Following the death of Paul Samuelson, I looked at his seminal paper `Proof that properly anticipated prices fluctuate randomly’. I found it intriguingly related to my beloved puzzlement about the meaning of probability, and I have something to say about it. Since I didn’t look at the forty-five years of literature that followed this paper I am probably going to flaunt my ignorance in public, but this is just a blog so what harm can it do.

The purpose of Samuelson’s paper is to formalize the intuition that competitive prices must, in some sense, look like random walk: Let {\dots,X_{t-1},X_t,X_{t+1},\dots} be a stochastic process that represents the spot price of some product, say wheat: {X_t} is the price in which you can buy wheat at day {t}. Fix a day {T^\ast} and let {Z(t)}, for {t\leq T^\ast}, be the price at day {t} of a contract that requires the delivery of wheat at day {T^\ast}. Samuelson proves the following theorem:

Theorem Assume that

\displaystyle Z(t)=\mathop{\mathbb E}\left(X_{T^\ast}\left|\dots,X_{t-1},X_t\right.\right),

Then the sequence {\dots,Z(t), Z(t+1),\dots,Z(T^\ast)} is a martingale

So, in day {t}, the properly anticipated price of wheat in day {T^\ast} is given by the conditional expectation of the currently unknown spot price of wheat in day {T^\ast}. Note that agents are assumed to be risk neutral and the interest rate is zero (Samuelson later dispenses with these assumptions). It’s a tribute to Samuelson that what he wrote and proved fifty years ago is now probably the first example of a martingale in your favorite probability textbook. Samuelson proves this theorem under some stationary assumptions about the distribution of the {X_t}-s. Today we know that the theorem is true for every process and the proof is immediate from the properties of conditional expectation. (Please remember that Samuelson’s paper is from 1965).

But anyway, the math is not my subject matter. For me, the main problem with Samuelson’s argument is its starting point: the stochastic process over spot prices.

I have not here discussed where the basic probability distributions are supposed to come from. In whose minds are they ex ante ? Is there any ex post validation of them ? Are they supposed to belong to the market as a whole? And what does that mean ? Are they supposed to belong to the “representative individual” and who is he ? Are they some defensible or necessitous compromise of divergent expectation patterns ? Do price quotations somehow produce a Pareto-optimal configuration of ex ante subjective probabilities ?

So, Samuelson views the probabilities as subjective. What I understand from this paragraph is that he is troubled by the identity of the agent whose mind these probabilities occupy. There is a good reason to be troubled: the assumption in the theorem about the quoted price {Z(t)} makes sense if the traders have the same beliefs. Otherwise you have to formalize some way in which the market creates a“representative individual” from the beliefs of the traders.

But I see a more disturbing problem here: Even if we assume for a moment that all the traders actually agree on the distribution of the process {X_t}, the only sense in which {Z(t)} is a martingale is with respect to this distribution. So shouldn’t we view the theorem as proving that properly anticipated prices are a subjective martingale, that is a martingale according to the subjective belief of the traders ? But then, does the theorem says anything about the anticipated prices of wheat in the world should this subjective probability turns out to be incorrect? Admittedly, I don’t know what correct probability means (more on that in another post), but clearly if all probabilities in the model are interpreted as subjective probabilities then there is no reason for us to expect — even as a theoretical conclusion of the model — that the quoted prices will look random; it only means that the traders expect them to look random. But we, the outside observers, might see something entirely different.