Novel Insights in the Levy-Levy-Solomon Agent-Based Economic Market Model
NNovel Insights in the Levy-Levy-Solomon Agent-BasedEconomic Market Model
Maximilian Beikirch ∗† , Torsten Trimborn ‡†§ February 25, 2020
Abstract
The Levy-Levy-Solomon model [14] is one of the most influential agent-based economicmarket models. In several publications this model has been discussed and analyzed. Es-pecially Lux and Zschischang [23] have shown that the model exhibits finite-size effects.In this study we extend existing work in several directions. First, we show simulationswhich reveal finite-size effects of the model. Secondly, we shed light on the origin of thesefinite-size effects. Furthermore, we demonstrate the sensitivity of the Levy-Levy-Solomonmodel with respect to random numbers. Especially, we can conclude that a low-qualitypseudo random number generator has a huge impact on the simulation results. Finally,we study the impact of the stopping criteria in the market clearance mechanism of theLevy-Levy-Solomon model.
Keywords:
Levy-Levy-Solomon, agent-based models, Monte Carlo simulations, finite-size effects, random number generator, econophysics ∗ RWTH Aachen University, Templergraben 55, 52056 Aachen, Germany † ORCiD IDs: Maximilian Beikirch: 0000-0001-6055-4089,Torsten Trimborn: 0000-0001-5134-7643 ‡ IGPM, RWTH Aachen, Templergraben 55, 52056 Aachen, Germany § Corresponding author: [email protected] a r X i v : . [ q -f i n . T R ] F e b Introduction
The Levy-Levy-Solomon model has been introduced by the same-named authors in 1994 [14].This model can be regarded as an early contribution in the field of econophysics. In severalsubsequent publications the authors Levy, Levy and Solomon have extesively studied theirmodel [13, 15, 16]. In addition, the model has been investigated by other scientists [11, 23].As emphasized by several authors, the Levy-Levy-Salomon model is regarded as one of themost influential agent-based economic market models [20].The model considers heterogeneous financial agents equipped with a personal wealth and theirinvestment decision. The agents have to decide whether they want to invest their money in asafe bond or in risky stocks. The stock price is determined by the market clearance conditionwhich ensures that supply matches demand. The dynamics is driven by a stochastic dividendprocess. By a utility maximization process, the investment propensity of agents is determined.The model behavior is studied by means of Monte Carlo simulations which is a classicaltool from statistical physics. The original goal of the Levy-Levy-Solomon model was to builda simple model able to reproduce realistic stock price data. In order to judge the qualityof artificial stock price data it is common practice to compare statistical quantities of finan-cial data (so-called stylized facts ), to the artificial data. Examples are the wealth inequalityobtained by Pareto [19] or volatility clustering discovered by Mandelbrot [18]. For an intro-duction to stylized facts we refer to [4, 5, 17]. The common goal of agent based economicmarket models, and to some extend also the Levy-Levy-Solomon model, is to find sufficientconditions for the appearance of stylized facts. In fact, Levy, Levy and Solomon first focusedon the reproducibility of chaotic prices and booms and crashes [14, 15]. In a subsequentpublication [13], Levy, Levy and Solomon claimed that their model was able to reproducestylized facts such as volatility clustering and fat-tails in asset returns.Further studies [11, 23] have raised doubt on the claimed ability of the Levy-Levy-Solomonmodel to reproduce stylized facts. More precisely it has been shown that the Levy-Levy-Solomon model exhibits finite-size effects, meaning that the output (and hence the abilityto reproduce stylized facts) heavily depends on the number of agents in the model. This isan undesireable model characteristic and has been discussed in the context of agent basedfinancial market models in [6, 8].In this work, we study the reasons for the appearance of finite size effects in the Levy-Levy-Solomon model and we conduct additional novel studies. It has been documented in [2]that the model behavior of the Levy-Levy-Solomon model is very sensitive to the noise levelof the dividend process. We show that a low quality random number generator leads to dif-ferent qualitative simulation results. Furthermore, we study the impact of different stoppingcriteria of the market clearance mechanism with respect to the investment decision of agentsand several stylized facts. All simulations have been conducted with the recently introducedSABCEMM tool [22].This paper is structured as follows: In the next section we give a detailed definition ofthe Levy-Levy-Solomon model. In section 3, we give a short introduction to the SABCEMMsimulator. In the subsequent section, we present our simulation results. First, we present2nite-size effects of the model and discuss their origin. Secondly, we show the impact of anon-reliable (pseudo) random number generator on the qualitative output of the model. Fi-nally, we discuss the impact of different stopping criteria in the clearance mechanism of themodel. We finish this work with a short conclusion.
