J Econ Interact Coord (2012) 7:223–248 DOI 10.1007/s11403-012-0099-0 REGULAR ARTICLE

A re-examination of the “zero is enough” hypothesis in the emergence of financial stylized facts Olivier Brandouy · Angelo Corelli · Iryna Veryzhenko · Roger Waldeck

Received: 8 September 2011 / Accepted: 16 July 2012 / Published online: 5 August 2012 © Springer-Verlag 2012

Abstract In recent years, a growing literature has claimed that the market microstructure is sufficient to generate the so-called stylized facts without any reference to the behaviour of market players. Indeed, qualitative stylized-facts can be generated with zero-intelligence traders (ZITs) but we stress that they are without any quantitative predictive power. In this paper we show that in most of the cases, such qualitative stylized facts hide unrealistic price motions at the intraday level and ill-calibrated return processes as well. To generate realistic price motions and return series with adequate quantitative values is out-of-reach using pure ZIT populations. To do so, one must increasingly constrain agents’ choices to a point where it is hard to claim that their behaviour is completely random. In addition we show that even with highly constrained ZIT agents, one cannot reproduce real time series from these. Except in a few cases, first order moments of ZITs never equal real data ones. We therefore claim that stylized facts produced by means of ZIT agents are useless for financial engineering.

The authors would like to thank funding from the Institut Mines-Telecom, grant “Future and rupture”. O. Brandouy (B) Sorbonne Graduate Business School, GREGOR (EA 2474, IAE de Paris), Paris, France e-mail: [email protected] A. Corelli IEI Department, Linköping University, Linköping, Sweden e-mail: [email protected] I. Veryzhenko ENSAM Paristech, GREGOR (EA 2474, IAE de Paris), Paris, France e-mail: [email protected] R. Waldeck Department LUSSI, Telecom Bretagne, Institut Mines-Telecom, Brest, France e-mail: [email protected]

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Keywords Agent based modelling · Time series analysis · Financial markets · Heterogeneity of agents · Market microstructure 1 Introduction Agent-based modelling is a powerful research tool for understanding how macro phenomena emerge. As such, it is a widely used methodology in the social sciences and economics in particular (Duffy 2006). As with the approach of the representative individual agent, it aims at explaining the macroscopic level from the microscopic point of view of agents, but with the additional step that interaction structures and rationality play a major role. Financial markets are particularly appealing applications for agent-based methods (Le Baron 2006; Hommes 2006) and the debate is on whether institutions or rationality is the most important factor in explaining stylized facts that are observed in real markets, but also replicated in artificial ones. Indeed, as Cont (2000) points out, the result of more than half a century of empirical studies on financial time series indicates that the seemingly random variations of asset prices do share some quite non trivial statistical properties. Such properties, common across a wide range of instruments, markets and time periods are called empirical stylized facts. Such stylized facts are usually formulated in terms of qualitative properties of asset returns, and may not be precise enough to distinguish among different parametric models. The major stylized facts are the following : i) for the return time series : absence of autocorrelation, except for very small intraday time scales, volatility clustering,1 autocorrelation of volumes and cross correlation volume volatility (Levy et al. 2000); ii) for the empirical distribution of asset returns, one observes that they are non-Gaussian but leptokurtic i.e. fat tailed distributions tending to Gaussianity when the time scale increases. The fact that many different assets share such common qualitative properties and that many artificial systems may reproduce the latter does not imply that they can also generate adequate quantitative values for these properties. This paper shows that in fact, this is almost out of reach within the framework of zero intelligence agents (Gode and Sunder 1993). Stylized facts are difficult to explain by the mainstream theory and the effort in empirical research to describe data lacks of a convenient theoretical foundation of these facts. However, an alternative approach has emerged from econophysics which describes the same financial facts as scaling laws. Indeed, this approach considers that physical systems, consisting of a large number of interacting particles, obey universal scaling laws that are independent of the microscopic details : economies (and social systems in general) can be considered in the same way Amaral et al. (1999). This approach may suggest that the rational individual choice is not important in explaining macroscopic facts. But as Lux (2009) points out, it might be the heterogeneity of market participants together with a few basic principles of interactions that exerts a dominating influence on the macroscopic market behaviour, whatever the institutional 1 Volatility clustering expresses the fact that large (small) changes in returns tend to be followed by large

(small) changes of either sign, for some considerable time.

