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Heterogeneity, spontaneous coordination and extreme events within large-scale and small-scale agent-based financial market models

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Abstract

We propose a novel agent-based financial market framework in which speculators usually follow their own individual technical and fundamental trading rules to determine their orders. However, there are also sunspot-initiated periods in which their trading behavior is correlated. We are able to convert our (very) simple large-scale agent-based model into a simple small-scale agent-based model and show that our framework is able to produce bubbles and crashes, excess volatility, fat-tailed return distributions, serially uncorrelated returns and volatility clustering. While lasting volatility outbursts occur if the mass of speculators switches to technical analysis, extreme price changes emerge if sunspots coordinate temporarily the behavior of speculators.

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Notes

  1. For a review of large-scale agent-based financial market models see, for instance, LeBaron (2006), while important contributions in this direction include LeBaron et al. (1999), Chen and Yeh (2001) and Raberto et al. (2001). See Hommes (2006) for a review of small-scale agent-based models and Day and Huang (1990), Brock and Hommes (1998), Lux and Marchesi (1999) and Chiarella et al. (2007) for some seminal benchmark contributions. The boundary between large-scale and small-scale agent-based models is not clear cut. Examples of models with intermediate complexity include Cont and Bouchaud (2000), Farmer and Joshi (2002) and LeBaron (2012). Moreover, Brock et al. (2005) and Diks and van der Weide (2005) also propose agent-based models with many different trader types and transform these models into small-scale models.

  2. To be precise, Franke and Westerhoff (2012) propose a family of agent-based models and check via an empirical model contest which member of this family has the greatest ability to mimic the behavior of actual financial markets. The model specification DCA-HPM is the winner of this model contest and, for simplicity, we call this model the FW model. Overall, the FW model has a remarkable ability to match the stylized facts of financial markets.

  3. It may be helpful to put these numbers into perspective. Suppose that the moments were independent from each other. Then we would expect the true data-generating process to imply a JMCR score of 0.9510 = 0.60. More conservatively, if we assume that five of the 10 moments are independent, then a JMCR score of 0.955 = 0.77 would be obtained.

  4. Since the performance of our large-scale model is practically identical to the performance of our small-scale model, the greater heterogeneity of the large-scale model does not seem to be very relevant in explaining the stylized facts of financial markets. As pointed out by an anonymous referee, this suggests that, in order to improve the matching of the stylized facts, one has to consider more trading rules or more complex trading rules or both.

  5. Note that there are some contributions in which random terms are added to the demand functions of speculators to capture the diversity of their trading strategies, e.g. in Westerhoff and Dieci (2006), Franke (2010) and Franke and Westerhoff (2016). Moreover, Schmitt and Westerhoff (2014) consider a multi-asset market model in which the random components consist of idiosyncratic, rule-specific, market-wide and general shocks.

  6. One example of a sunspot signal could be the investment advice provided by a financial guru. In fact, Shiller (2015, p. 109) argues that Joseph Granville, “a flamboyant market forecaster”, may have caused a couple of major market moves. Another example of a sunspot signal could be the emergence of a famous chart signal, such as a clear head-and-shoulders pattern. Analyzing the stock market crash on October 19, 1987, Shiller (2015, p. 118) ventures that two plots that appeared in the Wall Street Journal on the morning of the 1987 crash might have actually triggered the crash. One plot showed the evolution of the Dow in the 1980s, and the other one the evolution of the Dow in the 1920s. Together, both plots suggested that the crash of 1929 might be about to repeat itself. Clearly, chart traders who saw these plots might have thought along similar lines.

  7. For simplicity, we keep the fundamental value in our model constant. In future work, it may be interesting to link the coordination of fundamentalists to the evolution of the fundamental value. For instance, the coordination of fundamentalists may increase with the size of fundamental shocks. We are aware that such a modification stretches the original meaning of sunspots.

  8. The example about the price dynamics of the Dow in the 1920s and 1980s, given in footnote 6, may also influence fundamentalists since it indicates that a fundamental price correction may occur. Whether sunspots affect the behavior of chartists or fundamentalists more strongly is ultimately an empirical question that will be addressed in Section 5. An interesting model extension may be to include sunspots that affect fundamentalists and chartists alike.

