Long super-exponential bubbles in an agent-based model

Transcription

1 Long super-exponential bubbles in an agent-based model Daniel Philipp July 25, 2014 The agent-based model for financial markets proposed by Kaizoji et al. [1] is analyzed whether it is able to produce bubbles of realistic time scales of about one year. The model was found to exhibit some important stylized facts of financial markets such as volatility clustering and hyperbolically decreasing asset return autocorrelations. The task of finding price bubbles is broken down to the analysis of the parameter κ as a starting point to save computation time. With the choice of parameters it is up to the random components in the model possible to control the behavior of κ. The parameter analysis yielded a subspace of parameter sets that reliably produce realistic market simulations exhibiting the desired behavior of price and returns, i.e. moderately frequent bubbles of about one year and slowlydecaying absolute autocorrelations of the asset returns. 1 Introduction Financial bubbles are a natural occurrence on financial markets. They can be characterized as a moment in time in which an asset prize widely exceeds its intrinsic value. Famous examples among others are the tulip mania during the Dutch Golden Age in the 17th century, the dot-com bubble at the turn of the millennium and the US subprime mortgage crisis. But how do these bubbles emerge microscopically? In 2011, Kaizoji et al. [1] proposed an agent-based model for financial markets and the formation of bubbles. The results presented in the paper look promising: most stylized facts of financial markets can be reproduced by the model. However, the bubbles found from the simulation parameters suggested in the paper tended to be too short. This paper tries to answer the question whether the simulated market can generate longer bubbles while maintining the realistic market behavior. The model and the 1

2 most important theoretical concepts are explained in section 2. It is motivated that the search for bubbles can be broken down to the task of finding supercritical behavior in one parameter, κ. In section 3, first a sanity check is made by searching for long sequences of κ with the given parameters. Second, the parameters necessary to control the behavior of κ are varied and the connection between supercritical κ and the asset price is investigated. Last, it is checked that model indeed shows volatility clustering as a stylized fact of real financial markets. Section 4 wraps up and summarizes the results and gives a small outlook into which direction possible further investigations might go. 2 Theoretical Background In the model proposed by Kaizoji et al. [1], investors are assumed to buy either a riskfree asset (e.g. bonds) with a constant return rate R f or a riskier asset (e.g. stocks) with dividends d t at time t. Furthermore, it is assumed that there are two types of traders. The first is a group of rational investors that aim to maximize their expected utility of next period wealth (cf. [2], [3]). More detailed treatment shows that under certain assumptions (i.a. constant relative risk aversion) this leads to a strategy of investing a constant fraction x of the rational investors wealth in the risky asset. The second group is assumed to be noise traders whose behavior is governed by past performance of stocks and social components (imitation) (cf. [2]). Further information on the mathematical implementation can be found in Kaizoji et al. [1]. 2.1 Dynamical equations Balancing excess demand for the risky asset from both groups of investors (market clearing condition) leads to the following dynamical equations: Dynamics of noise traders opinion index s t = 1 N noise (1+s t 1 )/2 [ 1 2ζk (p + N t 1) ] + noise k=1 N noise (1 s t 1 )/2 j=1 [ 2ξj (p t 1) 1 ] (1) p ± t 1(s t 1, H t 1 ) = 1 2 [p κ (s t 1 + H t 1 )] (2) Dynamics of risky asset price P t /P t 1 = { (1 + s t ) [(1 + R f ) (1 s t 1 ) + (r + σ r u t ) (1 + s t 1 )] W n t 1 +4x [(1 + R f ) (1 x) + (r + σ r u t ) x] W t 1 } / (3) [ (1 + st 1 ) (1 s t ) W n t 1 + 4x (1 x) W t 1 ] 2

