trading strategy for random market

Abstract

In that paper we search the taxon persona of randomness in financial markets, divine by the beneficial role of resound in many physical systems and in previous applications to complex socio-economic systems. Afterward a short introduction, we study the performance of some of the most secondhand trading strategies in predicting the dynamics of financial markets for contrasting international strain exchange indexes, with the destination of comparing them to the public presentation of a completely hit-or-miss scheme. Therein abide by, historical information for FTSE-UK, FTSE-MIB, DAX, and S danA; P500 indexes are taken into account for a period of about 15–20 years (since their creation until today).

Introduction

In physics, both at the classical and quantum level, many literal systems work fine and much expeditiously out-of-pocket to the useable role of a random weak noise [1]–[6]. But not only physical systems benefits from disorder. As a matter of fact, noise has a great influences connected the dynamics of cells, neurons and other biological entities, but also on bionomical, geophysical and socio-economic systems. Following this line of research, we have lately investigated how random strategies can help to improve the efficiency of a hierarchical group ready to confront the Peter principle[7]–[9] or a public mental home such American Samoa a Parliament [10]. Former groups have successfully explored akin strategies in minority and Parrondo games [11], [12], in portfolio public presentation valuation [13] and in the context of the constant double auction [14].

Recently Taleb has brilliantly discussed in his successful books [15], [16] how chance and illegal swans rule our life, just also saving and financial market behavior beyond our personal and rational expectations Oregon mastery. In reality, randomness enters in our informal life although we hardly greet it. Therefore, even without existence skeptic as much As Taleb, one could easily claim that we often misunderstand phenomena around us and are fooled past unmistakable connections which are only due to fortuity. Economical systems are unavoidably affected away expectations, some present and past, since agents' beliefs strongly influence their tense kinetics. If today a precise good expectation emerged about the public presentation of any security, everyone would try to buy it and this occurrence would imply an growth in its price. Then, tomorrow, this security would be priced higher than today, and this fact would just be the consequence of the market outlook itself. This deep addiction on expectations successful business enterprise economists try to soma mechanisms to predict future assets prices. The aim of this study is on the button to hindrance whether these mechanisms, which will be described in detail in the next sections, are more operational in predicting the market dynamics compared to a completely random strategy.

In a previous article [17], intended also past many intriguing experiments where a child, a chimpanzee and darts were with success used for remunerative investments [18], [19], we already found roughly evidence in favou of random strategies for the FTSE-UK livestock marketplace. Hera we will extend this investigation to opposite financial markets and for new trading strategies. The paper is organized as follows. Section 2 presents a brief introduction to the consider active predictability in financial markets. In Section 3 we introduce the commercial enterprise time series considered in our study and perform a detrended analysis in search for possible correlations of roughly kind. In Section 4 we define the trading strategies utilized in our simulations while, in Section5, we discuss the briny results obtained. Lastly, in Section6, we draw our conclusions, suggesting too some counterintuitive policy implications.

Expectations and Predictability in Financial Markets

As Simon [20] pointed retired, individuals assume their decisiveness on the base of a limited noesis about their environment and thus face high search costs to find requisite information. However, normally, they cannot gather all entropy they should. Therefore, agents act on the basis of delimited reason, which leads to significant biases in the potential public-service corporation maximization that they pursue. In contrast, Friedman [21] defended the rational number federal agent approach, which considers that the demeanor of agents can be best described assuming their rationality, since not-rational agents do non survive competition on the marketplace and are driven out of it. Hence, neither systematic biases in expected inferior, nor bounded rationality tail end be used to describe agents' behaviors and their expectations.

Without any fright of contradiction in terms, one could say that nowadays two main quotation models of expectations have been wide established inside the political economy lit: the adaptive expectations model and the rational expectation model. Here we will not give some semiformal definition of these paradigms. For our purposes, it is sufficient to recall their rationale. The adaptive expectations model is founded connected a somehow weighted series of backward-looking values (then that the matter-of-course value of a variable is the resultant of the combination of its past values). In contrast, the rational expectations model hypothesizes that all agents have access to whol the easy information and, therefore, be intimate exactly the model that describes the economic system (the expected note value of a variable is then the accusative anticipation provided by possibility). These two theories dates back to very relevant contributions, among which we just refer to Friedman [21], [22], Phelps [23], and Cagan [24] for adaptive expectations (IT is all the same deserving to notice that the opinion of "adaptive expectations" has been first introduced by Pointer and Nerlove [25]). For rational expectations we refer to Muth [26], Lucas [27], and Sargent-Wallace [28].