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settings. Indeed a large literature has emerged which aims at discerning among two major explanations: market micro-structure and agents’ behaviour and heterogeneity. Papers in favour of the explanation by the microstructure initiate with the research on zero intelligent traders (here-after ZIT) of Gode and Sunder (1993) who show, in the spirit of Becker (1962), that traders acting randomly but within a budget constraint act comparably well in terms of convergence to equilibrium price and efficiency as compared to human subjects in experimental economics (see Smith 1962). Cliff et al. (1997) show that for special demand or supply function this may not be the case and a slightly more complex behavioural assumption is required to achieve equilibrium in continuous double auction markets. Ladley and Schenk-Hoppe (2009) in a similar framework as Gode and Sunder, but with a constant flow of traders entering the market, reproduce price movements and show that aspects of the order book such as the size of spreads and conditional probabilities of order submissions can be obtained by the interplay of ZITs and the book. However, the observed frequency of the submitted order types seems related to strategic behaviours based on the observed book state. Maslov (2000) introduces a model where traders randomly choose to trade either at the market price or by placing a limit order; limit orders are simply determined by offsetting by a random amount the most recent market price. Maslov shows that fat tails, long range correlation in the volatility and non-trivial Hurst exponents arise in such framework. One paper by Farmer et al. (2005) shows that a simple ZIT model working as a continuous double auction with both market and limit orders predicts well bid-ask spreads, price diffusion rates and market impact function related to the supply and demand of 11 stocks in the London Stock Exchange. Their conclusion is that the price formation mechanism strongly constrains the market, playing a more important role than strategic behaviour. They adjust their model to real data by making simple assumptions about order placement, cancelling process of limit orders and ticks (price increments). These quantities are estimated on a daily basis from the real stocks and serve for making predictions. Nevertheless, their framework exhibits odd assumptions such as an order issuing based on log prices rather than raw prices and various Poisson laws. Other studies show that stylized facts can be reproduced using heterogeneous behaviours in the agents population. Most of these models2 suggest that the aggregation of simple interactions at the micro level leads to complex non linear behaviour at the macro level. Typically, the heterogeneity of behaviours is due to different types of rationality (informational and computational) and heterogeneity in preferences as well. The experimental economics literature also tackles some issues concerning stylized facts. Early experiments show that bubbles can easily be reproduced in double auction markets (Smith et al. 1988). Bubbles are resilient to market conditions such as short selling, margin buying opportunities, limit price-change rules, informed insider trading (King et al. 1993). The only way to generate prices that approximately reflect the intrinsic value of a dividend share is by increasing the level of subject

2 Since there is a large literature the reader is refereed to the surveys of Le Baron et al. (1999), Hommes

(2006), Lux (2009) or to the book of Levy et al. (2000).

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experience. Plott and Sunder (1982) are the first to report some stylized facts within the lab without however providing any explanation about it. They show excess Kurtosis and the lack of autocorrelation of returns in prices. Kirchler and Huber (2007) present an explanation based on asymmetric information between traders in a double auction markets. They introduce a market characterized by traders receiving heterogeneous information. They show that the latter is the source of fat tails and absolute returns whereas higher noise trading (trading not based on fundamentals) does not explain absolute returns. Bottazzi et al. (2011) design an experimental market essentially adapted from Hommes (2006). In this research, each subject makes a prediction about future expected return and volatility which determines the investment strategy of an artificial speculator through the maximization of a CARA utility function; they reproduce some of the stylized facts regarding returns, i.e. excess volatility, skewness and fat tails. They conjecture that unstable markets are more destabilizing for traders since the latter rely more on the crowd which generates unstable dynamics. To the best of our knowledge, only one paper attempts to reconcile the two streams of literature presented above, by suggesting that different elements can be at the heart of the emergence of stylized facts for different time horizons. Liu et al. (2008) show that a market with a clearing house microstructure and zero intelligent agents is responsible for reproducing leptokurtic and fat-tailed distributions, autocorrelation and excess volatility for intraday data but it is only responsible for excess volatility in daily returns. Therefore, behavioural assumptions are required to explain other facts which they show by the use of an extended minority game. Our paper lies in the strand of literature of ZIT agents and contributes to the debate concerning the level of intelligence that is necessary for generating stylized facts and realistic price motions as well. Not only are we interested in coarse grain qualitative empirical regularities, but also in the actual ability of ZITs to generate quantitatively acceptable stylized facts. For that purpose, we produce several families of ZIT agents similar to those found in the literature, but calibrated using real market data. This paper is organized as follows: Sect. 2 presents the artificial market simulator used for replicating real data, defines the ZIT agents and the simulation methodology. Section 3 shows an introductory case study, presents the core results of the paper and proposes a sensitivity analysis of the latter. The punchline that can be drawn from our results is simple: ZITs are unable to generate realistic price motions, except when highly constrained, and fail most often to generate quantitatively stylized facts. It implies that aspects linked to rationality, especially heterogeneity in behaviour, should play a central role for this line of research to become useful in financial engineering. This is the essence of the concluding Sect. 4.

2 Simulation methodology In this section, we start with a general presentation of the simulation tool (Sect. 2.1). We then move to a presentation of the different behavioural assumptions used in the experiments (Sect. 2.2).