3 Wealth dynamics (rational investors & noise traders) ( ) W t Pt = x + (r + σ r u t ) + (1 x) (1 + R f ) (4) W t 1 P t 1 W n t W n t 1 = 1 + s t 1 2 ( ) Pt + (r + σ r u t ) + 1 s t 1 (1 + R f ) (5) P t 1 2 Momentum of risky asset price H t = θh t 1 + (1 θ) ( ) Pt 1 P t 1 (6) where (lower index t indicates time step t): s t : noise trader opinion index N noise : number of noise traders ζ k (p)/ξ k (p): random variables that take value 1 with probability p and 0 otherwise p ± t : probability for a noise trader to become bullish/bearish (i.e. invested in the risky/risk-free asset) H t : momentum of risky asset price p: 2/p days is the average holding time of positions for noise traders κ: quantifies the influence of social interactions s t and price momentum H P t : risky asset price R f : return rate of risk-free asset r: mean of stochastic process for dividend-price-ratio d/p σ r : standard deviation of stochastic process for dividend-price ratio d/p u t : series of standard i.i.d. random variables with distribution N(0, 1) W t : wealth of rational investors W n t : wealth of noise traders x: fraction of rational investors wealth invested in risky asset θ: 1/(1 θ) is the length of noise traders memory of past stock performance 3

4 Details on the derivation of the equations can be found in [1]. These equations can be used to simulate the behavior of the market. As mentioned in the introduction, it is crucial that the parameters of the model can be chosen such that characteristic features of real markets can be reproduced. This includes the emergence of bubbles and their length, along with the dynamics of price and wealth levels of the different investor types. 2.2 Bubble duration Particularly the typical duration of bubbles of a few tens of days with the parameters chosen in the paper was found to be too short to qualitatively fit real data. Following the theoretical treatment in [1], equation (3) can be simplified to: P t P t 1 = 1 + const (s t s t 1 ) + O ( r, R f, (s s 0 ) 2). (7) This is equivalent to an exponential growth of the price with s. Bubble regimes in financial markets have been found to exhibit super-exponential growth in time in the formation phase of a bubble. This would be the case if s increases with time as well. The opinion index s, along with the price momentum H, influences the dynamics via equation (2) through the probability p ± t of an investor to be invested in the risky/riskfree asset. The parameter κ reflects the impact that herding and price momentum have on behavior of noise traders. In Kaizoji et al. [1], κ is proposed to follow a discretized Ornstein-Uhlenbeck process: κ t = κ t 1 + η (µ κ κ t 1 ) + σ κ v t. (8) The last summand in the equation is white noise with σ κ and v t being the standard deviation and standard i.i.d. random variables with distribution N(0, 1), respectively. The mean is denoted by µ κ and the mean reversion rate by η > 0. From a more detailed stability analysis (found in ch. 4.2 of the same paper), the opinion index is found to exhibit exponential growth for κ > p. Stochastic fluctuations (white noise term in eq. (8)) can drive κ into this regime. Tuning of the parameter η controls the average speed with which the stochastic process returns to its mean. In the unstable regime for κ above p, the price thus grows super-exponentially. A starting point for bubble length analysis is to find parameters that lead to sequences κ t with intervals of about one year (order of magnitude of typical bubbles) in the critical regime above p. This reduction is necessary to gain a first insight into the feasibility to find such parameter sets and their frequency. From there, further investigations can be performed for the implications of these findings for the model dynamics. 4

5 The Ornstein-Uhlenbeck process expectation value is µ κ. It is chosen to be slightly below p such that the critical regime is not the norm. The distance to p influences the frequency of smaller sequences of κ in the critical regime. However this does not control the amount of time it takes for κ to revert to its mean value below p. This is controlled by the parameter η which can be connected to the average mean reversion time by calculating the expectation value of κ t with initial value κ 0 starting at time t = 0: E[κ t ] = κ 0 e ηt + µ κ ( 1 e ηt ). (9) We choose a mean reversion rate η such that a 2σ deviation returns to the subcritical regime in time T. If the initial value κ 0 is chosen in the critical regime, setting E[κ t ] to p (threshold value for the critical regime) allows for an estimate of η depending on the time T needed to revert to the subcritical regime, η = 1 T log ( κ0 µ κ p µ κ ). (10) In contrast to a Wiener process, Ornstein-Uhlenbeck processes admit a stationary probability distribution. This is mainly due to the drift term (second term in equation (8)). In this case, the stationary distribution is N ( µ κ, σ κ / 2η ), so the standard deviation for the distribution of the Ornstein-Uhlenbeck process in the long run can be written as σ stat κ = σ κ / 2η. (11) The choice of σ κ, as mentioned above, influences the frequency with which bubbles occur (or, strictly speaking, κ > p). If we choose κ 0 = µ κ + 2 σκ stat, we can compute the mean reversion rate for any desired behavior (choice of T ) numerically using equations (10) and (11): The behavior of κ is thus controlled by three parameters: average reversion time ( length of bubbles ) T mean reversion level µ κ Wiener process step size σ κ ( η = 1 ) T log σ κ 2/η. (12) p µ κ 5