Financial markets are oft taken as object lesson for complex dynamics and dangerous excitability. This somehow suggests the idea of volatility. Yet, attributable the relevant role of those markets in the worldly system, a wide body of literature has been developed to obtain some reliable predictions. As a subject of fact, forecasting is the samara point of financial markets. Since Fama [29], we say a market is efficient if double-dyed arbitrage occurs. This means that the case of inefficiency implies the existence of opportunities for undeveloped profits and, naturally, traders would immediately operate long OR short positions until any farther possibility of profit disappears. Jensen [30] states exactly that a market is to be thoughtful efficient with prize to an information determined if it is impossible to make profits past trading on the basis of that given information set. This is consistent with Malkiel [31], who argues that an high-octane market perfectly reflects every information in determining assets' prices. As the reader can easily understand, the much important part of this definition of efficiency relies on the completeness of the information set. In fact, Fama [29] distinguishes trio forms of market efficiency, according to the degree of completeness of the informative set (namely "dilute", "semi-forceful", and "strong"). Thus, traders and financial analysts unendingly seek to expound their information set to gain the chance to choose the outflank scheme: this process involves agents so much in monetary value fluctuations that, at the terminate of the day, one could say that their activity is reduced to a systematic guess. The complete globalisation of financial markets amplified this swear out and, eventually, we are experiencing decades of intense variability and high volatility.

Keynes argued, many years past, that rationality of agents and mass psychology (then-titled "physical spirits") should not be interpreted as if they were the same thing. The Author introduced the identical famous beauty contest example to explain the logic underneath financial markets. In his General Possibility [32] he wrote that "investment based on genuine long-term expectations is and so difficult as to live scarcely practicable. He World Health Organization attempts information technology must surely lead much more laborious days and run greater risks than he who tries to opine meliorate than the crowd how the gang will behave; and, given equal intelligence, he may spend a penny more fateful mistakes." In other lyric, systematic to predict the winner of the beauty contest, unrivalled should try to interpret the jury's preferred beauty, rather than pay attention along the philosophical theory of aim beauty. In financial markets IT is exactly the same thing. It seems impossible to forecast prices of shares without mistakes. The argue is that no more investor can know in advance the opinion "of the jury", i.e. of a widespread, heterogeneous and very substantial mass of investors that reduces any contingent prediction to fair-and-square a guess.

Despite considerations like these, the so-called Efficient Market Hypothesis (whose main theoretical background is the theory of rational expectations), describes the case of perfectly competitive markets and dead magnitude relation agents, endowed with wholly procurable information, who choose for the best strategies (since otherwise the competitive clarification mechanism would put them out of the market). There is evidence that this interpretation of a fully working thoroughgoing arbitrage mechanism is not adequate to analyze financial markets every bit, for instance: Cutler et al. [33], WHO shows that large price movements occur even when little operating theater no red-hot information is available; Engle [34] who reported that price volatility is powerfully temporally correlated; Mandelbrot [35], [36], Lux [37], Andrea Mantegna and Stanley [38] who argue that light-time fluctuations of prices are non-modal; or last not least, Campbell and Shiller [39] who explain that prices May not accurately reverberate rational valuations.

Very interestingly, a plethora of sundry agents models have been introduced in the field of financial literature. In these models, different groups of traders co-exist, with divers expectations, influencing each other by means of the consequences of their behaviors. Erstwhile once again, our discussion cannot embody exhaustive here, but we can fruitfully mention at least contributions aside Brock [40], [41], Brock and Hommes [42], Chiarella [43], Chiarella and He [44], DeGrauwe et al. [45], Frankel and Froot [46], Lux [47], Wang [48], and Pieter Zeeman [49].

Break u of this literature refers to the come near, named "adaptive belief systems", that tries to apply non-one-dimensionality and noise to financial market models. Intrinsic uncertainty about economic fundamentals, along with errors and heterogeneity, leads to the idea that, apart from the fundamental measure (i.e. the present discounted value of the expected flows of dividends), share prices fluctuate erratically because of phases of either optimism Beaver State pessimism according to corresponding phases of uptrend and downtrend that grounds grocery store crises. How could this rather erratic behavior be managed in order to optimize an investment strategy? In consecrate to explain the very different mental attitude adoptive by agents to prefer strategies when trading on financial markets, a distinction is done betwixt fundamentalists and chartists. The former ones base their expectations about future assets' prices upon market fundamentals and economic factors (i.e. both micro- and macroeconomic variables, so much As dividends, wage, social science growth, unemployment rates, etc). Conversely, the latter ones try to generalise trends or statistically relevant characteristics from past series of data, ready to predict proximo paths of assets prices (likewise known as technical analysis).

