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Journal of Economic Interaction and Coordination

Erschienen in:

04.03.2016 | Regular Article

Complexity and model comparison in agent based modeling of financial markets

verfasst von: Alexandru Mandes, Peter Winker

Erschienen in: Journal of Economic Interaction and Coordination | Ausgabe 3/2017

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Abstract

Agent based models of financial markets follow different approaches and might be categorized according to major building blocks used. Such building blocks include agent design, agent evolution and the price finding mechanism. The performance of agent based models in matching key features of real market processes depends on how these building blocks are selected and combined. For model comparison, both measures of model fit and model complexity are required. Some suggestions are made on how to measure complexity of agent based models. An application for the foreign exchange market illustrates the potential of this approach.

Vorheriger Artikel Voting for the distribution rule in a Public Good Game with heterogeneous endowments

Nächster Artikel Monetary policy and dark corners in a stylized agent-based model

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We will also use the acronym ABM for agent based models.

It should be noted that the term “complexity” is used here in a quite general meaning. We come back to the question which concepts and measures of complexity might be adequate in the context of ABM in Sect. 3.

It is also to be noted that the results are based on original reported data and not every ABM approach refers to the same list of stylized facts.

Usually, one of the factors is stochastic in order to reflect missing information or uncertainty about the strategy. It also introduces some heterogeneity of agents’ behavior.

The fundamental value can be perceived by the agents without or with errors. For example, in Fischer and Riedler (2014), evaluation errors persist only for a limited time, as agents will eventually become aware and correct them.

Arthur et al. (1997) underline this reflexive nature of the market, where agents’ expectations co-evolve in a world they co-create.

Each agent has at her disposal a set of strategies which is continuously evaluated—even if the rule was not actually executed, but would have been a valid selection based on its condition classifier part—and updated at random times by replacing a subset of the agent’s worst rules with new ones generated by crossing-over and mutating a selection of its own best rules.

A population of new trading rules is evolved by means of GP within a distinct “schooling” component, which is visited from time to time by under-performing single strategy holding agents.

Alternatively, in the case of intraday ABM, a continuous market is implemented where trading demand is disclosed asynchronously and orders are matched by means of a limit order book at various prices [see, e.g., Chiarella and Iori (2002), Daniel (2006), Chiarella et al. (2009), Mandes (2015)]. As a consequence, transactions do not take place only at the global equilibrium price. However, this higher-frequency framework deals with a finer level of details which are not within the scope of market features we want to capture in this paper.

A complexity count is added for each logical connective operator and and or within logical statements. Also, conditional expressions using the ternary operator ?: add one count to the complexity total.

At the start of each file the block level is zero and the block depth grows with each execution control statements such as if, case and while.

One way to achieve such a standardization consists in considering Flops instead of time. Given that this might not be feasible in practice, counting the counts to procedures, functions etc. might provide a way to obtain more reliable lower bounds for this type of computational complexity.

This view is in line with the intuition of Chen et al. (2012) who define complexity as the degree of heterogeneity (diversity).

If applied directly to the final (observable) output, this statistic could evaluate and even validate/reject a calibrated model with respect to specific data. In other words, it has the potential of becoming a stylized fact, given that certain values would be exhibited by a wide range of financial data. A first application is contained in Pincus and Kalman (2004), but more work is needed.

The chosen values for the two parameters, i.e. the block length \(m = 1\) and the tolerance window \(r = 0.2\,\sigma \), have been provided in Pincus and Kalman (2004).

After translating the target time-series into a sequence via a measuring channel, the ABM can be considered a “message” source, and the general idea is to construct a “parallel” computational model which is able to “predict” its behavior (statistically indistinguishable). The information content of the simplest such model determines the complexity class of the original system.

Barde (2015) develops an Universal Information Criterion which relies on mapping the target data-generated process to an underlying Markov process by applying a universal data compression algorithm, i.e. the Context Tree Weighting algorithm proposed by Willems et al. (1995). Once the Markov transition matrix is generated, the mean value of the benchmark observation-level score, i.e. the log-likelihood, can be computed giving the prediction accuracy which is equivalent to the Kullback–Leibler distance.

Each trading day is divided into \(n = 50\) micro-time intervals.

Farmer and Joshi (2002) show that even if the strategy coefficients would be heterogeneous among agents, when the strategies are linear, equivalent results can be achieved by using a “representative agent” for each strategy type with a coefficient equal to the group-mean. As a side note, Chen et al. (2012) classify this category as “few-type” models.

Some “fallacy of composition” might arise due to this aggregation.

Chen et al. (2012) identify the Lux model as an hierarchical two-type model, with two chartist subdivisions.

The fundamental value is assumed to remain constant over the entire simulation time span.

Agents’ portfolios are ignored and therefore they are able to accumulate unbounded inventories.

This means that there are \(n = 100\) microintervals per day, at which interaction and trading sessions occur.

When computing the effective net transitions between different clusters, we are not using the expected value as described in Lux (1998), rather we simulate the sampling data effect by drawing a random deviate from a normal distribution centered around the expected value and with a variance depending on the cluster size. Otherwise, due to rounding and very small probabilities per time unit, the number of agents changing their types would be zero most of the time.

For example, the flow of fundamentalists to the optimistic chartists group is \(N_F\,r^{F+} - N^+_C\,r^{+F}\), where \(r^{F+} \sim \mathcal {N}(\pi ^{F+},1/N_F)\) and \(r^{+F} \sim \mathcal {N}(\pi ^{+F},1/N^+_C)\).

The switching probabilities are based on the assumption of direct interaction between agents (herding effect) and, thus, their values depend on the sizes of agent groups pursuing a common strategy.

Other conditions regarding the thresholds are \(-T < \tau < T\) and \(|\tau _{min}| \le T_{min}\).

There is an exception in the case of the Kirman model where, because of missing original parameters, we use the parameters presented in Winker et al. (2007)

By employing a population based search heuristic, such as the Genetic Algorithm, we try to alleviate the difficulty of choosing sensible starting points, an issue underlined in recent work on estimation of ABMs (Chen and Lux 2015). However, a larger number of iterations and reruns would be necessary in order to account for both the stochastics of the heuristics and of the estimators, and to precisely report on their convergence.

In the case of DEM/USD there are 10,000 bootstrap replications, while in the case of all simulated models throughout this paper there are 100 replications with different random seeds.

We are using the set-up provided in Pincus and Kalman (2004) where \(m=1\) and r equals 20 % of the time-series standard deviation.

It is worth mentioning that the calibration of the Lux model—as opposed to the other two considered ABM—is highly sensitive to the underlying time-series to be fitted. We have run a similar calibration exercise for the DAX 30 (2003–2011) time-series and a non-trivial readjustment of the search intervals for the parameters is necessary in the Lux case. On the other side, the Farmer–Joshi model is able to better fit the DAX 30 time-series, both at the aggregate, as well as at the individual moments level.

Every Java application has a single instance of class java.lang.Runtime that allows the application to interface with the environment in which the application is running, including the virtual memory space assigned by the operating system to the Java process.

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Titel: Complexity and model comparison in agent based modeling of financial markets
verfasst von: Alexandru Mandes
Peter Winker
Publikationsdatum: 04.03.2016
Verlag: Springer Berlin Heidelberg
Erschienen in: Journal of Economic Interaction and Coordination / Ausgabe 3/2017
Print ISSN: 1860-711X
Elektronische ISSN: 1860-7128
DOI: https://doi.org/10.1007/s11403-016-0173-0

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