6 All three values determine the mean reversion rate η in equation (8). The larger η is chosen, the faster the process returns to its mean µ κ. The Wiener process step size σ κ also controls the extent of the fluctuations in κ and hence the likelihood for bubbly κ sequences. The latter two values mainly influence the choice of κ 0 in equation (10). For example, in [1], the parameters are chosen as σκ stat = 0.1p, κ 0 = µ κ + 2 σκ stat and T = 20. This means that η is chosen such that a distance of two standard deviations σκ stat above the mean will return to the subcritical regime within T days on average. The step size σ κ can then be computed from equation (11) to be able to fully describe the dynamics of κ in (8). Thus the length of a sequence of κ t in the critical regime and the frequency with which these sequences appear can be governed by above parameters. It has to be stressed again that these sequences do not necessarily give the exact length of a bubble but rather a good ballpark figure. The condition κ > p is not a definition of super-exponential growth but rather a good hint for further investigation. By reducing the problem to κ however, it is possible to sweep larger portions of parameter space in a reasonable amount of time. In section 3, the points made in this section are quantitatively or at least qualitatively validated. 2.3 Volatility clustering According to the semi-strong form of the efficient-market hypothesis (EMH), the autocorrelation of asset returns should be zero for time lags greater than zero. Non-zero autocorrelation would imply that it is possible to consistently beat the market. Equivalently, all information available at time t is incorporated in the asset price and it is thus impossible to correctly guess the price at time t + 1. Formally, the autocorrelation is defined as follows: ρ τ = Cov(r t, r t+τ ) σ t σ t+τ = E [(r t µ t )(r t+τ µ t+τ )] σ t σ t+τ. (13) Here r t denotes the returns at time t, σ t the standard deviation, E the expected value operator and µ the mean value of returns. Since asset returns are nearly uncorrelated (there are some statistically significant correlations that however cannot be exploited with trading strategies [3]), they are not i.d.d. stochastic processes. The degree of fluctuations is not uniform and there are more turbulent and more tranquil periods. This lack of i.d.d. properties is furthermore reflected in powers of absolute autocorrelations which tend to exhibit much higher and longer-lasting autocorrelations than expected. This means that the extent of fluctuations is somehow connected, i.e. that periods of large fluctuations tend to be followed by periods of large fluctuations and vice versa [4]. Note that this only connects the magnitude of fluctuations and not their sign. Absolute autocorrelations of returns can thus be used as a measure of volatility 6