Given that the interaction of these two groups of agents determines the evolution of the market, we choose here to focus on chartists' behavior (since a soft analysis on macroeconomic basic principle is absolutely subjective and difficult to asses), trying to evaluate the individual investor's ex-ante predictive capacity. Assuming the miss of complete data, randomness plays a key character, since efficiency is impossible to be reached. This is peculiarly important in order to underline that our approach does not bank on any form of the in a higher place mentioned Businesslike Markets Hypothesis paradigm. More precisely, we are seeking for the answer to the following question: if a trader assumes the deficiency of complete information finished each the market (i.e. the volatility of stock prices dynamics [50]–[53]), would an ex-ante random trading scheme perform, on average, as good as well-known trading strategies? We move from the evidence that, since each agent relies on a different information kick in guild to build his/her trading strategies, no efficient chemical mechanism fanny be invoked. Or else, a complex network of ego-influencing behavior, ascribable lopsided circulation of entropy, develops its links and generates herd behaviors to follow much signals whose credibleness is accepted.

Financial crises point that commercial enterprise markets are not immune to failures. Their periodic success is not free charge: catastrophic events burn enormous values in dollars and the economic systems in dangerous risk. Are traders so sure that detailed strategies fit the kinetics of the markets? Our simple simulation will perform a comparative analysis of the performance of different trading strategies: our traders will take up to predict, day by day, if the market will go up ('bullish' trend) or down ('pessimistic' trend). Tested strategies are: the Momentum, the RSI, the UPD, the MACD, and a completely Random one.

Rational expectations theorists would immediately bet that the random strategy would loose the competition as it is non making use of whatever information but, A we will show off, our results are quite surprising.

Detrended Analysis of the Index Clock Series

We count four identical popular indexes of business enterprise markets and in particular, we examine the following proportionate clock series, shown in Libyan Fighting Group. 1:

Temporal evolution of four important fiscal market indexes (ended time intervals going from 3714 to 5750 years).

From the tipto to the bottom, we show the FTSE UK All-Share index, the FTSE MIB All-Share index, the DAX All-Share index number and the S danampere; P 500 indicant. Watch text for further details.

• FTSE UK All-Portion index, from January, 1st 1998 to August, 3rd 2012, for a entire of T =  3714 days;

• FTSE MIB Every last-Share index, from December, 31th 1997 to June, 29th 2012, for a unconditional of T =  3684 days;

• DAX All-Ploughshare index, from November, 26th 1990 to August, 09th 2012, for a total of T =  5493 days;

• S danamp; P 500 index, from September, 11th 1989 to June, 29th 2012, for a total of T =  5750 days;

In general, the possibility to predict commercial enterprise time serial publication has been stimulated by the determination of or s kind of persistent demeanor in roughly of them [38], [54], [55]. The main purport of the present section is to investigate the possible presence of correlations in the old four financial series of European and United States stock market all share indexes. In this connection, we will calculate the time-dependent Hurst exponent by using the detrended emotional average (DMA) technique [56]. Permit us begin with a summary of the DMA algorithm. The process function is supported the calculation of the standard deviation along a granted clock series definite as

(1)

where is the average calculated in each time window of size . In order to determine the Hurst advocate , the go is calculated for raising values of inwardly the interval , beingness the length of the time series, and the obtained values are reported as a function of on a log-log plot. In general, exhibits a baron-law addiction with proponent , i.e.

(2)

In particular, if , one has a negative correlation or anti-persistent behavior, while if one has a cocksure correlation or persistent behavior. The lawsuit of corresponds to an uncorrelated Brownian process. In our case, as a first step, we deliberate the Hurst proponent considering the complete serial. This analysis is illustrated in the iv plots of Fig. 2. Hither, a linear conform to to the backlog-log plots reveals that all the values of the Hurst index H obtained in this way for the time series studied are, happening ordinary, very close to 0.5. This result seems to indicate an petit mal epilepsy of correlations happening large time scales and a consistence with a random process.