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2.1 A brief description of ATOM In this research, we use an artificial market simulator called “ArTificial Open Market” (here-after ATOM). In this simulator, price formation directly emerges from the interaction of agents at the micro-level. The market microstructure in ATOM is duplicated from the Paris NYSE-Euronext stock exchange with the following features:3 1. It can emulate several order-books in parallel. Each order-book runs a continuous, double auction mechanism. 2. All types of NYSE-Euronext orders are allowed: limit, market, cancel or update orders, as well as sophisticated combinations such as stop-limit orders or limit orders with ‘‘iceberg” execution.4 3. The philosophy ruling the platform is such that an agent is an abstract entity that can be instantiated both by an autonomous process (developing its own strategy), or through an interface allowing humans to interact with the system. This latter possibility is not used in this research. 4. Concerning virtual agents, their possible behaviour is flexible and can be designed to fit the researchers’ requirements. At each time step, the scheduler system offers agents the possibly to decide if they will send or not a new order for each of their stocks to the corresponding order book. Particular attention has been paid to the accuracy of calculations made by the platform so as to ensure the trustworthiness of prices it delivers. Notably, no real types are used or arithmetic divisions are allowed in the code. A logging system systematically historicizes all agents’ decisions, as well as their impacts on the market. This is an essential feature to exploit the “replay-engine” ability of the simulator: if a population of agents simply re-processes real orders submitted to the market on a given day, the resulting price series is exactly the same as the one produced by the market.5 So to speak, the simulator is unbiased with regard to the microstructure it claims to be grounded on. For example, Fig. 1 illustrates such an experiment with a set of 83616 real-world orders concerning the French blue-chip France-Telecom (FTE), and submitted to the NYSE-Euronext market on June 26th 2008 between H9.02 .14 .813 and H17.24 .59 .917 . Figure 1a presents the price produced with ATOM and the real-order flow while Fig. 1b presents the corresponding exchanged volumes. In each of these figures, two sets of data are plotted. The upper set corresponds to the series generated by ATOM in processing the real-world order flow. The bottom part displays those actually observed on the NYSE-Euronext market. One clearly sees that 3 For technical details describing the simulation platform, see Mathieu and Brandouy (2010). 4 See NYSE-Euronext rule-book, at http://www.euronext.com In this paper we only use “Limit”, “Market”

and “Cancel” orders. 5 Technically speaking, the replay-engine configuration of the platform operates as follows: it first reads a log file in which orders, with their characteristics -and notably the different time stamps-, are stored. It then creates as many assets,order-books and agents as necessary to reprocess exactly this file. Once this has been done, each order is parsed from the log file and attributed to the agent issuing the order, taking into account the time stamp of the order. The platform can read most of the logs in text or FIX formats through wrappers.

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the replay-engine included in our artificial platform can process the real-world order flow in the same way it is treated by the NYSE-Euronext engine. Beyond this simple case, ATOM can use various kind of agents with their own behaviours and intelligence. Thousands of these agents can evolve simultaneously,

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creating a truly heterogeneous population. Once designed, agents evolve by themselves, learning and adapting to their (financial) environment in the most sophisticated cases. In this research, we do not use such agents but rather artificial entities using random generators to make their decisions and unable to learn or evolve: Zero Intelligence Traders. As we show below, ZITs can come in various forms, from the simplest to the most constrained ones. 2.2 Agent’s behaviour : from unconstrained to highly-constrained ZITs In this section we introduce four different types of ZITs.6 For the sake of possible replication of the results presented in this research, the pseudo-code describing each agent is available in “Appendix”. The most basic ZITs we use (called hereafter “Unconstrained ZIT”) are directly inspired by the work of Maslov (2000). From this starting point, we add, step by step, constraints to their behaviour (in accordance with real market data) with the aim of reproducing quantitatively some stylized facts. Notice that we denote by capital letters all real market data (P, V, R, for example resp prices, volumes and returns) and by small letters all simulated data. Except when a cancel order is issued, the trading activity consists in sending to the market an order made of a direction, a price limit (except when this order is a market order) and a quantity. The common characteristics for all of our ZITs, whatever their level of constraints, are as follows: – As said previously, there are three possible order types: limit, market and cancel. The proportions between these different order types are 80 % of limit orders, 15 % of market orders and 5 % of cancel orders which reflects realistic characteristics of real markets. These figures are estimated from a data set of real order flows gathering around 36,000 observations of intraday trading (courtesy of Calyon SA, here-after “CDS”, for Calyon Data Set). However, they may vary from a given stock to another and across categories of investors (Foucault 1999; Handa and Schwartz 2006). Accordingly, these proportions are modified in the sensitivity analysis (see page 241). – Each agent can submit both orders, Buy and Sell. – Buy and Sell orders arise with equal probability ( p = 0.5). – A single asset is traded. – Within each ZIT family, we define two subcategories of agents with respect to the real market average volume observed for a given stock. ‘‘Big fishes” draw volumes between the mean and the maximum real volume , while ‘‘Small fishes”, draw this value between the minimum and the mean real volume. The reason for this choice is that it may generate a realistic picture of contemporary markets, where, in a 6 What “zero intelligence” means is questionable: in our simulations, we denominate ZIT artificial agents that mostly use random number generators to determine prices and volumes in their orders. These agents are more or less sophisticated but none of them use artificial intelligence methods such as classifiers or learning mechanisms to adapt their behaviour and/or to evolve. We thus do not claim to investigate all possible declinations of ZITs but a series of ZIT instantiations that are common in the literature.