In this section, we briefly define the Levy-Levy-Solomon (LLS) model. For detailed informa-tion regarding the modeling and motivation we refer to the original papers [14, 15]. The timestep ∆ t > t k = k · , k ∈ N .The model considers N ∈ N financial agents who can invest γ i ∈ [0 . , . , i = 1 , ..., N of their wealth w i ∈ R > in stocks and have to invest 1 − γ i of their wealth in a safe bondwith interest rate r ∈ (0 , γ i are determined by a utilitymaximization and the wealth dynamic of each agent at time t ∈ [0 , ∞ ) is given by w i ( t k ) = w i ( t k − ) + (1 − γ i ( t k − )) r w i ( t k − ) + γ i ( t k − ) w i ( t k − ) S ( t k ) − S ( t k − ) + D ( t k ) S ( t k − ) (cid:124) (cid:123)(cid:122) (cid:125) =: x ( S,t k ,D ) . The dynamics is driven by a multiplicative dividend process given by: D ( t k ) := (1 + ˜ z ) D ( t k − ) , where ˜ z is a uniformly distributed random variable with support [ z , z ]. The price is fixedby the so-called market clearance condition , where n ∈ N is the fixed number of stocks and n i ( t ) the number of stocks of each agent. n = N (cid:88) i =1 n i ( t k ) = N (cid:88) k =1 γ k ( t k ) w k ( t k ) S ( t k ) . (1)The utility maximization is given bymax γ i ∈ [0 . , . E [log( w ( t k +1 , γ i , S h ))] . where E [log( w ( t k +1 , γ i , S h ))] = 1 m i m i (cid:88) j =1 U i (cid:32) (1 − γ i ( t k )) w i ( t, S h ) (1 + r )+ γ i ( t k ) w i ( t k , S h ) (cid:16) x (cid:0) S, t k − j , D (cid:1)(cid:17)(cid:33) . The constant m i denotes the number of time steps each agent looks back. Thus, the numberof time steps m i and the length of the time step ∆ t defines the time period each agentextrapolates the past values. The superscript h indicates that the stock price is uncertainand needs to be fixed by the market clearance condition. In practice the agents derive theirinvestment proportions γ i ( t k ) according to the utility maximization which depends on the3ypothetical stock price S h . Provided the market clearance is satisfied the stock price getsfixed, otherwise the hypothetical stock price gets updated and the procedure repeats. Finally,the computed optimal investment proportion gets blurred by a noise term. γ i ( t k ) = H ( γ ∗ i ( t k ) + (cid:15) i ) , where (cid:15) i is a Gaussian distributed random variable with mean zero and standard deviation σ γ . Here, H denotes the cutoff function which ensures that γ i ∈ [0 . , .
99] holds. After thenoising process, the price is updated. Since the investment fraction is constant we are ableto compute the stock price explicitly: S ( t k ) = n N (cid:80) i =1 γ i ( t k ) (cid:16) w i ( t k − ) + w i ( t k − ) (cid:0) γ i ( t k − ) D ( t k − ) − S ( t k − ) S ( t k − ) + (1 − γ i ( t k − )) r (cid:1)(cid:17) − n N (cid:80) i =1 γ i ( t k ) γ i ( t k − ) w i ( t k − ) S ( t k − ) . Utility maximization
Thanks to the simple log utility function and linear dynamics wecan compute the optimal investment proportion in the cases where the maximum is reachedat the boundaries. In these cases, the solution is found after two evaluations of f , i.e. inconstant time. The first order necessary condition is given by: f ( γ i ) := ddt E [log( w ( t k +1 , γ i , S h ))] = 1 m i m i (cid:88) j =1 ( x (cid:0) S, t k − j , D (cid:1) − r )( x (cid:0) S, t k − j , D (cid:1) − r ) γ i + 1 + r . Thus, for f (0 . < γ i = 0 .