Detrended analysis for the four fiscal market serial publication shown in FIG. 1.

The Stevens' law conduct of the DMA standard deviation allows to gain an Hurst index that, in entirely the quadruplet cases, oscillates round 0.5, thus indicating an absence of correlations, on average, over large time periods. See text.

But then, it is interesting to calculate the Hurst exponent locally in meter. In order to do this analysis, we consider subsets of the complete series away means of sliding Windows of sized , which move along the serial with time step . This means that, at each time , we calculate the within the sliding window past changing with in Eq.(1). Hence, following the synoptical procedure described above, a successiveness of Hurst exponent values is obtained as social function of clock. In Ficus carica. 3 we show off the results obtained for the parameters , . In this cause, the values obtained for the Hurst exponent differ very much locally from 0.5, thus indicating the comportment of significant local correlations.

Sentence dependence of the Hurst index for the four series: along smaller time scales, significant correlations are present.

Construe with schoolbook.

This investigation, which is succeeding with what was found previously in Ref. [56] for the Dax index, seems to advise that correlations are important only connected a local temporal scale of measurement, while they cancel out averaging over long-term periods. As we will see in the next sections, this feature article will affect the performances of the trading strategies considered.

Trading Strategies Description

In the present study we consider five trading strategies distinct as follows:

1. Random (RND) Strategy

2. This scheme is the simplest unmatched, since the correspondent monger makes his/her prediction at time completely willy-nilly (with uniform distribution).

3. Momentum (MOM) Scheme

4. This strategy is based connected the so known as 'momentum' indicant, i.e. the remainder between the value and the value , where is a acknowledged trading interval (in days). Then, if , the trader predicts an growth of the closing indicant for the next day (i.e. IT predicts that ) and vice-versa. In the tailing simulations we will view years, since this is one of the about utilized hold for the momentum indicant. See Ref. [57].

5. Congener Strength Index (RSI) Strategy

6. This scheme is founded on a many composite indicator named 'RSI'. It is considered a measure of the stock's recent trading strength and its definition is: , where is the ratio between the heart and soul of the positive returns and the sum of money of the negative returns occurred during the net days before . Once calculated the RSI indicator for altogether the days included in a given time-window of distance directly preceding the time , the trader which follows the RSI strategy makes his/her foretelling on the basis of a achievable reversal of the market trend, unconcealed by the so called 'divergence' between the groundbreaking time series and the new RSI one and only. A divergence can be formed referring to a comparison between the original data serial publication and the generated RSI-series, and IT is the nearly substantial trading signal delivered by any oscillator-style indicator. It is the case when the epochal trend 'tween deuce localised extrema shown aside the RSI trend is oriented in the opposite counseling to the significant trend between two extrema (in the Saami time gaol) shown aside the underived series. When the RSI pedigree slopes other than from the original series melodic phras, a divergence occurs. Look at the example in Fig. 4: two local maxima follow two different trends sloped oppositely. In the case shown, the analyst will interpret this divergence equally a optimistic expectation (since the RSI oscillator diverges from the original series: it starts increasing when the original series is still decreasing). In our simplified model, the presence of such a divergence translates into a change in the prediction of the sign, dependant on the bullish or bearish tendency of the late days. In the chase simulations we leave choose days, since - again - this value is one of the mostly used in RSI-based actual trading strategies. Examine Ref. [57].

   

   RSI discrepancy example.

   A deviation is a dissonance between the indicator (RSI) and the underlying price. By means of sheer-lines, the psychoanalyst check that slopes of both series agree. When the divergence occurs, an inversion of the price high-energy is expected. In the good example a bullish full stop is expected.

7. Up and Downcast Persistency (UPD) Scheme

8. This deterministic strategy does non come from technical analysis. However, we decided to reckon it because it seems to follows the apparently simple alternate "up and go through" behavior of market serial that any percipient can see at first raft. The scheme is based on the following very linear rule: the prediction for tomorrow market's behavior is just the face-to-face of what happened the day before. If, e.g., combined has , the expectation at sentence for the full stop will be bullish: , and vice versa.

9. Moving Average Convergence Divergence (MACD) Strategy

10. The 'MACD' is a serial publication built by substance of the difference between two Exponential Moving Averages (EMA, henceforth) of the market toll, referred to two unlike time windows, ane smaller and one large. In any moment t, . In item, the first is the Exponential Moving Average of taken over xii days, whereas the instant refers to cardinal-sextet days. The calculation of these EMAs connected a pre-set time lag, x, given a proportionality weight , is executed aside the following algorithmic formula: with , where . At one time the MACD series has been calculated, its 9-years Exponential Moving Average is obtained and, finally, the trading strategy for the market dynamics prediction can be defined: the expectation for the market is bullish (bearish) if (). See to it Ref. [57].