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typical experiment, the ratio of big to small fishes is 1 over 5. This proportion is also obtained from the “CDS”. – Budget constraints are implemented: agents cannot make a trade that yields a negative profit, i.e., buyers cannot buy at a price higher than their reservation value and sellers cannot sell for a price below their marginal cost. Note that all the parameters chosen for the initial settings of the simulations are calculated based on real market data. These parameters are inferred for each stock on each day. As mentioned previously, four types of ZITs are defined from these basic elements: 1. Unconstrained ZIT (U Z I T ) with two types of price-generating processes: – Uniform price distribution (U Z I TU , see Algorithm 1, page 246): pt is drawn from a Uniform distribution in [Pmin , Pmax ] where, for a reminder of the convention, Pmin and Pmax are the minimum, respectively maximum price observed on the real market. – Normal price distribution (U Z I TN , see Algorithm 2, page 246): pt follows a Normal N (Pmean , Psd ). 2. Statistically calibrated ZIT (S Z I T , see Algorithm 3, page 246) are kind of bounded U Z I TU , meaning that i) they still perform a random draw from a Uniform distribution ii) the price range is limited by [Pmin , Pmax ] and iii) the range of admissible price is different between Sellers and Buyers, since we took for Sellers a simulated range for ask prices amin and amax and for bid prices bmin and bmax . These simulated boundaries are obtained from the “CDS” as mentioned previously. To go into the details, we first separate Bids and Asks and then find for each of these subsets the minimum and the maximum values observed on a given day. Similarly, the same procedure was applied to volume data. 3. Trend calibrated agents (T Z I T , see Algorithm 4, page 247) are S Z I T with the following additional feature: when they issue a new order, they pick a price that is formed using two additional parameters γt and δt . γt is geared at reproducing the tendency of a given series. δt generates some additional randomness. More precisely, T Z I T agents respect the following procedure : (a) Divide the day into n sub-intervals (b) Find min and max prices within each sub-intervals (c) Estimate the price series according to Eqs. (1) to (4): for this, choose the number of time period t for which you want an estimation of price (by fixing the parameter θ ) and apply Eqs. (1) to (4) to estimate prices. Let’s consider the following example : on a given period, one observes for a given stock a slow decay from a maximum to a minimum price. This slow decay can be described using Eqs. (1) to (4). γ0 = 0 1 γt = γt−1 + with t ∈ [0, θ ] θ δt ∼ log N (0, 1) Pt = Pmax + (Pmin − Pmax ) × γt × δt

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Fig. 2 Example of calibration procedure, Renault SA. S Pn : Subperiod “n”

Note that in Eq. 2, γt is an increasing step function from zero to one with step size equal to 1/θ . θ allows to track the underlying trend with a certain level of accuracy. The bigger θ , the more accurate this fitting. Thus, the behaviour of these agents necessitates to specify within a given day n sub-periods and,7 for each of these sub periods, the maximum and the minimum price. Note that Eq. 4 implies that the first price set by this category of agents is close to Pmax , for this example of a slow decay in prices (Fig. 2). Based on these information, agents can track the global price tendency so as to algorithmically grasp the underlying motions. So to speak, we endow TZITs with limited foreseeing capabilities in the very short run. 4. Derived from a procedure introduced by Farmer et al. (2005), the fourth category of ZIT agents is characterized by their relative aggressiveness (AZ I T , see Algorithm 5, 26):8 – Patient agents (AZ I TP ): send limit buy orders with prices drawn from a uniform distribution between zero and amin and limit sell orders from bmax to ∞.9 These agents are parametrized in such a way that they cannot trade one with another. Trades cannot occur within this population. The quantity is determined randomly using a uniform distribution between two integers min(V ) and max(V ). – Impatient ZIT ( AZ I TI ): send market orders using the same distribution for quantities as Patient ZIT. There are 85 % of AZ I TP and 15 % of AZ I TI in a typical simulation10 and the quantity posted by AZ I TP is twice the quantity posted by AZ I TI . 7 The precision in this procedure can be tuned so as to track more or less closely the underlying dynamics. n = 50 is the value arbitrarily chosen for the experiments. 8 Note that in Farmer et al. (2005) log prices and not prices where drawn from a Uniform distribution. 9 a min and bmax are inferred as described for S Z I T . 10 These figures corresponding roughly to the proportion of market versus limit orders in real markets.

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3 Empirical design and results We introduce a case study to illustrate the main concepts and statistics used in the empirical part of this research (Sect. 3.1) and then generalize our results (Sect. 3.2). We also run a sensitivity analysis to explore how our results depend on various experimental settings (Sect. 3.3). 3.1 An introductory case study In this section, we focus on a single stock (Renault SA) for a single day (August 1, 2002) and compare simulated versus real data in terms of price motions and stylized facts. It is clear that the return series coming from Renault S.A. prices is not normal. Moreover, it exhibits fat tails and the autocorrelation in absolute returns decays slowly in log-log scale (see Fig. 3a–d). To assess the Volume-Volatility relationship we use the following, simplified framework: 1. We first slice the series in non overlapping time windows containing 300 observations. Depending upon the length of the series under investigation, the number of time windows may vary slightly. 2. We calculate for each of these slices, for each asset and/or each artificial series produced by the agents, two indicators: the mean volume over this period (mvt ) and the standard deviation of these returns (sdt ).