01 holds. In the same manner, we get γ i = 0 .
99, if f (0 . > f (0 . > . , . f (0 . > f (0 . <
0. This coincides with theobservations in [20]. 4
The SABCEMM Simulator
The simulation results in this study have been created by the recently introduced open sourcesimulator SABCEMM [22]. SABCEMM is especially designed for large-scale simulations ofagent based computational economic market (ABCEM) models, which is essential for thestudy of finite-size effects. This simulator implements an object oriented design leveraginga generalized structure of ABCEM models as defined in [22]. The implementations of theindividual ABCEM building blocks are well-separated and the ABCEM model is assembledfrom the building blocks via an XML-based configuration file. Hence, the evaluation of anABCEM model using a different building block, such as the market mechanism, requires onlya change in the configuration file. If the changed building block does not already exist, onlythis single block has to be implemented. In the following, we present the main conceptualideas behind the simulator.SABCEMM is well suited for any economic market model which consists of at least one agent and one market mechanism . An agent is an investor who has a supply of or demand fora certain good or asset, which is traded at the market. The market mechanism determines theprice from the demand and supply of all market participants. More precisely, we differentiatebetween the so-called price adjustment process and the excess demand calculator . The latterone aggregates the supply and demand of all market participants (agents) to a single quantity,the excess demand. The former one represents the method of how the market price is fixedbased on this excess demand. A schematic picture, which illustrates the presented ideas isshown in Figure 1. The concept of an environment has been first introduced by the authors in
EnvironmentMarket MechanismAgent I Agent IIExcess DemandClearance Mechanism
Figure 1: Schematic picture of the abstract ABCEMM model [22].[22]. An environment represents possible additional coupling between the agents. Probably,the most famous example for an environment is herding, which is frequently used in ABCEMmodels. We emphasize that such an environment is not mandatory. For a rigorous math-ematical definition of the meta-model which is the foundation of the SABCEMM simulator5nd a detailed discussion of technical details and computational aspects of SABCEMM, werefer to [22]. We finish the presentation of the SABCEMM simulator by an short introductionof the supported pseudo random number generators.In SABCEMM, we provide two pseudo random number generators (RNG) based on thewidely popular Mersenne Twister family of RNGs. First, we include the
MT19937 (64-bit)Mersenne Twister, introduced into the C++ standard library with the C++11 standard.
MT19937 provides an extremely long period of 2 − ≈ . and particularly goodequidistribution (623 dimensions of approximate equidistribution [1]). Secondly, we supportthe MT2203 (64-bit) multi-stream Mersenne twister implemented within the Intel Math KernelLibrary (MKL) providing a period of 2 − ≈ . , very good equidistribution (68 di-mensions of approximate equidistribution [1]) and 6024 parallel streams. Note that MT2203 isincluded as it outperforms the
MT19937 implementation within the C++11 standard libraryby far when combined with pooling of generated random numbers. For further details onpooling of random numbers within SABCEMM, the interested reader is refered to [22]. Since
MT19937 and
MT2203 fail only on two and four tests of the BigCrush battery of statisticaltests [12], we accept these highly popular RNGs as high-quality RNGs. In comparison, insection 4.2, we show that the low-quality RNG
RANDU [9], failing already 14 tests within theSmallCrush and 125 tests within the Crush battieries [12], is of insufficient quality for appli-cation within ABCEM models. RNGs within SABCEMM are seeded from the /dev/urandom device on Linux/Unix systems. If the device is not available, the current time stamp is used. https://software.intel.com/en-us/mkl (may require extra license) std::chrono::high resolution clock::now() Simulation Results
In this section, we perform several numerical tests on the LLS Model. First, we show finite-size effects in the LLS model and study their origins. Furthermore, we present an examplewhich reveals the sensitivity of the LLS model with respect to (pseudo) random numbers andespecially the impact of a low quality pseudo random number generator. Finally, we showthat the termination condition in the root finding algorithm of the LLS model has a hugeimpact on the model output.All presented results have been generated by the SABCEMM simulator which is freely avail-able on GitHub [21]. For the pseudo random number generators used in each simulation,please consult Table 3. Furthermore, the simulation data is published [3] such that the readercan reproduce the presented results.