7 and thus the volatility can be predicted well. This phenomenon is known as volatility clustering. The hyperbolic decay of absolute autocorrelations has been found to be a stylized fact in financial markets [3] and therefore a good model should be able to reproduce this behavior. This will be further examined in section Methods & Results The model was implemented in Python using the dynamical equations (1) to (6) for price P, price momentum H, returns R, opinion index s and wealth levels W. The overall goal of all investigations was to find out if the model is able to reproduce characteristics of real financial markets, namely: Price bubbles of the range of about one year (sections 3.2 to 3.5) Volatility clustering, i.e. hyperbolic decay of absolute return autocorrelation (section 3.6) In order to understand the model better, the influence of the initial wealth ratio ν = W n 0 /W 0 on price and final wealth of both investor types will be examined first. 3.1 Influence of wealth ratio on price The wealth that the two types of investors start with in the beginning of the simulation can be controlled by the parameter ν = W n 0 /W 0, i.e. the wealth ratio of noise trader and rational investor at time t = 0. A larger ratio leads to a stronger impact of noise traders on the market price. As the price feeds back into other quantities described by the dynamical equations in section 2.1, it is interesting to investigate the impact on the price of the risky asset. Figure 1 shows the price of the risky asset and the wealth of noise traders/rational investors for different values of ν. Increasing initial wealth leads to higher final wealth for the noise traders which is not surprising. Interestingly, this increase is not proportional to ν: the final normalized noise trader wealth in Figure 1 shows higher values for high ν. For a proportional dependence, these values should be the same. The qualitative development of both investor types wealth remains similar for different ν. The noise traders strategy seems to be slightly favorable when comparing the final wealth for e.g. ν = 1. The impact of increasing ν is naturally much higher on the noise traders wealth. Curiously, the rational investors also seem to profit from larger initial wealth of the noise traders. Considering that their wealth similar to the noise traders diverges only after a bubble leads to the conclusion that this is mainly due to the impact of ν on the price. Looking at the evolution of the price, we can distinctly 7

8 Price P ν = ν = ν = 2 ν = ν = Noise trader wealth W n t Rational investor wealth W t Time [days] Figure 1: Price and wealth of both investors compared for different values of ν. Note the semilogarithmic scale. The starting value for the wealth of rational traders W 0 was set to 100 and the random seed to 15. Based on this, W0 n was computed. The noise traders wealth for different ν was normalized to 100 for bettter comparison. For all other parameter values the one proposed in [1] were used. 8

9 see two larger bubbles and a few smaller ones. It is noticeable that the price is similar before the first bubble is reached. This means that the effects of bubbles are amplified by choosing a higher ν. This makes sense as the creation of bubbles is mostly due to the noise traders whose influence on the market is enhanced by increasing their initial wealth. This difference in final wealth for both investor types is be due to a larger influx of money into the system in bubble regimes. The dividends at time t are proportional to the price P t 1 in the preceding time step. For strong changes in the price (bubble behavior) the returns become larger and the system experiences a non-proportional influx of wealth. The rational investors (who invest a constant fraction of their wealth in the risky asset) profit from this influx as well. Consequently, this behavior should vanish for simulations where the price only exhibits smaller bubbles. This notion could be established by running multiple simulations. 3.2 Searching for long bubbles variation of random seed The stochastic equation for κ includes a term that introduces fluctuations (last term in eq. (8)). This however means that an implementation in a computer program depends on pseudo-random number generators. Using an algorithm, sequences of approximately evenly distributed random numbers can be generated. The algorithm uses a starting number, usually called random seed from which all other number are computed. Computing a sequence with the same seed value yields the same result. State of the art in most languages (and also in Python) is the Mersenne twister algorithm [5],[6]. By setting the seed number in the algorithm it is possible to reproduce the same sequence of numbers. In a first step, κ sequences of days (i.e. 40 years) were generated with random seeds between 0 and All other parameters were chosen following the suggestions in Kaizoji et al. [1] with a starting wealth ratio of ν = W n 0 /W 0 = 2. In a Python script, every generated κ was checked for intervals in which the value of κ exceeded p for 150, 200 and 250 days ( 1 year). It was not even possible to find one random seed that produced sequences of 200 and/or 250 days of κ in the supercritical regime. For a threshold of 150 days, 5 of two million generated sequences were found to meet the criticality condition. The suggested parameters were thus not optimized towards the desired bubble length of about one year. 3.3 Searching for long bubbles variation of mean In the next step, µ κ, which was set to 0.98p in [1], was chosen closer to p. Naïvely, one would assume that changing the mean should only shift κ by a small amount. However, as discussed in section 2.2, the choice of µ κ also influences the mean reversion rate η. Figure 2 shows κ t and the price P t for the same parameter set and random seed but with 9