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be discussed, is the T Z I T example that produces a price dynamics that looks like, at coarse grain, the real one. In summary, we point out two shortcomings in the use of ZIT for financial markets simulations: 1. The qualitative adequacy of stylized facts generated by ZIT can be accepted, at least at coarse grain. However, it is clearly more questionable if one considers their quantitative values. 2. The underlying price motion remains most of the time totally unrealistic from a qualitative point of view. This remark will be emphasized later-on (see Sect. 4) but will not be specifically investigated further in the following developments. In the following section, we present a procedure enabling us to make a comparison between a set of 37 real data and different simulations from the five ZIT families described Sect. 2.2. 3.2 Beyond the case study: ‘‘zero is not enough” Our data consists in intraday prices collected from the Paris NYSE-Euronext Stock Exchange covering 37 stocks in August 2002 (22 trading days). These stocks are

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components of the CAC 40 index and are therefore amongst the most traded within the French market. For each day and each of the i = 37 stocks in the sample we produce a set of 5 simulations with the 4 different ZIT families described in Sect. 2.2.12 Notice again that ZIT agents are calibrated using real values calculated from the sample. For each simulation, we produce one concatenated return series based on the 22 simulated days. To avoid closing-to-opening jumps due to overnight information accumulation, we exclude returns that can be computed using closing prices at date t and opening prices at date t + 1. In other terms, the 22 days for each stock are summarized in j = 1 long time series. Note that the same concatenation procedure is run over the 12 2 unconstrained ZITS, statistically calibrated, Trend ZITs, Aggressive ZITs.

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real dataset. We thus have k = 6 subsets, 1 from real data, 5 from simulated ones. The dataset is therefore equivalent to a i = 37 × j = 1 × k = 6 tensor. For each of these simulations, we estimate, over the 37 observations, the distribution for the following statistics calculated on returns: 1. 2. 3. 4. 5.

Mean Standard deviation Skewness Kurtosis ρ1 and ρ2 , the first two values for autocorrelation coefficients computed using raw returns 6. The slope of the decay function for autocorrelation coefficients calculated over absolute returns. 7. The value of φˆ (see Eq. 5) indicating the direction and the strength of the volume volatility relationship. We then run two series of non-parametric tests using one simulation and real data as a benchmark to: – test equality in population distribution (two-sample KS test). For two distributions D1 and D2 , the null is that ‘‘D1 and D2 come from the same distribution.” – test equality in means (Fligner–Policello test13 and paired Wilcoxon test). For two distributions D1 and D2 , the null is that ‘‘D1 and D2 have the same mean value.” These tests are geared at appreciating whether the quantitative stylized facts are reproduced or not by means of ZITs. For illustration purposes, a limited example of the distribution over the 37 observations of each characteristic value for SF calculated on TZIT is plotted against the corresponding values calculated with real data in Fig. 7. We first report a series of non-parametric tests geared at examining whether the whole distribution of each stylized fact (summed-up with a single parameter) is similar to the distribution of the corresponding stylized facts calculated from the real world distributions. Results of the Kolmogorov–Smirnov test of equality in distribution is presented in Table 2. The results are rather unambiguous: except for the single case of the distribution of mean returns generated by Trend Calibrated ZIT (T Z I T ),14 and for the volume/volatility relationship for U Z I TU and U Z I TN , the two-sample Kolmogorov–Smirnoff test can be rejected with high levels of confidence. The interpretation is straightforward: neither higher moments nor autocorrelation-based stylized facts can be matched by any of our ZIT families. However, unconstrained ZITs, probably due to their high level of freedom, generate a realistic relationship between the average traded volume and the resulting average volatility observed within the same time window. One can also notice that when ZITs are more and more constrained, this stylized fact, if still noticeable, does no longer fit the values of our benchmark. 13 Fligner–Policello is the equivalent to the Mann–Whitney test but without assuming equality in variance. 14 For example, an illustration confirming this result for T Z I T can be found in Fig. 7a.

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0

0

500

1000

1500

2000

50000 100000 150000 200000

A re-examination of the “zero is enough” hypothesis

−2.0e−05

−1.5e−05

−1.0e−05

−5.0e−06

0.0e+00

5.0e−06

0.0005

0.0010

0.0015

0.0020

0.0025

(b)

0.0

0.00

0.01

0.5

0.02

1.0

0.03

1.5

0.04

(a)

−1

0

1

2

0

50

100

(d)

0

0

5

5

10

15

10

20

25

15

(c)

−0.30

−0.25

−0.20

−0.15

−0.10

−0.30

−0.25

−0.20

−0.15

−0.10

−0.05

(f)

0

0

200

600

1000

100000 200000 300000

1400

(e)

−0.005

−0.004

−0.003

(g)

−0.002

−0.001

0.000

0e+00

1e−05

2e−05

3e−05

(h)

Fig. 7 Stylized facts distributions for T Z I T , real (plain lines) versus simulated (dash lines). a Mean. b SD. c Skewness. d Kurtosis. e ρ1 . f ρ2 . g Slope. h Vol/Vol

If we restrict our attention on first order moments, and check equality of means for these moments between simulated and real data using a test proposed by Fligner and Policello (1981), we obtain the following results (see Table 3).