Finite-size effects in ABCEM models have been documented by several authors [6, 8], inparticular for the LLS model [11, 23]. First, we demonstrate finite-size effects and secondlydiscover the reasons for that behavior.In our simulations, we observe two different effects caused by varying the number of agents.First, we obtain that the tail behavior of the wealth distribution changes for an increasingnumber of agents. Secondly, we find that dependent on the number of agents, different agenttypes (differing w.r.t. their memory mechanism) dominate the market. A population isconsidered dominant if it attains maximum wealth. Levy at al. [16] claimed that an investorgroup with higher memory span always dominates a group with a smaller memory. Wehave conducted simulations with 99 agents (33 short memory, 33 medium memory, 33 largememory, for details we refer to appendix A.1) and with 999 (333 short memory, 333 mediummemory, 333 long memory, details in appendix A.1) agents, respectively. The aggregatedwealth evolution of the three agent groups (Figure 2) shows that the group ranking changesfor different number of agents. Furthermore, the quantile-quantile plots in Figure 2 clearlyindicate the change in tail behavior for different number of agents. Thus, we can concludethat the qualitative output of the model changes with respect to the number of agents. Wehighlight that this is an utmost undesirable model characteristic.We aim to understand why changes of agent count lead to changes in the model behavior.Before discussing the appearance of finite-size effects, we need to understand the origins of the usual oscillatory model behavior. Studies by Levy, Levy and Solomon [15], Zschischang andLux [23], and Otte et al. [2] emphasize the importance of white noise on the characteristicoscillatory model behavior. In fact, in the zero noise case the oscillatory model characteristiccompletely vanishes. The reason is that the stock return is always better than the bond return(in the original parameter setting) and thus the investors allocate their money in stocks. Thisleads to an exponential increase in the price and to constant investment propensities. In thecase of noise on the investment fraction it is possible that the return on stocks is worse thanthe return on bonds which may lead to changes in the investment propensity. These rapidchanges in the investment propensity lead to oscillatory price behavior. As shown in [2], thewave period of the wealth evolution heavily depends on the noise level. Clearly, in the case ofno noise, we have no oscillatory behavior since the price is in equilibrium, which is computedby the clearance mechanism: 7 = N (cid:88) i =1 n i ( t ) = N (cid:88) i =1 w i ( t ) S ( t ) γ i ( t ) . In the noisy case one adds to every optimal γ ∗ i a noise term: γ i = H ( γ ∗ i + (cid:15) i ) , where (cid:15) is aGaussian random variable. Then finally one can update the final stock price by an explicitcomputation since the investment fraction γ i is constant. For details we refer to section 2. Infact the stock price is computed as a quantity proportional to S ( t ) ∝ N N (cid:88) i =1 H ( γ ∗ i ( t ) + (cid:15) i ) , since the number of stocks n scales with the number of agents. In order to quantify theinfluence of the noise we define the difference of investment fractions before and after noiseapplication: d Nγ ( t ) := (cid:12)(cid:12)(cid:12) N N (cid:88) i =1 H ( γ ∗ i ( t ) + (cid:15) i ) − N N (cid:88) i =1 γ ∗ i ( t ) (cid:12)(cid:12)(cid:12) . Note that the difference is not additive due to the additional cutoff function H . Figure 3depicts average difference of investment fractions d Nγ and the stock price for different number ofagents. The simulation results clearly indicate that the variance of d Nγ decreases for increasingnumber of agents. Especially, we are able to deduce that large deviations of the mean of the d Nγ are needed in order to obtain the typical oscillatory behavior. Hence, we can concludethat an increasing number of agents reduces the variance of fluctuations caused by additionalnoising in the LLS model and heavily influences the stock price behavior, resulting in fewermarket crashes. Therefore we claim that this scaling behavior with respect to the number ofagents causes the finite-size effects in the LLS model.8 S a m p l e Q u a n t il e s Raw Return Quantiles vs. Standard Normal Quantiles (a) QQ-plot of returns with N = 99. Heavy tailsare visible. S a m p l e Q u a n t il e s Raw Return Quantiles vs. Standard Normal