تغییرات زمانی لحظه های بالاتر در یک بازار مالی با عوامل نامتجانس: یک رویکرد تحلیلی

A growing body of recent literature allows for heterogenous trading strategies and limited rationality of agents in behavioral models of financial markets. More and more, this literature has been concerned with the explanation of some of the stylized facts of financial markets. It now seems that some previously mysterious time series characteristics like fat tails of returns and temporal dependence of volatility can be observed in many of these models as macroscopic patterns resulting from the interaction among different groups of speculative traders. However, most of the available evidence stems from simulation studies of relatively complicated models which do not allow for analytical solutions. In this paper, this line of research is supplemented by analytical solutions of a simple variant of the seminal herding model introduced by Kirman [1993, Ants, rationality, and recruitment. Quarterly Journal of Economics 108, 137–156]. Embedding the herding framework into a simple equilibrium asset pricing model, we are able to derive closed-form solutions for the time-variation of higher moments as well as related quantities of interest enabling us to spell out under what circumstances the model gives rise to realistic behavior of the resulting time series.

Until very recently, theoretical research in finance has largely ignored some of the really universal stylized facts of practically all available financial data. In fact, a glance at frequently used textbooks like the ones by O’Hara (1995) and Barucci (2003) shows that even their glossaries lack entries for some of the prevalent technical terms of the empirical finance literature. For example, while many developments in empirical finance are essentially motivated by the observation of non-Gaussian returns distributions with their ‘fat tails’ and temporal dependence of second moments leading to ‘volatility clustering’, these notions have been almost entirely absent from the theoretical literature.1 While these phenomena have spurred the development of such seminal innovations like GARCH type and stochastic volatility models in empirical finance, their behavioral origins have apparently remained an almost unaccessible puzzle for a long time. Two reasons might be responsible for this neglect: first, the above features characterize the behavior of financial time series as a whole, while the interest in economic theory has typically been to spell out the effect of a change of one (endogenous) economic variable on other, exogenous variables. Even when allowing for an ensemble of traders, such a comparative statics approach is not appropriate for explaining universal conditional and unconditional stochastic properties. Second, the prevalent efficient market paradigm did, in fact, provide a very simple implicit answer to the question of the origin of all stylized facts of returns: since, in this framework, prices would reflect forthcoming news in an unbiased and immediate manner, any property of the returns distribution would simply reflect a similar feature of the distribution of new information items. As a corollary of the efficient market hypothesis, the ‘news arrival process’ would, therefore, have to come along with fat tails and clustering of important news. Unfortunately, this corollary can hardly be subjected to econometric scrutiny. On the other hand, enough evidence had been collected against the universal validity of the efficient market paradigm to motivate alternative, behavioral approaches which then mushroomed over the nineties. First analyses of complex data-generating mechanisms based on interacting agents can be found in Kirman, 1991 and Kirman, 1993 and DeGrauwe et al. (1993). While they did not focus then on the above stylized facts (not broadly acknowledged at that time among theoretical researchers), they both already showed that their models could mimic the random walk nature of asset prices and exchange rates although their data-generating processes were clearly different from a true random walk. Notably, both studies also investigated what might be described as secondary stylized facts: they applied certain popular econometric analyses to their simulated data and found similar behavior as with empirical records providing a possible explanation of, e.g. the forward premium puzzle of foreign exchange markets. Evidence for volatility clustering as an emergent phenomenon of a multi-agent model appeared first in Grannan and Swindle (1994). While a large body of subsequent models studied artificial markets with heterogenous autonomous agents often endowed with some sort of artificial intelligence (classifier systems, genetic algorithms), any consideration of empirical stylized facts is also curiously absent in the first wave of such papers (Levy et al., 1994, Arifovic, 1996 and Arthur et al., 1997). In fact, much of this early literature had been preoccupied with the question of convergence or not of their learning algorithms to the benchmark of rational expectations rather than considering empirical applications. However, subsequent research has shown that relatively simple models of interacting traders could produce realistic time series sharing the ‘stylized facts’ of fat tails and clustering volatility even up to numerical agreement with key quantities of empirical data (Lux and Marchesi, 1999 and Lux and Marchesi, 2000). Similar investigations of the dynamic properties of alternative models revealed that many agent-based approaches share a certain tendency of generating fat tails and volatility clustering although their quantitative manifestations are not always identical to the very robust numbers obtained with empirical data (cf. LeBaron et al., 1999, Chen and Yeh, 2002, Kirman and Teyssière, 2002 and Lux and Schornstein, 2005). Often ‘realistic’ dynamics are identified as the consequence of some kind of critical or intermittent process. In Lux and Marchesi, 1999 and Lux and Marchesi, 2000 as well as in Giardina and Bouchaud (2003) and Lux and Schornstein (2005) the dynamics are characterized by endogenous changes in the population composition triggered by both systematic factors (i.e. some fitness criterion like profits or utility obtained by a particular trading strategy) and noise (idiosyncratic components or experimentation). However, the stability of the entire dynamic system also depends on the strategy configuration within the population and with its random dynamics the population composition switches between stable and unstable configurations leading to intermittent bursts of volatility. Similar mechanisms probably prevail in the artificial markets of Raberto et al. (2003) and Chiarella and Iori (2002). A related branch of agent-based models with realistic time series properties has been launched by Cont and Bouchaud (2000) adapting the framework of percolation models from statistical physics. In these models, traders are situated in a lattice and form clusters of agents with the same ‘strategy’ (buy, sell, remain inactive). Contributions to this literature have been inspired by the well-known critical behavior of percolation models at a certain threshold of the connectivity level and have added various additional features for preserving the critical power-law fluctuations for a wider range of parameters or adding temporal dependency (cf. Bornholdt, 2001; Stauffer and Sornette, 1999; Iori, 2002). Another closely related approach is the discrete choice framework for the choice of agents’ strategies adopted by Gaunersdorfer and Hommes (2007). Giving rise to a dynamic system with coexisting attractors (e.g. a stable fix point together with a limit cycle) they obtain realistic dynamics by allowing for additional noise that triggers intermittent switching between different basins of attraction. While in most of these models the general appearance of simulated data seemed to be quite robust with respect to most of the underlying parameters, it often also turned out that the potential of generating stylized facts depends crucially on the system size (i.e. number of agents). Realistic dynamic patterns are typically observed with the (probably natural) initial choice of a few hundred or thousand agents. However, with an increasing number of market participants one often experiences a fading away of the fat tails and volatility clustering beyond a certain threshold (cf. Egenter et al., 1999, Lux and Schornstein, 2005 and Challet and Marsili, 2004). The present paper attempts to shed light on both of the findings detailed above. Within a relatively simple type of herding model (broadly along the lines of Kirman, 1993) we derive closed-form solutions for auto-covariances of returns and their higher moments together with other statistics such as mean-passage times. Inspection of the results allows to infer in how far and under what conditions the model could mimic the empirical findings of fat tails and clustering of volatility. Investigation of different specifications of the model also allows us to point out why – in certain scenarios – increasing system size would lead to vanishing stylized facts. Our approach is broadly complementary to recent attempts at studying asymptotic properties of related agent-based models (Horst, 2004 and Föllmer et al., 2005). These authors provide conditions under which the limiting distribution of the price process exists in models with both global and local interactions of agents (Horst, 2004) and models with feedback from the price process on the group dynamics (Föllmer et al., 2005). Since our model can be viewed as a special case of the class of models studied in Horst (2004) his results on convergence of equilibrium prices to a unique equilibrium distribution also apply in our case. However, instead of focusing on the properties of the price process alone we extend the analysis to returns to which the famous empirical regularities of fat tails and clustered volatility apply. We also go beyond asymptotic convergence results by working out various properties of the stationary distribution which are of interest in the light of empirical findings. The main results of our paper are: •• the derivation of closed-form solutions for conditional and unconditional moments which provide insights into the origins of leptokurtosis and heteroskedasticity in this model. •• the derivation of results on the dependency of their realistic features on system size. As it will turn out, whether or not we observe a fading away of stylized facts depends on rather subtle details of the formalization of transition rates (extensive vs. non-extensive transition rates, see below). While both variants can be clearly distinguished in the present framework, it might not be easy to determine whether certain more complicated models fall into the first or second category. The paper is organized as follows: Section 2 introduces different variants of herding models inspired by Kirman's ant process (Kirman, 1993). Section 3 incorporates the recruitment mechanism of the ant model into an asset pricing framework. Section 4 details our analytical results for both the unconditional and conditional distribution of returns. In Section 5 we explain different techniques for simulating these agent-based models, while Section 6 points out under which conditions the interesting dynamics survives for large numbers of agents. Section 7 concludes. Technical details are relegated to various appendices.

