Abstract
We study the Johansen–Ledoit–Sornette (JLS) model of financial market crashes (Johansen et al. in Int J Theor Appl Financ 3(2):219–255, 2000). On our view, the JLS model is a curious case from the perspective of the recent philosophy of science literature, as it is naturally construed as a “minimal model” in the sense of Batterman and Rice (Philos Sci 81(3):349–376, 2014) that nonetheless provides a causal explanation of market crashes, in the sense of Woodward’s interventionist account of causation (Woodward in Making things happen: a theory of causal explanation. Oxford University Press, Oxford, 2003).
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Notes
For more on the relationship between physics, finance, and econophysics, see Weatherall (2013); for further technical details and overviews of recent work, see Mantegna and Stanley (1999), McCauley (2004), and Cottrell et al. (2009). There is also a small literature in philosophy of science dealing with econophysics, including Rickles (2007) and Thébault et al. (2017).
Despite the prevalence of this sort of criticism, it is far from clear that physics is more guilty of oversimplification than economics when it is applied to economic facts.
Gallegati et al. (2006) has sparked a small debate, with responses by McCauley (2006) and Rosser Jr. (2008) among others. It is worth noting that financial markets, which are the focus of the present paper, are one of the areas of economic activity that Gallegati et al. seem to think are amenable to the methods of econophysics, and so it is not clear that the model we consider here is touched by these criticisms.
These stylized facts are often treated as qualitative laws or as descriptions of lawlike behavior, capturing “set[s] of properties, common across many instruments, markets, and time periods” (Cont 2001, p. 223).
This has also been noted by Mandelbrot (1963, p. 418).
As we hope will be clear in what follows, we do not mean to disagree with Batterman (2000, 2002) and others, such as Reutlinger (2014), who have argued that explanations of why universal phenomena occur that draw on RG methods are generally non-causal. Instead, we mean to argue that RG methods may be used for multiple purposes, and that in the present case, the salient explanations have a different character than in the case of statistical physics. The explanandum is not the existence of universality, and the explanation is causal.
The EMH has been a topic of considerable controversy. For instance, Shiller (1984) has argued that the argument behind the EMH is invalid. The main worry is that current models neglect (i) agent psychology and (ii) interactions amongst agents as key causal and explanatory features of asset price variations. Once these factors are considered, it seems markets may well be random irrespective of how efficiently markets process information or how accurately prices reflect fundamental values. Meanwhile, as Ball (2009) and others have argued, over-reliance on the assumption of efficiency may affect how market participants synthesize information regarding possible asset bubbles. But we will not weigh in on such controversies; our purpose here is not to endorse the EMH, but rather to describe the context of the JLS model and to emphasize its continuity with mainstream economic modeling methods.
Note that this argument appears to suppose that news that will positively affect price is equally likely as news that will negatively affect price. But if there were any information available that would indicate that positive (resp. negative) news was more likely, then that fact alone would count as tradeable information that would affect price.
Note that we mean “self-organization” in the informal sense of coordinated action between agents without any apparent external mechanism. We do not intend to invoke any specific theories of self-organization or self-organized criticality.
More precisely, if \(F\left( t \right) \) is the cumulative distribution function of a crash occurring at or before time t, then \(h\left( t \right) =F^{\prime }\left( t \right) /\left( {1-F\left( t \right) } \right) \), where \({F}''\) is the probability density function. Conversely, one can define a cumulative probability function from a hazard rate by integrating both sides of this equation with respect to t. See, for instance, (Cleves et al. 2004, Ch. 2) for further details on interpreting hazard rates.
It is tempting to interpret the right hand side of Eq. (1) as representing the probability of a crash occurring during the period from \(t_0\) to t, but this would be incorrect: the integral of \(h\left( t \right) dt\) does not yield a probability. (For instance, it may exceed 1.) Instead, this quantity should be understood as a measure of accumulated risk, in the sense that it represents the total number of times you should have experienced a crash during this period, supposing the crash were repeatable. Once again, see Cleves et al (2004, Ch. 2).
Sornette (2003) also considers the possibilities of “anti-crashes”, wherein a large number of traders suddenly transition to “buy” states; these are taken to be the ends of “anti-bubble” regimes. However, it is important to note that neither Sornette (2003) nor Johansen et al. (2000) explain the fact that crashes are generally caused by “sell” states instead of “buy” states.