Quantiles (b) QQ-plot of returns with N = 999 agents.Heavy tails are visible but changed their shape(cmp. fig. 2a). W e a l t h Wealth m=10m=141m=256 (c) Aggregated wealth of each agent group with N = 33 agents. W e a l t h Wealth m=10m=141m=256 (d) Aggregated wealth of each agent group with N = 333 agents. The group ranking has changed. Figure 2: Simulation of the LLS model with 99 agents (left) and 999 agents (right). Theagents are divided in three groups with different memory sizes. Figure 2a and 2b reveala change in the tail behavior, while fig. 2a and 2b shows a change in the group ranking.Parameters as in table 2. For colored plots, please refer to the online version.9
25 50 75 100 125 150 175 200Time Step0.120.130.140.150.160.170.180.190.20 a v e r a g e g a mm a d e v i a t i o n average gamma deviation P r i c e Prices (Log-Scale) (a) 200 agents a v e r a g e g a mm a d e v i a t i o n average gamma deviation P r i c e Prices (Log-Scale) (b) 500 agents a v e r a g e g a mm a d e v i a t i o n average gamma deviation P r i c e Prices (Log-Scale) (c) 1000 agents
Figure 3: Prices and gamma differences d Nγ for different agent counts.10
50 100 150 200 time steps l ogp r i ce (a) C++ MT19937 RNG (64 bit) time steps l ogp r i ce time steps (b) RANDU generator (32 bit)
Figure 4: Simulations of the LLS model conducted with different pseudo random numbergenerators. Parameters as in Table 1 with σ γ = 0 . As discussed in [22], ABCEM models require the generation of huge amounts (commonly inthe order of several million) of high-quality pseudo random numbers. Furthermore, it is welldocumented that many ABCEM models are sensitive to the precise noise level [2], i.e. thevariance of the pseudo random numbers used. Thus, the question whether low-quality pseudorandom numbers influence the qualitative results of ABCEM models is legit. We answer thisquestion examplarily by applying the well-known linear congruential pseudo random numbergenerator
RANDU , [7, 10] to the LLS model.
RANDU is known for poor performance whenused for the generation of many pseudo random numbers. We compare an implementationof
RANDU on a ARMv7 32 bit processor to the C++11 standard pseudo random numbergenerator on a x86 64 64 bit processor. As shown in Figure 4, changing the pseudo randomnumber generator drastically changes the qualitative model output. Thus, we may concludethat models sensitive to the choice of random variables, generate different model outputswith respect to different pseudo random number generators. Therefore we emphasize that weexplicitly do not recommend using the
RANDU generator. Due to this, the published versionof SABCEMM does not support the
RANDU generator as we have only used this generator forthis special test. 11 .3 Stopping Criteria in Clearance Mechanism
In this section, we study the impact of the stopping criterion applied in the root findingalgorithm of the LLS model. This is employed in the clearance mechanism which fixes thestock price at each time step. The clearance mechanism of the LLS model n = N (cid:88) k =1 γ k ( t ) w k ( t ) S ( t )clearly models an equilibrium market where supply equals demand. In order to computethe equilibrium price at each time step, the clearance condition is solved numerically usinga root finding algorithm. The crucial parameter is the chosen stopping criterion, defined as (cid:12)(cid:12)(cid:12)(cid:12) N (cid:80) k =1 γ k ( t ) w k ( t ) S k − n (cid:12)(cid:12)(cid:12)(cid:12) ≤ ξ, ξ >
0. Thus, the algorithm terminates if the clearance mechanism forthe stock price S k is satisfied with respect to precision ξ . Note that the original publications[14, 15] do not report values or references for the tolerance used.The choice of stopping criterion is also important from an economical view on the model. Forrelatively small values of ξ , the market price clearly is forced close to equilibrium resulting inan approximately rational market. For larger ξ , the market may become irrational at times.Note that the market is not inherently irrational for larger tolerances - most of the time, weobserve low excess demand despite high tolerances. It is only around crashes that we findhigh excess demands that actually exhausts the tolerance. From a computational perspective,it is desirable to increase ξ as much as possible to lower computational cost. Figures 5, 6,7 (200 agents) demonstrate that the choice of stopping criterion ξ may strongly impacts themodel behavior.In Figure 5, we present results with tolerance levels ξ = 0 .
