Not Just Any Formula: Self-Confirming Equilibria and the Performativity of Black-Scholes-Merton
June 21, 2012
[Cross-posted from my personal blog as I think the readership here might have rather a lot to say on the subject.]
One of the most successful, but still controversial, papers in recent economic sociology is MacKenzie and Millo’s (2003) Constructing a Market, Performing Theory. M&M trace the history of Chicago Board Options Exchange and its relationship to a particular economic theory – the Black-Scholes-Merton (BSM) options pricing model. One of the main findings is summarized nicely in the abstract:
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.
Economics is thus performative (in what MacKenzie would later call a “Barnesian” sense), because the economic theory altered the world in such a way to make itself more true. M&M elaborate a bit more in the conclusion:
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.
Elsewhere, I have emphasized these other aspects of performativity – the legitimacy, the creation of implied volatility as a kind of economic object that could be calculated, etc. These are what I think of as Callonian performativity, a claim about how economic theories and knowledge practices produce economic objects (what Caliskan and Callon now call “economization“). But at the heart of M&M – and at the heart of the controversy surrounding the paper – is the claim that Black-Scholes-Merton “made itself true.” This claim summoned up complaints that M&M had given dramatically too much power to the economists – their theories were now capable of reshaping the world willy-nilly! Following M&M’s analysis, would any theory of options-pricing have sufficed, if it had sufficient backing by prominent economists, etc.? And if not, aren’t M&M just saying that BSM was a correct theory?
One way of out of this problem is to invoke a game theoretic concept: the self-confirming equilibrium (Fudenberg and Levine 1993).* In game theory, an equilibrium refers to consistent strategies – a strategy which no player has a reason to deviate from. There are lots of technical definitions of different kinds of equilibria depending on the kind of game (certain or probabilistic, sequential or simultaneous, etc.) and various refinements that go far above my head. The most famous, the Nash equilibrium, can be thought of as “mutual best responses” – my action is the optimal response to your action which is your optimal response to my best action. The traditional Nash equilibrium, like many parts of economics, assumes a lot – particularly, that you know all possible states of the world, the probabilities they will obtain (in a probabilistic game) and your payoffs in each. The self-confirming equilibria is one way to relax these knowledge assumptions. The name gives away the basic insight: my action is the best response to your action, and vice-versa, but not necessarily to all possible actions you might take. Here’s the wikipedia summary:
[P]layers correctly predict the moves their opponents actually make, but may have misconceptions about what their opponents would do at information sets that are never reached when the equilibrium is played. Informally, self-confirming equilibrium is motivated by the idea that if a game is played repeatedly, the players will revise their beliefs about their opponents’ play if and only if they observe these beliefs to be wrong.
So, if we think of different traders all using BSM, checking the model to see if it was working, and then choosing to use it again, we can see how BSM could work as a self-confirming equilibria.** And, in turn, the concept might help restrict the sets of theories that could have been self-confirming. A radically different theory might not have produced consistent outcomes – but many other such theories could have. I don’t know enough about options pricing to say for sure, but logically I think it works: given all the kinds of imperfect information and expectations one could have, there were probably a wide range of formulas that would have worked (coordinated traders’ activities in a self-confirming way) but not just any formula would do. So, a possible amendment of M&M’s findings would be to say that in addition to all the generic/Callonian ways that BSM was performative (legitimizing the market, creating “implied volatility” as an object to be traded), it also was in a class of theories capable of coordinating expectations and thus once it was adopted, it pushed the market to conform to its predictions. Until the 1987 crash, of course, when it broke down and was replaced with a host of follow-ups that attempted to account for persistent deviations. But that’s another story!
*I thank Kevin Bryan for the suggestion.***
**I may be butchering the technical definition here, apologies if so. The overall metaphor should still work though.
***Kevin offers some additional useful clarification. First, here’s a link to a post discussing self-confirming equilibria (SCE) on Cheap Talk (about college football of all things). Second, I should have pointed out that the SCE concept only makes a difference in dynamic games (which take place over time). In one shot games, there is no chance to learn, and thus nothing to be self-confirmed. Third, here’s Kevin’s take on how the SCE concept could apply:
Here’s how it could work in BSM. SCE requires that pricing according to BSM be a best response if everyone else is pricing according to BSM. But option pricing is a dynamic game. It may be that if I price according to some other rule today, a group of players tomorrow will rationally respond to my deviation in a way that makes my original change in pricing strategy optimal. Clearly, this is not something I would just “learn” without actually doing the experiment.
My hunch, given how BSM is constructed, is that there are probably very few pricing rules that are SCE. But I agree it’s an appropriate addendum to performativity work.