The argument here is subtle. JLS first present their model generically, without making any assumptions about the details of the network. They then observe that if the network has certain features—in particular, if it is hierarchical in a sense to be explained in Sect. 4.2—then it will exhibit complex critical exponents, and hence log-periodic oscillations near criticality. They give some plausibility argument for considering hierarchical lattices, but leave the actual lattice structure open until they consider historical data—at which point they conclude that, given the presence of oscillations, the network must be approximately hierarchical and the critical exponents must be complex. It is in this sense that introducing complex critical exponents is “phenomenological”. One can also run the argument in the other direction, however, and argue that on the basis of a plausible assumption concerning the hierarchical nature of trader networks, the critical exponents should be expected to be complex; at times, Sornette and collaborators appear to prefer this version of the argument.
An early discussion of log-periodicity and self-similarity is given by Barenblatt and Zel’dovich (1972). Extensive work on the existence of complex critical exponents with log-periodic oscillations has been carried out by Sornette and his collaborators (e.g. Sornette 1998; Arnéodo et al. 1998; Gluzman and Sornette 2002; Zhou et al. 2005; Sornette 2006).
There are various criticisms of the JLS model that also stress the disanalogies between the JLS model of financial crises and critical phase transitions. For example, Ilinski (1999) casts doubt on a main component of the JLS model: crashes are principally caused by imitative dynamics between individual traders. He objects that different market participants may act over different time horizons (e.g. minutes for speculators, years for managers), so that the instantaneous long-range interactions between traders postulated by the JLS model are implausible. We will not engage with this criticism or others; instead, we want to see how far the analogy goes if we assume that the model is well-motivated and well-supported empirically.
As will become clear in what follows, by “universality class” we mean the basin of attraction of a given non-trivial fixed point under some RG flow. In cases of critical phase transitions, these correspond to systems with the same critical exponent near the transition point, though RG methods may be applied more generally. Batterman and Rice (2014) suggest a still-broader definition of “universality class” that applies to systems outside of physics where the RG does not apply; as we will see below, market crashes will turn out to form a universality class in this more general sense, but one needs to be careful about the role that the RG plays in the argument for this.
Note that our description of RG methods here follows the “field space” approach, in the sense of Franklin (2017).
Note that it is not essential, here, to begin with an empirical observation—though that is what happened in the physics of phase transitions. In principle, one can demonstrate that two systems are in the same universality class and thereby predict their behavior near critical points.
In addition to what we argue in what follows, Sornette (personal correspondence) points out that market crashes should be understood as dynamical (i.e., non-equilibrium) phase transitions, wherein a parameter diverges at a critical time. In these systems, one generally finds universality to be much weaker than in equilibrium systems.
Note however that this does not mean that RG methods cannot be applied at all in the context of the JLS model. Zhou et al. (2003), for instance, use renormalization group methods to obtain an extension of Eq. 5 that gives an account of larger time scales. Moreover, as we will see, RG methods will reappear in our analysis below, although they will play a different role than in statistical physics.
Note that, while he continues to argue that DSI and LPPLs are important features of crashes that signal the end of bubble regimes, in more recent work Sornette has suggested that both of these may be secondary, with the fundamental signature of a crash instead being positive feedbacks, leading to power law singular behavior (of some sort or other) (Sornette 2015b; Sornette and Cauwels 2015a; Leiss et al. 2015). These arguments seem to us to move beyond the JLS model as we have presented it, though we take it they are broadly compatible with the picture we sketch here of the sort of explanation these models seek to give. In particular, on this alternative view it would be the inference from LPPLs to positive feedback loops that forms the explanatory core of the model. We are grateful to Didier Sornette for drawing our attention to these more recent arguments.
Sornette also speaks of this explanation as “causal”, for instance, when he writes “...the market anticipates the crash in a subtle self-organized and cooperative fashion, hence releasing precursory “fingerprints” observable in the stock market prices.... we propose that the underlying cause of the crash must be searched years before it in the progressive accelerating ascent of the market price, reflecting an increasing build-up of the market cooperativity” (Sornette 2003, p. 279). As we noted in footnote 8, we do not take the claim that this explanation is causal to be in conflict with the views defended by Batterman (2000, 2002), Reutlinger (2014), or others. The claim is not that there is an explanation of universality in this model that is causal. Rather, the claim is that the explanation of a given crash, or even crashes in general, is causal, because the JLS model identifies how to intervene to produce a crash, or to prevent one—namely, by changing network structure.