10 10 2 Price P κ and p Time [days] Figure 2: Price and κ for different µ κ over 5000 days (20 years). The numbers in the legend denote p/µ κ, i.e. how far away from p the mean of the Ornstein-Uhlenbeck process is chosen. For the different plots the same random seed was used to obtain comparable data. It was chosen to appropriately show most characteristic features. In the lower figure, p = 0.2 is additionally plotted to show the border between subcritical and critical regime. The simulation parameters were set to the values proposed in Kaizoji et al. [1] with ν = 2, except for µ κ, which was varied (the value proposed in the paper of µ κ = 0.98p is also plotted for reference). The random seed was set to a value that yielded a sequence of κ that best incorporated the qualitative features found during the analysis. different mean µ κ. While the general trends stay similar, there are notable diffences in the price evolution. As expected, the general length of bubbles is not affected much. Choosing µ κ closer to p leads to larger mean reversion rates (cf. eq. (12)). This explains the more jerky behavior of κ for large means. This is especially evident in Figure 3, where lower values of µ κ are also explored. These lower values also lead to sequences of κ well below the critical regime. The price thus does not show the formation of bubbles whereas for the higher values this is the case. This nicely demonstrates the connection between κ and bubble formation. It is important to note that the mean reversion time is set to T = 20 in these plots which explains the short duration of bubbles. 3.4 Searching for long bubbles parameter sweep Following the treatment in section 2.2, the length of κ sequences in the critical regime should be controllable by choosing the mean reversion time T. The frequency of bubble occurence is connected to µ κ and σκ stat. To show that the model is indeed capable of producing realistic bubbles, parameters need to be found that produce κ s with longer sequences in the critical regime. In an effort to find such parameters, κ was computed analyzed for different parameters. This was again done via a Python script that swept 10

11 Price P κ and p Time [days] Figure 3: Same figure as figure 2, but also exploring lower values of µ κ. the intervals µ κ /p = [0.75, 0.99] (step size 0.02), T = [20, 300] (step size 20) and σκ stat = [0.05, 0.175] (step size 0.025). Again all other parameters were set to the values proposed in Kaizoji et al. with an initial wealth ratio of ν = 2. Parameter sets for which κ exhibited at least one sequence of at least 300 days with κ > p and the number of times these sequences occurred for one κ were saved in a log file. Figure 4 shows the number of times a parameter set was encountered with 3 or more bubbles of length of over one year in 20 years. Each plot shows the histogram along one dimension in the three-dimensional parameter space that was investigated. As expected, the histogram showed a positive influence of µ κ close to p on the formation of more than three κ bubbles. Also, the standard deviation of the Ornstein-Uhlenbeck process shows a correlation. The same goes for T, however this is probably not due to the impact on the frequency of bubbles but to the length of average bubbles. To confirm this notion, κ was again simulated for multiple random seeds and σκ stat with the other parameters fixed. The bottom right plot in Figure 4 shows the results of grouping the κ instances resulting from simulations by the number of contained critical κ sequences. Even though the difference is not always clearly visible (note the semilogarithmic scale!), the amount of κ sequences rises for any bubble number. This nicely confirms the considerations from section 2.2 where a higher amount of κ bubbles was predicted the higher the standard deviation of the stationary distribution is chosen. In a similar way, the predictions for the dependence of κ bubbles on T and µ κ could be verified. The general trend in Figure 4 can be linked to η. Figure 5 shows the mean reversion time for different values of σ κ and µ κ. Increasing σ κ leads to an increase in η, however the influence similar to the lower left plot in Fig. 4 levels off. Also, the claims made 11

12 Ratio µ κ /p Std. dev. σκ stat of stat. process Mean reversion time T σ st κ =0.016 σ st. κ =0.02 σ st. κ = σ st. κ = Number of bubbles Figure 4: Histograms showing the number of parameters sets corresponding to values of µ κ /p, T and σκ stat for which more than 3 sequences of κ longer than 300 days were found in 20 years. The bottom right plot shows the effect of changing σκ stat for fixed µ κ = 0.95 and T = 280 over different random seeds on the frequency with which aforementioned sequences are found. The number of bubbles increases with increasing stationary standard deviation (note the semilogarithmic scale to visualize the lower ). in section 3.3 about the influence of µ κ on η are nicely shown: values close to p = 0.2 lead to larger η and thus the more jerky behavior of κ in e.g. Figure Searching for long bubbles connection between κ and actual price bubbles So far, it was verified that the frequency and length of κ sequences in the critical regime can be controlled by three parameters in the model. To make the connection to price bubbles, it is interesting to investigate the influence of these sequences on the actual price. In a qualitative approach, a whole simulation was done for one of the parameter sets found in the previous section. 12