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0.9459

0.0000***

p value

0.5279

D

0.1892

p value

0.0017**

D

0.4324

p value

0.0000***

D

0.5676

p value

0.0002***

p value

D

0.4865

D

*** Sign. at 0.1 %, ** 1 %, * 5 % and “.” 10 %

AZ I T

T ZIT

SZ IT

U Z I TN

U Z I TU

Mean

Table 2 Two-sample Kolmogorov Smirnov tests

0.0201*

0.3514

0.0000***

0.6486

0.0000***

0.9459

0.0000***

0.6757

0.0000***

0.9189

SD

0.0000***

0.5676

0.0094**

0.3784

0.0402*

0.3243

0.0094**

0.3784

0.0017**

0.4324

Skewness

0.0000***

0.9189

0.0017**

0.4324

0.0001***

0.5135

0.0007***

0.4595

0.0002***

0.4865

Kurtosis

0.0000***

0.7027

0.0000***

0.8919

0.0000***

0.9189

0.0000***

0.9459

0.0000***

0.9459

ρ1

0.0000***

0.7027

0.0000***

0.9189

0.0000***

1.0000

0.0000***

1.0000

0.0000***

1.0000

ρ2

0.0402*

0.3243

0.0000***

0.7838

0.0000***

0.9730

0.0000***

1.0000

0.0000***

1.0000

Slope

0.0000***

0.6757

0.0201*

0.3514

0.0757.

0.2973

0.5279

0.1892

0.3565

0.2162

φˆ

238 O. Brandouy et al.

Mean

1.276E−7

Mean

0.0130∗

0.0012

−2.656E−6

2.4828

0.0000∗∗∗

U*

p value

−4.035E−5

155.6127

Mean

0.0010

0.1441

0.0000∗∗∗

4.3272

−16.9185

0.0007∗∗∗

0.3194

0.0000∗∗∗

p value

3.3761

7.3446

−0.9957

U*

−0.1787

0.0008

0.3648

0.9062

−7.5817

0.1034

1.6285

−0.0119

0.1221

1.5459

−0.0001

Skewness

−1.747E−6

0.0000∗∗∗

−39.8479

0.0040

0.0000∗∗∗

−8.9075

0.0022

0.0000∗∗∗

−40.3570

0.0035

SD

Mean

0.0000∗∗∗

−4.2283

U*

p value

−1.395E−6

0.0000∗∗∗

−5.9756

1.754E−7

0.0000∗∗∗

−5.6787

Mean

p value

U*

Mean

p value

U*

*** Sign. at 0.1 %, ** 1 % and * 5 %

Real

AZ I T

T ZIT

SZ IT

U Z I TN

U Z I TU

Mean

Table 3 Fligner Policello test

25.8839

−0.2137

0.0078∗∗∗

−2.6619

0.0000∗∗∗

−0.1996

−20.2753

0.0000∗∗∗

−16.1067

−0.1526

0.0000∗∗∗

12.4847

−0.2809

0.0000∗∗∗

25.4974

−0.3001

0.0000∗∗∗

25.6899

−0.3015

ρ1

2739.8962

0.0001∗∗∗

−3.8624

40.6920

0.0002∗∗∗

3.6821

2221.9776

0.0026∗∗∗

3.0100

19.6185

0.0003∗∗∗

3.6420

17.7071

Kurtosis

−0.2137

0.0078∗∗∗

−2.6619

−0.1997

0.0000∗∗∗

−32.4546

−0.1349

0.0000∗∗∗

−∞

−0.0800

0.0000∗∗∗

−∞

−0.0746

0.0000∗∗∗

−∞

−0.0781

ρ2

−0.0018

0.0150∗

−2.4313

−0.0017

0.0000∗∗∗

8.1609

−0.0030

0.0000∗∗∗

17.5064

−0.0033

0.0000∗∗∗

∞

−0.0035

0.0000∗∗∗

∞

−0.0035

Slope

2.816E−6

0.0000∗∗∗

−8.8679

6.623E−7

0.0184∗

2.3564

2.781E−6

0.9960

0.0049

5.725E−6

0.3385

−0.9570

1.912E−6

0.7127

−0.3682

4.288E−6

φˆ

A re-examination of the “zero is enough” hypothesis 239

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Mean

p value

4.0000

0.0121∗

0.0012

0.0000∗∗∗

516.0000

0.0010

0.0000∗∗∗

677.0000

0.0008

0.0000∗∗∗

−2.656E−6

−4.035E−5

703.0000

Mean

W

0.0023∗∗

154.0000

W

p value

−1.747E−6

Mean

0.0002∗∗∗

115.0000

p value

W

0.0040

0.0000∗∗∗

0.0000∗∗∗

−1.395E−6

p value

Mean

44.0000

0.0022

76.0000

1.754E−7

W

Mean

0.0000∗∗∗

0.0000∗∗∗

p value

0.0035

5.0000

1.276E−7

69.0000

W

Mean

SD

*** Sign. at 0.1 %, ** 1 % * 5 % and “.” 10 %

Real

AZ I T

T ZIT

SZ IT

U Z I TN

U Z I TU

Mean

Table 4 Paired Wilcoxon tests

0.1441

0.0000∗∗∗

616.0000

−16.9185

0.0056∗∗

532.0000

−0.1787

0.1448

449.0000

−7.5817

0.0800.