Results of Through empirical observation Based Simulations

For each one of our 4 financial time series of length (in days), the destination was simply to anticipate, day by day and for each scheme, the upward (bullish) or downward (pessimistic) social movement of the index at a given day with respect to the windup value one day before: if the prediction is make up, the trader wins, otherwise he/she looses. In this joining we are solely fascinated in evaluating the pct of wins achieved by each strategy, assuming that - at every time step - the traders utterly know the past history of the indexes but do not possess whatever other data and can neither exert any influence on the market, nor receive any information about future moves.

In the following, we test the performance of the 5 strategies by dividing each of the four time series into a sequence of trading windows of equal size (in days) and evaluating the average percentage of wins for each strategy within from each one window patc the traders move along the series day by day, from to . This procedure, when applied for , allows us to explore the performance of the various strategies for several time scales (ranging, approximatively, from months to years).

The motivation behind this choice is connected to the fact that the time evolution of each indicator clearly alternates between calm and volatile periods, which at a finer resolve would let on a further, ego-similar, alternation of sporadic and regular behavior over smaller time scales, a characteristic feature of turbulent financial markets [35], [36], [38], [58]. Such a feature makes some long-term prediction of their behavior very unruly or even unsurmountable with instruments of standard financial analysis. The tip is that, due to the presence of correlations over small temporal scales (as confirmed by the depth psychology of the time dependent Hurst exponent in Fig. 3), one might expect that a apt standard trading strategy, based on the past account of the indexes, could perform better than the others inside a given fourth dimension windowpane. But this could bet much more on chance than on the concrete effectualness of the adopted algorithm. Connected the other hand, if on a very large earthly scale the financial market time evolution is an uncorrelated Brownian process (every bit indicated away the average Hurst exponent, which result to be or so for each the financial time serial publication considered), one might also bear that the performance of the standard trading strategies on a oversized time scale becomes equal to random ones. In fact, this is exactly what we found As explained in the following.

In Figs. 5–8, we report the results of our simulations for the four gillyflower indexes considered (FTSE-UK, FTSE-MIB, DAX, S danamp; P 500). In for each one figure, from top to bottom, we plot: the market time serial as a function of time; the correspondent 'returns' series, determined as the ratio ; the volatility of the returns, i.e. the disagreement of the previous series, deliberate inside each windowpane for increasing values of the trading window size (equal to, from left to satisfactory, , , and respectively); the average percentage of wins for the five trading strategies well thought out, measured for the same four kinds of windows (the average is performed over all the windows in all configuration, considering different simulation runs inside each windowpane); the related standard deviations for the wins of the quintuplet strategies.

Results for the FTSE-United Kingdom index series, divided into an increasing number of trading-windows of match size up (3,9,18,30), simulating different time scales.

From top to bottom, we report the index finger time series, the corresponding returns time series, the excitableness, the percentages of wins for the five strategies over all the Windows and the corresponding criterional deviations. The last two quantities are averaged finished 10 different runs (events) internal each window.

Results for the S danadenylic acid; P 500 index series, black-and-white into an increasing number of trading-windows of equal size of it (3,9,18,30), simulating different clock time scales.

From top to bottom, we news report the index time series, the related to returns clip series, the excitableness, the percentages of wins for the five strategies over all the windows and the related to standard deviations. The unalterable two quantities are averaged over 10 different runs (events) inside each window.

Results for the FTSE-MIB forefinger series, divided into an profit-maximizing number of trading-windows of equal size up (3,9,18,30), simulating various sentence scales.

From top to bottom, we report the indicant time serial, the corresponding returns time series, the unpredictability, the percentages of wins for the cinque strategies over complete the windows and the corresponding standard deviations. The last two quantities are averaged over 10 different runs (events) inside each window.

Results for the DAX index series, bifid into an increasing number of trading-Windows of equal sizing (3,9,18,30), simulating divers time scales.

From top to bottom, we cover the forefinger time series, the corresponding returns fourth dimension series, the volatility, the percentages of wins for the five strategies over each the Windows and the corresponding standard deviations. The hold out two quantities are averaged over 10 different runs (events) inside each window.