See Lange (2015) for a different critique of Batterman and Rice (2014) than we give here. Lange argues that Batterman and Rice cannot sustain the distinction they draw between their account and “common feature” accounts such as Weisberg’s (discussed below). We take it that one can sustain a distinction between different explanatory goals, one of which might well be to explain why many different systems should be expected to be saliently similar to some highly idealized model, and we think that Batterman and Rice do an adequate job of explaining both how that explanatory goal can be met, and why the strategies for meeting it do not look like they are appealing to common features of a model and a target system. That said, as we will argue, in some cases a single model, including the JLS model, can be used to achieve more than one explanatory goal.
This is not to say that the model could not be reconfigured as one that is invariant across some scales, but not under arbitrary scale transformations. In other words, we do not mean to deny what is sometimes known as “Earman’s principle”, that idealized models can only be explanatory if one can imagine removing the idealization and still being able to explain the same phenomenon (Earman 2004; Butterfield 2011). But doing so would require substantial changes in the analysis, and would effectively produce a different model from the one under consideration. Our interest is in the explanatory role of the infinite idealization in the present version of the model.
See also Morrison (2006) for a related point.
Here there is a relationship both to “Earman’s principle”, as noted in footnote 29, and also to Butterfield (2011), who argues that in cases where one takes an unrealistic infinite limit, one should expect to see the qualitative behavior that arises in the limit appearing already on the way to the limit.
Of course, one might consider stronger senses in which an explanation could be reductive. For instance, one might require that a reductive explanation gives us information about the details concerning the behavior of the micro-constituents of the system, or that a reductive explanation elucidate why the microscopic details are causally relevant for the phenomena under study. One might even insist that an explanation is reductive only if it appeals to fundamental physics—in which case, no explanation in the social sciences, and few in biology, chemistry, or even physics could ever be reductive. As we hope is clear from the text, we have in mind a weaker sense of an explanation being “reductive”; it is not essential to our purposes that this sense of reductive contravene Batterman and Rice. We are grateful to an anonymous referee for pushing us on this point.
This point mirrors one made by O’Connor and Weatherall (2016): there are many different purposes for which models may be constructed, and to which they may be put. This includes different explanatory purposes, and so one should be cautious about attempts to classify or taxonomize models on the basis of how they may be used to explain.
We tend to think that they are convincing, or at least, we agree that explanations of universality of the sort Batterman and Rice discuss are non-causal. (See also Reutlinger 2014 for a different argument concerning why these explanations are non-causal.)
We should emphasize that we do not take the claim that different questions call for different kinds of explanation to be in tension with Batterman and Rice’s view. Our point, rather, is to resolve the apparent tension between our arguments and Batterman and Rice’s view by distinguishing the why questions at issue. We are grateful to an anonymous referee for encouraging us to clarify this.
Note that there is another interpretation of “Why do stock markets crash?” that does not demand a causal explanation, but rather another minimal model explanation: namely, “Why do markets fall into a universality class of systems that exhibit crashes, as opposed to tamer sorts of transitions?” Of course, this is a legitimate explanatory demand, and the answer, invoking the JLS model, would look more like the answer to the first question than the second. The difference between these two understandings of the question “Why do stock market crash?” invokes Van Fraassen’s (1980) analysis of the logic of why questions. Explanatory demands, van Fraassen convincingly argues, involve, in addition to the explinandum, both a contrast class and a relevance relation.
Other exchanges, e.g. the Chicago mercantile exchange, have similar measures in place.
This is actually a refinement of an old circuit breaker that would halt that stock’s trade entirely for 5 min, but it caused too much administrative trouble to be usable.
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Acknowledgements
This paper is partially based upon work supported by the National Science Foundation under Grant No. 1328172. Previous versions of this work have been presented at the at a workshop on the Physics of Society and a conference on Infinite Idealizations, both hosted by the Munich Center for Mathematical Philosophy; we are grateful to the audiences and organizers for helpful feedback. We are also grateful to Didier Sornette for helpful discussions concerning his work and for detailed feedback on a previous draft of the paper, to Alexander Reutlinger for detailed comments on an earlier draft, and to two anonymous referees for their helpful comments.
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Jhun, J., Palacios, P. & Weatherall, J.O. Market crashes as critical phenomena? Explanation, idealization, and universality in econophysics. Synthese 195, 4477–4505 (2018). https://doi.org/10.1007/s11229-017-1415-y
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DOI: https://doi.org/10.1007/s11229-017-1415-y