (2) would be the so-called Kolmogorov-Chapman equation. But I do not assume any special Markov property. The generality of (1) and (2) must be emphasized. Nothing necessarily Gaussian or Normal is assumed about any distribution."

A fat-tails distributed doesn't, by itself, disprove random walk. It could merely means that news also has fat tails.

Commenterlein,

I do understand statistics. And a fat-tailed distribution is not a standard normal distribution.
If you think otherwise, you're the one with the less than high-school understanding of statistics.

Jesus, what a fucking moron Bill Stepp is.

Samuelson's premise of perfectly anticipated prices can't exist in the real world, or even in a laboratory, unlike a perfect vacuum. His paper is vacuous, just like the boy Flattus Maximus.

I think that some people here are confusing random walk, Wiener process, and geometric Brownian motion. In continuous time and continuous space it is very possible to construct a random walk composed entirely of jumps.

anon,

You are wrong. It is not trivial to test this, and those who
claim to have done so have made some restricting assumption.
This is the same issue as that of the so-called "misspecified
fundamental" problem in testing for speculative bubbles. The
only way to do it is to have some independent measure of
expectations, which is not likely to be econometric, but something
unreliable such as a survey.

anon.

I do not buy your argument, having just scanned through Chap. 2 of Campbell et al. The only mention
of Samuelson's result is early on, where it is identified as being basically a martingale result.
He does use the martingale argument in his proof, but he does not tie it to IID outcomes or to
efficiency, specifically saying that is not an implication, even as others have interpreted as
such, with Roberts labeling it "weak efficiency." I think you are misinterpreting, as have so
many others.

B the M,

Why should I tell you what a "change in measure in probability theory is"? I did not mention any such concept here. If you would love to hear somebody tell it, I suggest you do it yourself. I think the assembled audience might even let you use a wikipedia link, if you want to.

fineng,

Not sure what planet you are on, but a) we have not been discussing risk neutral pricing per se,
and b) the term "change of measure" does not appear in either of the two leading grad level fin
econ textbooks, Cochrane or Campbell, Lo, and MacKinlay, which anon above claimed incorrectly
contains a disproof of Samuelson's results.

BR,

You did not read what I wrote. I said that you can check out Samuelson's hypothesis in data, with some mild assumptions. Exactly how to accomplish that is what those books tell you. And they even report some tables of results.

Samuleson's results are theorems, correct given his assumptions. The proofs are correct, as far as I know.

As far as I can tell, you are a random math word generator. You seem to think that martingales require iid innovations, or somehow that Samuelson proved that you need iid innovations. But maybe I am giving you too much credit here.

anon,

"with some mild assumptions" Exactly.

As for your two questions of me, no and no.

fineng,

Um, sure, there are lots of theorems that support all kinds of things, and, yes, there are even some important ones that do not get mentioned unless one wants to dig very deeply. However, this one that you and B the M are all worked up about looks to be only useful for a certain set of fairly narrow results. Duh.

If either of you wishes to see my views more clearly stated than here, check out my posting fresh up on EconoSpeak on this matter, which links to this discussion.

If you wish to see my more extensive views on math and econ, well, you can read my paper on my website, "On the Foundations of Mathematical Economics," which is forthcoming in some random math and comp sci journal, or one of my more mathy books.

Samuelson had a profound influence. Unfortunately for everyone, it was a very negative influence. He taught generations of students to be Keynesians and, in doing so, gave government bureacrats just what they were looking for: theoretical excuses for mercantilism, interventionism and legalized theft.

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