05 and ξ = 0 .
1. There areonly subtle differences in return distribution and price trend. In Figure 6, simulation resultsfor tolerances ξ = 0 . ξ = 0 . ξ = 0 .
75, and ξ = 1 are compared with respect to averageinvestment proportions. There are moderate differences in the plots for ξ ≤ .
75. Notably,these results feature peaks at the same times and the trends look generally alike. For ξ = 1however, average investment proportion is overall very low and it never exceeds 0.1. Hence, ξ = 1 is found to represent too much of a relaxation of the rational market assumption. Figure7 presents the autocorrelation of logarithmic stock returns and absolute logarithmic stockreturns for multiple tolerance levels. Notably, the highest autocorrelation can be observedfor ξ = 0 . ξ = 0 .
5, and ξ = 0 .
75. From qq-plots of logarithmic stock returns (see Figure8) one can conclude that increased tolerances result in heavier tails. From an economic pointof view, a non-negligible tolerance represents a relaxation of the rational market hypothesisand therefore results in an irrational market. This raises the question whether increasingtolerance, and by this more irrational markets, generally result in heavier tails.12
100 200 300 400 500 600 700 800Time Step10 P r i c e Prices (Log-Scale) S a m p l e Q u a n t il e s Raw Return Quantiles vs. Standard Normal Quantiles (a) ξ = 0.1 P r i c e Prices (Log-Scale) S a m p l e Q u a n t il e s Raw Return Quantiles vs. Standard Normal Quantiles (b) ξ = 0.05 Figure 5: Price trends and return distribution for low tolerances.13 a v e r a g e g a mm a average gamma (a) ξ = 0.1 a v e r a g e g a mm a average gamma (b) ξ = 0.5 a v e r a g e g a mm a average gamma (c) ξ = 0.75 a v e r a g e g a mm a average gamma (d) ξ = 1 Figure 6: Average investment proportion for various tolerances.14
100 200 300 400 500Lag l1.000.750.500.250.000.250.500.751.00 A u t o c o rr e l a t i o n Autocorrelation of Returns
ReturnsAbs. Returns (a) ξ = 0.05 A u t o c o rr e l a t i o n Autocorrelation of Returns
ReturnsAbs. Returns (b) ξ = 0.1 A u t o c o rr e l a t i o n Autocorrelation of Returns
ReturnsAbs. Returns (c) ξ = 0.25 A u t o c o rr e l a t i o n Autocorrelation of Returns
ReturnsAbs. Returns (d) ξ = 0.5 A u t o c o rr e l a t i o n Autocorrelation of Returns
ReturnsAbs. Returns (e) ξ = 0.75 A u t o c o rr e l a t i o n Autocorrelation of Returns
ReturnsAbs. Returns (f) ξ = 1 Figure 7: Autocorrelation for various tolerances. For colored plots, please refer to onlineversion. 15 S a m p l e Q u a n t il e s Raw Return Quantiles vs. Standard Normal Quantiles (a) ξ = 0.1 S a m p l e Q u a n t il e s Raw Return Quantiles vs. Standard Normal Quantiles (b) ξ = 0.25 S a m p l e Q u a n t il e s Raw Return Quantiles vs. Standard Normal Quantiles (c) ξ = 0.5 S a m p l e Q u a n t il e s Raw Return Quantiles vs. Standard Normal Quantiles (d) ξ = 1 Figure 8: Qantile-quantile plot of returns for various tolerances.16
Conclusion