13 0.40 Mean reversion rate η Wiener process step size σ κ Mean µ κ Figure 5: Three-dimensional plot of the mean reversion rate η as a function of the Wiener process step size σ κ and the mean µ κ following equation (12) for T = 20. As expected from the equation, choosing µ κ close to p leads to large η. Larger η lead to faster mean reversion and shorter critical κ sequences. Note that changing T would change the z-scale linearly but not the qualitative behavior (cf. eq. (12)). 13

14 10 2 Price Fraction of wealth in risky asset/opinion index Noise traders Rational traders Absolute wealth levels (normalized at t=1) Noise traders Rational traders Asset returns κ and p Time [days] Autocorrelation of returns Raw autocorrelation Absolute autocorrelation 95 % confidence bounds Time [days] Figure 6: Simulated price P t, opinion index s t, wealth levels W t and Wt n, asset returns R t, parameter κ t and the autocorrelation of returns for a certain parameter set (ν = 2, µ κ /p = 0.97, T = 240, σκ stat = 0.01) for 5000 days simulation time. The bottom left plot shows the evolution of κ with time. The black dashed line represents the level p = 0.2, dividing critical and subcritical regime. In the bottom right plot, the autocorrelation of asset returns and its absolute value are plotted against the time lag. The green dashed line confines the 95% confidence interval (computed as ±2/ T, [7]). 14

15 Figure 6 shows the most important quantities to characterize the simulated markets behavior. It features a number of bubbly κ sequences over longer periods. It is interesting to see that critical κ over longer periods of time can be directly mapped to price explosions or crashes. Also, all price bubbles are extended over longer periods of time in the order of magnitude of one year. This means that the considerations from the previous section have been more or less accurate. Approaching the formation of bubbles via κ was thus a valid approach. When examined closely, the opinion index tends to align with the price in its qualitative characteristics which also was the reason to leave out a plot of the price momentum H t : in the current model, its influence is by far overshadowed by the opinion index s. Choosing a different parameter was not done in Kaizoji et al. for the sake of simplicity of the model. It would make the examination of bubbles harder because the other parameter could be chosen such that the influence of the price momentum gets stronger. Studying κ would not give such a good insight into the price dynamics. 3.6 Volatility clustering The middle right plot in Figure 6 shows the asset returns. From this plot alone, clusters of higher volatility can be seen next to neighborhoods of lower volatility. Especially towards the end, there is a neighborhood of strong fluctuations in asset returns or a cluster of volatility. Naturally, this neighborhood coincides with a strong change in price slope (the narrow peaks towards the end in the top left plot). Looking at the autocorrelation in the lower right plot, the raw autocorrelation drops quickly from one and remains, as expected, largely in the 95% confidence interval. The absolute autocorrelation, however, shows a slower, hyperbolic decay. This is a stylized fact of financial markets and a crucial indicator for the plausibility of the model. Even for lags well above 100 days the absolute autocorrelation remains largely outside of the confidence bounds. Figure 7 once more shows the dependence of price and investor wealth on the initial wealth ratio ν. In contrast to before, a parameter set is used that leads to more realistic bubble behavior. Not surprisingly, the higher the initial wealth of the noise traders is chosen, the higher the volatility of the the asset returns. The relation between the volatility and the initial wealth ν is plotted in Figure 8. As the initial wealth of the noise traders increases, the strength of the impact on the volatility decreases. 4 Summary & Outlook Inspecting different values of ν did not lead to a significantly different price or wealth dynamics but a rise in volatility. However, the final noise trader wealth increased. Surprisingly, rational traders showed an increase of similar magnitude. Since rational 15