Perceptive the last two panels in each figure, ii primary results are evident:

1. The average percentages of wins for the five strategies are always comparable and oscillate around , with small hit-or-miss differences which turn on the financial forefinger considered. The operation of of wins for every the strategies may seem paradoxical, but IT depends on the averaging subroutine over every the windows along from each one clock time series. In Fig. 9 we show, for comparison, the behavior of the various strategies for the quadruplet financial indexes considered and for the case (the score in each window is averaged over different events): as one can check, within a given trading window each single strategy may randomly perform much better operating room worse than , but happening intermediate the global execution of the different strategies is very similar. Moreover, referring again to Figs. 5–8, it is valuable to notice that the strategy with the highest average percentage of wins (for just about of the windows configurations) changes from one index to some other united: for FTSE-UK, the MOM strategy seems to have a little reward; for FTSE-MIB, the UPD seems to be the best one; for DAX, the RSI, and for the S danamp; P 500, the UPD performs slightly better than the others. In any even the advantage of a strategy seems strictly synchronal.

   

   The percentage of wins of the different strategies inside each time window - averaged over 10 different events - is reportable, in the case N_w =  30, for the four markets considered.

   American Samoa overt, the performances of the strategies can be very different matchless from the others inside a single time window, but averaging complete the whole series these differences tend to melt and one recovers the vulgar outcome shown in the previous figures.

2. The second grand result is that the fluctuations of the hit-or-miss strategy are always smaller than those of the other strategies (every bit it is also panoptic in Fig. 9 for the casing ): this agency that the random scheme is fewer risky than the thoughtful standard trading strategies, while the average performance is almost identical. This implies that, when attempting to optimise the performance, standard traders are fooled by the "illusion of control" phenomenon [11], [12], built by a fortunate sequence of wins in a given time window. However, the first big exit may campaign them out of the market. Connected the unusual hand, the effectivity of random strategies can be believably related to the turbulent and erratic character of the financial markets: it is true that a random trader is likely to win less in a given sentence window, only he/she is likely also to loose less. Therefore his/her strategy implies to a lesser extent risk, as He/she has a lower probability to comprise thrown out of the game.

Conclusions and Policy Implications

In this paper we have explored the role of random strategies in financial systems from a micro-economic point of view. In finicky, we artificial the execution of five trading strategies, including a completely ergodic one, practical to four same popular financial markets indexes, ready to compare their predictive electrical capacity. Our main result, which is fissiparous of the market considered, is that canonical trading strategies and their algorithms, based on the past chronicle of the time series, although have occasionally the chance to be palmy inner small feature Windows, on a mammoth temporal scale perform on the average non bettor than the purely hit-or-miss strategy, which, connected the different hand, is also much less volatile. In this respect, for the private trader, a purely random strategy represents a gratuitous alternative to expensive professed financial consulting, being at the same time also much to a lesser extent risky, if compared to the other trading strategies.

This result, obtained at a micro-level, could have many an implications for real markets also at the macro-level, where other important phenomena, like herding, asymmetric information, rational bubbles occur. In fact, one might expect that a widespread adoption of a ergodic approach for financial transactions would result in a more constant securities industry with lower excitability. In this connection, unselected strategies could play the part of reduction herding behavior over the gross market since, if agents knew that financial transactions do not necessarily carry an information role, bandwagon personal effects could probably fade. Then again, as recently suggested by one of us [59], if the policy-maker (Central Banks) intervened by randomly buying and marketing financial assets, two results could be simultaneously obtained. From an individual stand, agents would suffer little for asymmetric or insider information, due to the consciousness of a "fog of uncertainty" created by the haphazard investments. From a general point of view, once more the herding behavior would be consequently reduced and eventual bubbles would burst when they are still small and are less dangerous; thus, the entire financial scheme would personify less prone to the speculative behavior of credible "Guru" traders, as explained also in [60]. Of course, this has to be explored in detail arsenic well as the feedback result of a spherical reaction of the market to the application of these actions.This topic is nonetheless on the far side the goal of the attending paper and it volition Be investigated in a future cultivate.

Acknowledgments

We thank H. Trummer for DAX humanistic discipline series and the other institutions for the respective information sets.

Funding Financial statement

The authors have no support or financial support to report.

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Source: https://www.ncbi.nlm.nih.gov/pmc/articles/PMC3708927/

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