Not Just Any Formula: Self-Confirming Equilibria and the Performativity of Black-Scholes-Merton

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.

EDIT: 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.

Advertisements

8 Comments

  1. Great post! A few years later (in Do Economists Make Markets), Callon tries to explain how performativity (turn to the more percise ‘co-performation’) is different from self-fulfillment (and from other notions such as expression or prescription).

    “we can agree to call performation the process whereby sociotechnical arrangements are enacted, to constitute so many ecological niches within and between which statements and models circulate and are true or at least enjoy a high degree of verisimilitude”.

    Thus, I believe you are correct to say that B&S “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.”; it’s just that Callonian performativity is (or at least claims to be) in a position to point to the content of that very coordination – on what were market participants coordinated? etc. Also, because coordination is an outcome of a process there’s room to talk about the struggles of performativity and whether or not B&S was * successful* in aligning the world to the one implied by the theory.

  2. I actually spent some time thinking a few years back about how to generalize the MacKenzie and Millo story using game theory for exactly the reasons you mention — to set a philosophical bound on the kinds of beliefs that could become true by being widely believed/used (although I was thinking more along the lines of Morris and Shin’s “global coordination games”, but your approach sounds simpler). I ultimately abandoned this line of thinking, however, because I came to believe that it’s based on a false analogy: that the Black-Scholes model represents some kind of codified “belief” or “prophecy” (in the Mertonian sense) that becomes self-fulfilling. Moreover, I think confusion over this simple point is the source of an unnecessary amount of controversy in the performativity literature.

    Let me explain: a self-fulfilling prophecy either has a null truth value (in the case of a performative utterance) or is actually false before it is uttered. Oedipus would most likely *not* have married his mother and murdered his father unless the Oracle had told him that these things were going to happen. Bank runs generally don’t happen until there’s widespread fear that one is about to occur. However, Black-Scholes is a different case all together.

    It’s helpful to draw on a distinction that Emmanuel Derman (a quant who happened to publish with Fischer Black back in the 90s) makes between “right” models and “true” models. A “right” model is one that is internally consistent from the standpoint of logic. A “true” model is one that has a high degree of similarity with the world, that closely “represents” the world (or is a good “analogy” of the world in Hesse’s sense). Many models are “right” that are not necessarily “true”. Newtonian physics is unequivocally “right” (in the sense that it is internally self-consistent), but is not “true” since it fails to predict things like how large gravitational bodies distort light. The Black-Scholes equation is unequivocally “right”. The pricing equation indubitably follows from the assumptions. This is what Callon means when he says in What does it mean to say Economics is performative? that Black-Scholes “is true in its own world.” Initially, though, Black-Scholes wasn’t true. As we all know from Mackenzie and Millo, the crucial thing that happened was: (1) the model became more “true” as things like bans on short selling were lifted; (2) moreover, this “right” (but initially “false”) model could be used for arbitrage, which in turn made the model more “true” than it was before (but it had always been “right”!). I suspect that Callon perfectly understands this distinction, because there’s in a section in that paper where he wonders whether the Black-Scholes equation was in any sense arbitrary and concludes that it was not.

    That said, textbook cases of self-fulfilling prophecies of the kind we study in sociology can *only* be true or false, but not “right” or “wrong”. This is because they are simple propositions about the real world. To speak of the degree of internal consistency of a proposition like “There is a bank run happening” doesn’t really make sense. Ultimately, much of the performativity literature ends up comparing apples and oranges.

    • I was hoping that I could edit my comment, but I’ll just post a quick addendum —

      I strongly agree with you that “coordination of expectations” is an extremely important part of the story of why financial models (particularly no arbitrage models like Black-Scholes) get used, and I’m afraid that point of agreement wasn’t clear in my comment. We sociologists could also benefit from internalizing some of the lessons of game theory — particularly the variants where there’s a relaxation of the assumption of common knowledge in situations of strategic interaction (this I think, is essential to understanding how/why social conventions become/remain stable, a point alluded to by none other than Parsons).

      I just think much of the philosophical controversy over performativity (in the case of Black-Scholes at least) comes down to the repeated use of a false analogy between scientific models and self-fulfilling beliefs.

      • Taylor – Thanks so much for your detailed thoughts!

        As a brief reply, I agree that moving performativity away from self-fulfilling prophecy is an important intervention – and ultimately, why I tend to think almost not at all in my own work about the “truth” of economic models, theories or data, but about their effects in the world (particularly at constructing new economic objects). I think these stories are more interesting and usually more compelling – e.g., without BSM, there might have been no options market as such, and certainly no implied volatility, and thus its predictions could not have been true or false.

        Unfortunately, because BSM was the first prominent example of the approach that “hit big”, the conflation is somewhat understandable: one of the most novel claims of M&M03 was that not only did BSM exhibit some generic forms of performativity as a model, but that the specific predictions of the model became true because they were predicted (to put it how the paper is read, not the more careful language of the authors). So, this post had that particular conversation in mind, and not the larger agenda of performativity research.

      • PS I very much enjoyed the new piece with MacKenzie on Gaussian Copula, but was a bit disappointed at Felix Salmon’s response. Have you been in contact with him at all?

  3. does SCE really only make sense for dynamic games? I think even in a one-shot game, if both players have incorrect beliefs about the other but are mutually best responding to those incorrect beliefs and both actions don’t contradict those false beliefs, that still counts as an SCE. I haven’t read that paper in a long time though. (But for example, maybe something like the fairness equilibrium examples in Rabin 1993: If I’m a nice person but believe you’re a mean person, so I punish you, and you likewise are a nice person and believe I’m a mean person and want to punish me, we’ll both be mean to each other in a one-time game, both will be optimally behaving given our tit-for-tat-ish preferences, and both will have no reason to update their beliefs about the other. I’m not sure that satisfies the definition in Fudenburg/Levine, since really it’s just a nash equilibrium when preferences are over beliefs, but something like that.)

    • afinetheorem

       /  June 30, 2012

      Vera, I just happened to notice your comment when looking at Dan’s new post today. In the original Fudenberg-Levine model, they note that in one-shot games, every information set is reached in every “path” of the game, hence off-equilibrium path beliefs do not exist. In standard Nash, beliefs must be correct at every information set by definition. There absolutely are other equilibrium concepts where you could have heterogeneous beliefs among players with the same information, but these aren’t Nash – perhaps someone has incorporated SCE into these?

      • ah, thanks Kevin, I misunderstood 🙂 I read “In one shot games, there is no chance to learn, and thus nothing to be self-confirmed” and focused on ‘one-shot’, which I would have phrased as ‘single-move’ instead. Although, my probably false example was also single move, but that’s appealing to psychological game theory (which isn’t nash, true, but is essentially the same concept but with different allowed inputs to utility, which complicates things.)