16 ν = Price P 10 3 ν = 0.5 ν = 1.0 ν = 1.5 ν = 2.5 ν = 3.5 ν = 4.5 Asset returns R Noise trader wealth Wt n Rational investor wealth W t ν = Time [days] Figure 7: Influence of initial wealth ratio ν on price, returns and wealth. Similar to Figure 1, but for more realistic parameters (random seed 860, µ κ /p = 0.97, T = 240, σκ stat = 0.1p). It can be visually confirmed that the volatility in returns increases for higher ν. The noise traders wealth has been normalized at time t = 0 to show the positive impact of the wealth ratio on overall wealth. Note the semi-logarithmic scale. Interestingly, the rational investors also profit from a more volatile market. 16

17 0.05 Daily volatility as a function of wealth ratio ν Wealth ratio ν Figure 8: Volatility plotted against wealth ratio ν for the same parameter set as in Figure 7. Higher relative wealth of noise traders leads to a more volatile behavior. The qualitative assumption about the influence of ν on the volatility in Figure 7 is thus confirmed. 17

18 investors always allocate a certain percentage of wealth to the risky asset, they should not be able to benefit from bubbles as much as noise traders. The price dynamics equation (3), shows, at first sight, only a non-fixed dependence on the dividends d t. Since d t is proportional to the price of the preceding day, an explosive behavior of the price leads to a larger influx of wealth into the system. This explains the wealth gains of the rational investors. A search for long bubbly κ sequences turned out to be futile with the original parameters: out of 2 million unique κ sequences it was only possible to find 5 sequences that showed supercritical behavior for more than 150 consecutive days. It was quickly found that the parameters proposed in Kaizoji et al. were not adequate to reproduce the desired behavior. Tuning of the three parameters µ κ, T and σκ stat and looking for long and frequent κ sequences in the supercritical regime was successfully done: with the right parameters, it is possible to reliably produce these κ sequences. The connection between supercritical κ and price bubbles in the actual model could be well established. Choosing one of the parameter sets found in the analysis led to longer and more frequent bubbles and volatility clustering in the asset returns. The autocorrelation and its absolute value further confirmed the emergence of volatile neighborhoods next to less volatile ones. The influence of the three main parameters (average reversion time T, mean reversion level µ κ and Wiener process step size σ κ ) on κ on the number of bubbles and their length was established. The average reversion time can be thought of as a bubble length parameter, determining the time after which the Ornstein-Uhlenbeck process governing κ returns to its mean µ κ. This means that sensible values for T lie in the range of expected bubble lengths. The mean value µ κ should not lie in the critical regime, values between 95% to 99% of the threshold value separating critical and sub-critical regime yield acceptable results. Tuning the mean has effect on both frequency and length of bubbles. The volatility of κ is controlled by σ κ and with this the frequency with which bubbles appear. The investigations presented in this paper show that indeed the model is able to reproduce some stylized facts of financial markets, even though the underlying assumption of two investor types and two investment opportunities is rather simplified. References [1] Taisei Kaizoji, Matthias Leiss, Alexander Saichev, and Didier Sornette. Superexponential endogenous bubbles in an equilibrium model of rational and noise traders. ArXiv e-prints, sep [2] Thomas Lux and Michele Marchesi. Scaling and criticality in a stochastic multi-agent model of a financial market. Nature, 397: ,

19 [3] Carl Chiarella, Roberto Dieci, and Xue-Zhong He. Handbook of Financial Markets: Dynamics and Evolution. Elsevier, [4] Benoît B. Mandelbrot. The variation of certain speculative prices. The Journal of Business, 36(4): , [5] Makoto Matsumoto and Takuji Nishimura. Mersenne twister: A 623-dimensionally equidistributed uniform pseudo-random number generator. ACM Transactions on Modeling and Computer Simulation, 8(1):3 30, January [6] NumPy Community. NumPy Reference (Release 1.8.1), March [7] William Egan. Testing for autocorrelation in the stock market, January