468.0000

−0.0119

0.1105

458.0000

−0.0001

Skewness

25.8839

0.0000∗∗∗

1.0000

2739.8962

0.0001∗∗∗

98.0000

40.6920

0.0003∗∗∗

584.0000

2221.9776

0.0046∗∗

536.0000

19.6185

0.0004∗∗∗

576.0000

17.7071

Kurtosis

−0.2137

0.0101∗

183.0000

−0.1996

0.0000∗∗∗

12.0000

−0.1526

0.0000∗∗∗

665.0000

−0.2809

0.0000∗∗∗

702.0000

−0.3001

0.0000∗∗∗

702.0000

−0.3015

ρ1

−0.2137

0.0106∗

184.0000

−0.1997

0.0000∗∗∗

0.0000

−0.1349

0.0000∗∗∗

0.0000

−0.0800

0.0000∗∗∗

0.0000

−0.0746

0.0000∗∗∗

0.0000

−0.0781

ρ2

−0.0018

0.0913.

239.0000

−0.0017

0.0000∗∗∗

679.0000

−0.0030

0.0000∗∗∗

702.0000

−0.0033

0.0000∗∗∗

703.0000

−0.0035

0.0000∗∗∗

703.0000

−0.0035

Slope

4.288E−6

2.816E−6

0.0000∗∗∗

1191.0000

6.623E−7

0.0219∗

473.0000

2.781E−6

1

684.0000

5.726E−6

0.338

774.0000

1.913E−6

0.7068

720.000

φˆ

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241

0.0

0.5

1.0

1.5

2.0

2.5

A re-examination of the “zero is enough” hypothesis

−1

0

1

2

Fig. 8 Distribution of Skewness for U Z I TU —dash line versus real world series—solid line

Here again, the tests reject the ability of our ZIT families at reproducing quantitative stylized facts. The only cases where these tests cannot be rejected are those of U Z I TU , U Z I TN , S Z I T for the skewness distribution and for the value of the volume/volatility relationship, and T Z I T for the mean. Said differently, the first three categories of agents might do a relative good job at generating realistic third-order moments for the return distribution and, as mentioned previously, in delivering a realistic volume/volatility interplay. However, in these tables only the mean of each parameter distribution is tested against its real-world counterpart. If the test cannot be rejected the only thing we can conclude is that the central values of the distribution are not so different. In our opinion, this is a necessary but not sufficient condition to accept a family of Agents. For example, if one considers the distribution of the Skewness for U Z I TU against the real series, the equality of the means cannot be rejected (see Tables 3, 4) although it is clear that the whole distribution is different (see Fig. 8). Rejection for T Z I T may be more surprising since they were constrained to generate more realistic price motions and even with that, they were not able to reproduce first orders moments beyond the mean. Even if skewness is an important feature of financial distributions (notably important for asset managers), given the overall negative conclusions drawn on other moments and correlations of the distributions, this is a rather weak result. If we go further in the analysis with a paired Wilcoxon rank test, T Z I T is now rejected while none of the skewness-related tests are rejected at the 5 % threshold.

3.3 Sensitivity analysis: the importance of model parameters A variety of factors may have an effect on the results presented above. This section is dedicated to a discussion on two factors that may affect notably our results: i) the impact of the proportion of “Big fishes” versus “Small fishes” and ii) the role of the ratio “Limit” to “Market” orders. To address these points, we first vary only one of

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these parameters while the other stay constant and report the impact of this variation on the market. Note that T Z I T are not significantly influenced by the variation in the proportions of the two factors, but rather by the price trend itself. Amongst the two remaining families, S Z I T is the more driven by the dynamics of order flows, and thus is the best potential candidate for an efficient calibration through both factors. We have chosen to restrict the presentation of the sensitivity analysis to the latter and to the AZ I T inspired by the Farmer model (Farmer et al. 2005). For the latter, we vary the proportion of “patient” versus “impatient” agents in the population. In Table 5, we report the correlations between the model parameters and the statistical properties tracked throughout this paper. i) Big fishes/Small fishes The results in Table 5 clearly show that except for the mean, which is insensitive to the modification of the model parameters, all other stylized facts do react in some way to the latter. Even if the correlation coefficients may be relatively small (ranging from −0.5781 for the Kurtosis to 0.5876 for the “slope”), all of them are significant at the 1 % level. However, if volatility tends to increase with

Table 5 Linear correlation coefficient of model parameters and stylized facts Statistics

Mean SD Skewness

Big/small fishes proportiona

ρ1 ρ2 Slope φˆ

Impatient agent

Patient agent

Corr.

