# Mathematics that makes itself

Can something be true, just because you say it?

One example might be a promise. If you promise somebody that you’ll feed their cats…well, all of the sudden there is a promise there. The act of promising creates a promise. All of the sudden, there it is. It makes itself.

Anyway, maybe mathematics can sometimes pull off a trick like that. In 2003, MacKenzie and Millo argued that this is precisely what happened in financial markets with the Black-Scholes formula, a highly successful mathematical model used to find “correct” prices for a stock option:

Option pricing theory—a “crown jewel” of neoclassical economics—succeeded empirically not because it discovered preexisting price patterns but because markets changed in ways that made its assumptions more accurate and because the theory was used in arbitrage.

In other words, the use of the formula itself made the formula more reliable. It was a self-fulfilling mathematical model, a piece of mathematics that reshaped the world to conform to its assumptions. Wow.

(I found this interesting blog post that dives a bit deeper into the logic of a self-fulfilling equilibrium.)

If this feels eerie, it’s only because we’re forgetting how strange and self-referential the notion of predicting the markets really is: markets are hard to predict because they are predictions. This is a way that finance and economics is fundamentally unlike the natural sciences. In finance there is always the possibility that the scientist will influence the subject.

Black, Scholes, and Merton’s model did not describe an already existing world: when first formulated, its assumptions were quite unrealistic, and empirical prices differed systematically from the model. Gradually, though, the financial markets changed in a way that fitted the model. In part, this was the result of technological improvements to price dissemination and transaction processing. In part, it was the general liberalizing effect of free market economics. In part, however, it was the effect of option pricing theory itself. Pricing models came to shape the very way participants thought and talked about options, in particular via the key, entirely model‐dependent, notion of “implied volatility.” The use of the BSM model in arbitrage—particularly in “spreading”—had the effect of reducing discrepancies between empirical prices and the model, especially in the econometrically crucial matter of the flat‐line relationship between implied volatility and strike price.

To be clear, Ed Throp used option pricing to make a killing before the markets were influenced by Black-Scholes. So it’s not like the formula created its own reality entirely. The claim can only be one of degrees — that the model became more reliable, that the markets grew more like what the model predicted. I am unable to evaluate the evidence on its own and haven’t dived deeper into any of this literature but, huh, it makes you think doesn’t it?

It reminds me of Ben Blum-Smith’s excellent post about voting theory, where he suggests that mathematicians have at times gotten lost in their models and believed in them too strongly, more because of their mathematical properties than for any of their use in application. But what if — only at times, and only by degrees — your mathematical model could be its own fulfillment by changing the world to more closely accord to its predictions? Wouldn’t that be something.