We have introduced the Levy-Levy-Solomon model and have conducted several numericaltests. As documented previously in [11, 23], we have shown that the Levy-Levy-Solomonmodel exhibits finite-size effects. This work indicates that finite-size effects are caused by awrong scaling which leads to a variance reduction of the investment decision for large numberof agents. In addition, we have verified that a high quality pseudo random number generatoris essential to guarantee correct simulation results. This is of major importance in ABCEMmodels since many pseudo random numbers are needed and they are sensitive with respectto different noise levels as e.g. the Levy-Levy-Solomon model. Furthermore we studied theimpact of different stopping criteria of the clearance mechanism on the model behavior. Wehave seen that the auto-correlation function of absolute returns and the heaviness of thefat-tails are influenced by the stopping criteria. Especially it seems that a higher toleranceleads to heavier tails, which maybe translated economically as follows: The more irrationalthe market is, the heavier are the tails of the stock return distribution of the Levy-Levy-Solomon model. This observation is in agreement with economic hypotheses on the influenceof irrational markets on stock price behavior. Clearly this finding deserves a more detailedstudy which is left open for future research. 17 cknowledgement
T. Trimborn was funded by the Deutsche Forschungsgemeinschaft (DFG, German ResearchFoundation) under Germany’s Excellence Strategy – EXC-2023 Internet of Production –390621612. T. Trimborn gratefully acknowledges support by the Hans-B¨ockler-Stiftung andthe RWTH Aachen University Start-Up grant. T. Trimborn acknowledges the support bythe ERS Prep Fund - Simulation and Data Science. The work was partially funded by theExcellence Initiative of the German federal and state governments.18arameter Value N m i σ γ . r . z = z . (a) Parameters of LLS model. Variable Initial Value µ h . σ h . γ ( t = 0) 0 . w i ( t = 0) 1000 n i ( t = 0) 100 S ( t = 0) 4 D ( t = 0) 0 . (b) Initial values of LLS model. Table 1: Basic setting of the LLS model.Parameter Value N m i , (cid:54) i (cid:54) , (cid:54) i (cid:54) , (cid:54) i (cid:54) σ γ . r . z = z . , (a) Parameters of LLS model. Variable Initial Value µ h . σ h . γ i ( t = 0) 0 . w i ( t = 0) 1000 n i ( t = 0) 100 S ( t = 0) 4 D ( t = 0) 0 . (b) Initial values of LLS model. Table 2: Setting for the LLS model (3 agent groups).
A Appendix
A.1 Parameter sets
LLS Model
The initialization of the stock return is performed by creating an artificialhistory of stock returns. The artificial history is modeled as a Gaussian random variable withmean µ h and standard deviation σ h . Furthermore, we have to point out that the incrementsof the dividend is deterministic, if z = z holds. We used the C++ standard random numbergenerator for all simulations of the LLS model if not otherwise stated.19imulation Random Number GeneratorFigures 2a to 2d C++ MT19937 RNG (64 bit)Figures 3a to 3c C++ MT19937 RNG (64 bit)Figure 4a C++ MT19937 RNG (64 bit)Figure 4b RANDU generator (32 bit)Figures 5a to 5b C++ MT19937 RNG (64 bit)Figures 6a to 6d C++ MT19937 RNG (64 bit)Figures 6a to 6d C++ MT19937 RNG (64 bit)Figures 7a to 7f C++ MT19937 RNG (64 bit)Figures 8a to 8d C++ MT19937 RNG (64 bit)Table 3: Random Generators for the Simulations20 eferences [1]
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