0.3216

0.1408

−0.1551

0.1848

p value

0.1250

0.2414

0.2493

0.1942

Corr.

0.5744

−0.9759

0.2259

−0.6677

p value

0.000∗∗∗

Corr. p value

Kurtosis

Limit/market proportionb

Corr.

−0.05648 0.000∗∗∗ −0.5781

0.000∗∗∗

0.0911∗

0.1688

0.0990

0.1593

0.4637

−0.1872

−0.5669

p value

0.000∗∗∗

Corr.

0.4663

p value

0.000∗∗∗

0.000∗∗∗

Corr.

0.1470

0.0580

pvalue

0.000∗∗∗

0.000∗∗∗

Corr.

0.5876

p value

0.000∗∗∗

Corr. p value

−0.3429 0.000∗∗∗

0.1179 −0.7495

−0.5572 0.000∗∗∗

0.000∗∗∗ 0.6469 0.000∗∗∗ −0.7164 0.000∗∗∗ 0.0339 0.8021

0.0571

0.0084

0.6358

0.9502

0.000∗∗∗ −0.0597 0.000∗∗∗ 0.2283 0.1070 −0.7771 0.000∗∗∗ 0.6859 0.000∗∗∗ −0.2688 0.0564∗ −0.3503 0.0117∗∗

100 Replications for each parameter set have been conducted. Each value is an averaged result *** Sign. at 1 %, ** 5 %, * 10 % a Experiment settings: basic design of S Z I T . The number of Small Fishes is fixed as 200, while the number of Big Fishes varies from 0 to 200 b Experiment settings: basic design of S Z I T . The proportion of Cancel orders is fixed,  = 5. The C proportion of Limit orders  L varies from 60 to 95 %, and proportion of Market orders is defined according to formula  M = 100 −  L − C

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A re-examination of the “zero is enough” hypothesis

243

this ratio (the correlation coefficient being equal to 0.5744), kurtosis tends to decrease (coefficient = −0.5781 ). Big fishes, due to the large trading volume they can generate, provide a large amount of liquidity that directly benefits to Small fishes. In other words, Big fishes “feed” Small fishes. Thus, Small fishes can easily buy or sell stocks with a price close to the current market price, until a big order is completely executed. When the number of Big fishes increases significantly, these big players loose their role of liquidity providers, because they trade more frequently within their own group. Increasing the proportion of Big fishes has also a positive effect on the slope of the decay function for autocorrelation coefficients calculated over absolute returns (correlation coefficient equals 0.5876). This result may suggest that the more “Big fishes” in the market, the more likely the emergence of volatility clusters. In summary the fact that we observe significant correlations in that case indicates that finding the appropriate proportion of Big fishes versus Small fishes might be a route to quantitatively fitting stylized facts observed on real markets by using ZITs. However, we were unable in our simulations to find the “ideal mix” which would have led to the perfect emergence of the whole set of studied stylized facts. In fact it seems a rather impossible task to fit most of the stylized fact by varying only one parameter. Moreover, the point is not to find the proportion which would fit these stylized facts but rather to show that within a reasonable range corresponding to what is commonly observed on the market, this is not the case. ii) Limit / Market orders From the same Table 5, one can notice that the proportion of “limit” to “market” orders has a significant effect on most of the studied stylized facts except the Skewness, the Kurtosis and the volume/volatility relationship. Concerning the standard deviation (SD), the correlation coefficient is close to −1. This result was expected: on the one hand, traders supply liquidity by posting limit orders and, on the other hand, demand liquidity by submitting market orders that yield immediate partial or full execution. Thus, a large proportion of limit orders provides an important liquidity on both sides of the order book (Bids and Asks). On the contrary, a market order is immediately executed against order(s) standing in the limit order book: it moves the market by walking up or down the limit order book. Clearly, the proportion of Limit/Market orders has a significant impact on market volatility. A higher proportion of limit orders stabilizes the market by decreasing the standard deviation (correlation coefficient = −0.9759). The negative coefficient for ρ1 (−0.7495) suggests that the more Limit orders, the lower the auto-correlation of raw returns: this is a well-known fact related to the BidAsk bounce. To summarize, some important stylized facts (Skewness, Kurtosis and volume/volatility relationship) seem to be insensitive to the modifications of the ratio Limit to Market orders. This indicates that fitting with accuracy real-world stylized facts is probably out of reach using this ratio alone. iii) Patient/Impatient agents One can observe that increasing the number of these two categories of agents has a similar impact on various stylized facts as the variation of the proportion between Limit and Market orders does (see Table 5). This result was expected since “patient” agents provide liquidity while “impatient” agents demand liquidity.

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Table 6 Regression results for effects of model parameters on stylized facts Model

Statistics

1

Mean

2 3 4 5 6

SD Skewness Kurtosis ρ1 ρ2

7

Slope

8

φˆ

α

β1

β2

R2 0.301

1.859E−4

1.573E−8

−2.137E−6