  9. The fitness functions in Brock and Hommes (1998) include a constant that may be regarded as a predisposition parameter. In the model of Lux (1995), speculators’ rule selection is subject to herding behavior, while in that of de De Grauwe et al. (1993), the market impact of fundamental analysis depends on misalignments.

  10. Recall that the variance of the sum of correlated random variables, say \(\delta _{t}^{C,i}\), is given by the sum of their covariances, i.e. \(Var({\sum }_{i=1}^{N} \delta _{t}^{C,i})={\sum }_{i=1}^{N}{\sum }_{j=1}^{N} Cov(\delta _{t}^{C,i}, \delta _{t}^{C,j})={\sum }_{i=1}^{N} Var(\delta _{t}^{C,i})+2{\sum }_{i=1}^{N-1} {\sum }_{j=i+1}^{N} Cov(\delta _{t}^{C,i}, \delta _{t}^{C,j})\). In our case, random variables \(\delta _{t}^{C,i}\) have equal variances \(Var(\delta _{t}^{C,i})=(\sigma ^{C})^{2}\) and their covariances are given by \(Cov(\delta _{t}^{C,i}, \delta _{t}^{C,j})={\rho _{t}^{C}}(\sigma ^{C})^{2}\). Therefore, we can write \(Var({\sum }_{i=1}^{N} \delta _{t}^{C,i})=N(\sigma ^{C})^{2}+2\frac {N(N-1)}{2}{\rho _{t}^{C}}(\sigma ^{C})^{2}=(\sigma ^{C})^{2}(N+N(N-1){\rho _{t}^{C}})\). Accordingly, \(Var({\sum }_{i=1}^{N} {I_{t}^{i}}\delta _{t}^{C,i})=(\sigma ^{C})^{2}{N_{t}^{C}} (1+({N_{t}^{C}}-1){\rho _{t}^{C}})\) . The same argument applies for the derivation of \(Var({\sum }_{i=1}^{N} \lvert {I_{t}^{i}}-1 \rvert \delta _{t}^{F,i})=(\sigma ^{F})^{2}{N_{t}^{F}} (1+({N_{t}^{F}}-1){\rho _{t}^{F}})\).

  11. By “virtually independent” we mean that we want to express the dynamical system of our model independently of N. Of course, restrictions \({X_{t}^{C}}\le N\) and \({X_{t}^{F}}\le N\) remain, i.e. \({X_{t}^{C}}\) and \({X_{t}^{F}}\) implicitly define a lower limit for N.

  12. Since our calibration strategy relies on the moments’ confidence intervals, different methods to compute these intervals may lead to different parameter settings, different performance values and, eventually, different model rankings.

  13. For other studies which rely on the method of simulated moments to estimate small-scale agent-based models see, for example, Winker et al. (2007), Franke (2009a) or Chen and Lux (2015).

  14. By paying greater attention to the tail behavior of the distribution of returns, these values may be further improved. However, we determine the model parameters by trying to maximize the JMCR(10) score, and in this sense there is a trade-off between matching of two tail indices and matching the other eight summary statistics.

  15. To give an example, we use Mathematica 10 to simulate our models. The time taken to compute a single time series with 6750 observations increases from about 0.4 seconds to about 45 seconds if we switch from a small-scale model to a large-scale model with 100 speculators, and to about 800 seconds if we set N = 400.

  16. We thank an anonymous referee for pointing this out.

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Authors and Affiliations

  1. Department of Economics, University of Bamberg, Feldkirchenstrasse 21, 96045, Bamberg, Germany

    Noemi Schmitt & Frank Westerhoff

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  1. Noemi Schmitt
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  2. Frank Westerhoff
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Correspondence to Frank Westerhoff.

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This study was funded by ISCH COST Action IS1104: “The EU in the new complex geography of economic systems: models, tools, and policy evaluation”. The authors declare that they have no conflict of interest.

Additional information

Presented at the Dynamic Macroeconomics Workshop, Kiel Institute for the World Economy, November 2015. We thank Simone Alfarano, Reiner Franke, Thomas Lux and Christian Proaño for their valuable feedback and comments. The paper also benefitted from a number of constructive remarks from two anonymous referees.

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Schmitt, N., Westerhoff, F. Heterogeneity, spontaneous coordination and extreme events within large-scale and small-scale agent-based financial market models. J Evol Econ 27, 1041–1070 (2017). https://doi.org/10.1007/s00191-017-0504-x

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