The aim of the present paper was to contribute to a better understanding of the statistical properties of agent-based models of financial markets. To this end, we attempted a thorough analysis of different types of herding models in the tradition of Kirman's seminal ant model. Our novel insights are the following: (i) We were able to derive closed-form solutions for various moments as well as the autocorrelations of raw and squared returns. Our results demonstrate that both leptokurtosis of returns and temporal dependence of volatility are generic features of this model and are intimately linked to the herding component. This underscores the importance of boundedly rational behavior as a potential explanation of the stylized facts. Admittedly, our model did not produce power laws (hyperbolic decay) of the cumulative distribution nor of the autocorrelation function. However, in a companion paper (Alfarano and Lux, 2007), we also show that pseudo-empirical analysis of time series for such a model could easily produce ‘spurious’ or ‘apparent’ power laws, which is a typical outcome within a pre-asymptotic regime with a length of the time series below some threshold value. (ii) We have also investigated the dependency of the ‘interesting’ dynamics and stylized facts on system size, revisiting earlier findings of trivial large N limits in similar models. As it turns out, trivial or non-trivial behavior of large markets depend on the exact formalization of transition rates. Ironically, a literal interpretation of pairwise interactions leads to some sort of ‘global’ coupling of agents, which preserves all interesting features of the model for any system size. On the contrary, a more conventional formalization of the transitions rates in terms of concentrations leads to a convergence to uninteresting (and unrealistic) Gaussian dynamics in the limit of large markets.