Agent-Based Financial Market Model: Direct Interactions & Profit Comparisons, Dispense di Economia Politica

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Terms of use: The ZBW grants you, the user, the non-exclusive right to use the selected work free of charge, territorially unrestricted and within the time limit of the term of the property rights according to the terms specified at → http://www.econstor.eu/dspace/Nutzungsbedingungen By the first use of the selected work the user agrees and declares to comply with these terms of use. zbw Leibniz-Informationszentrum WirtschaftLeibniz Information Centre for Economics Westerhoff, Frank Working Paper A simple agent-based financial market model: Direct interactions and comparisons of trading profits BERG Working paper series on government and growth, No. 61 Provided in Cooperation with: Bamberg Economic Research Group, Bamberg University Suggested Citation: Westerhoff, Frank (2009) : A simple agent-based financial market model: Direct interactions and comparisons of trading profits, BERG Working paper series on government and growth, No. 61, ISBN 978-3-931052-67-6 This Version is available at: http://hdl.handle.net/10419/38759 A simple agent-based financial market model: direct interactions and comparisons of trading profits Frank Westerhoff Working Paper No. 61 January 2009 k* b 0 k B A M AMBERG CONOMIC ESEARCH ROUP B E R G Working Paper SeriesBERG on Government and Growth Bamberg Economic Research Group on Government and Growth Bamberg University Feldkirchenstraße 21 D-96045 Bamberg Telefax: (0951) 863 5547 Telephone: (0951) 863 2547 E-mail: public-finance@sowi.uni-bamberg.de http://www.uni-bamberg.de/vwl-fiwi/forschung/berg/ ISBN 978-3-931052-67-6 1 Introduction In the recent past, a number of interesting agent-based financial market models have been proposed. These models successfully explain some important stylized facts of financial markets, such as bubbles and crashes, fat tails for the distribution of returns and volatility clustering. These models, reviewed, for instance, in Hommes (2006), LeBaron (2006), Chen et al. (2009), Lux (2009a) and Westerhoff (2009), are based on the observation that financial market participants use different heuristic trading rules to determine their speculative investment positions. Note that survey studies by Frankel and Froot (1986), Taylor and Allen (1992), Menkhoff (1997), and Menkhoff and Taylor (2007) in fact reveal that market participants use technical and fundamental analysis to assess financial markets. Agent-based financial market models thus have a strong empirical foundation. As is well known, technical analysis is a trading philosophy built on the assumption that prices tend to move in trends (Murphy 1999). By extrapolating price trends, technical trading rules usually add a positive feedback to the dynamics of financial markets, and thus may be destabilizing. Fundamental analysis is grounded on the belief that asset prices return to their fundamental values in the long run (Graham and Dodd 1951). Buying undervalued and selling overvalued assets, as suggested by these rules, apparently has a stabilizing impact on market dynamics. In most agent- based financial market models, the relative importance of these trading strategies varies over time. It is not difficult to imagine that changes in the composition of applied trading rules – such as a major shift from fundamental to technical trading rules – may have a marked impact on the dynamics of financial markets. 2 One goal of our paper is to provide a novel view on how financial market participants may select their trading rules. We do this by recombining a number of building blocks from three prominent agent-based financial market models. Let us briefly recapitulate these models: - Brock and Hommes (1997, 1998) developed a framework in which (a continuum of) financial market participants endogenously chooses between different trading rules. The agents are boundedly rational in the sense that they tend to pick trading rules which have performed well in the recent past, thereby displaying some kind of learning behavior. The performance of the trading rules may be measured as a weighted average of past realized profits, and the relative importance of the trading rules is derived via a discrete choice model. Contributions developed in this manner are often analytically tractable. Moreover, numerical investigations reveal that complex endogenous dynamics may emerge due to an ongoing evolutionary competition between trading rules. Note that in such a setting, agents interact only indirectly with each other: their orders have an impact on the price formation which, in turn, affects the performance of the trading rules and thus the agents’ selection of rules. Put differently, an agent is not directly affected by the actions of others. - In Kirman (1991, 1993), an influential opinion formation model with interactions between a fixed number of agents was introduced. In Kirman’s model, agents may hold one of two views. In each time step, two agents may meet at random, and there is a fixed probability that one agent may convince the other agent to follow his opinion. In addition, there is also a small probability that an agent changes his opinion independently. A key finding of this model is that direct interactions between heterogeneous agents may lead to substantial opinion swings. Applied to a financial 3 market setting, one may therefore observe periods where either destabilizing technical traders or stabilizing fundamental traders drive the market dynamics. Note that agents may change rules due to direct interactions with other agents but the switching probabilities are independent of the performance of the rules. - The models of Lux (1995, 1998) and Lux and Marchesi (1999, 2000) also focus on the case of a limited number of agents. Within this approach, an agent may either be an optimistic or a pessimistic technical trader or a fundamental trader. The probability that agents switch from having an optimistic technical attitude to a pessimistic one (and vice versa) depends on the majority opinion among the technical traders and the current price trend. For instance, if the majority of technical traders are optimistic and if prices are going up, the probability that pessimistic technical traders turn into optimistic technical traders is relatively high. The probability that technical traders (either being optimistic or pessimistic) switch to fundamental trading (and vice versa) depends on the relative profitability of the rules. However, a comparison of the performance of the trading rules is modeled in an asymmetric manner. While the attractiveness of technical analysis depends on realized profits, the popularity of fundamental analysis is given by expected future profit opportunities. This class of models is quite good at replicating several universal features of asset price dynamics. Each of these approaches has been extended in various interesting directions. There are also alternative strands of research in which the dynamics of financial markets is driven, for instance, by nonlinear trading rules or wealth effects. For related models see, for instance, Day and Huang (1990), Chiarella (1992), de Grauwe et al. (1993), Li and Rosser (2001), Chiarella et al. (2002), Farmer and Joshi (2002), Li and Rosser (2004), Rosser et al. (2003), de Grauwe and Grimaldi (2006), Westerhoff and Dieci (2006) or 4 extrapolation of the current price trend. The reaction parameter is positive and captures how strongly the agents react to this price signal. The second term reflects additional random orders to account for the large variety of technical trading rules. As in (1) we assume that shocks are normally distributed, i.e. b β is an IID normal random variable with mean zero and constant standard deviation . βσ Fundamental analysis (see Graham and Dodd 1951 for a classical contribution) presumes that prices may disconnect from fundamental values in the short run. In the long run, however, prices are expected to converge towards their fundamental values. Since fundamental analysis suggests buying (selling) the asset when the price is below (above) its fundamental value, orders generated by fundamental trading rules may be formalized as ttt F t PFcD γ+−= )( , (3) where is a positive reaction parameter and is the log of the fundamental value. Note that we assume that traders are able to compute the true fundamental value of the asset. In order to allow for deviations from the strict application of this rule, we include a random variable c F γ in (3), where γ is IID normally distributed with mean zero and constant standard deviation . γσ For simplicity, the fundamental value is set constant, i.e. 0=tF . (4) Alternatively, the evolution of the fundamental value may be modeled as a random walk. However, in order to show that the dynamics of a financial market may not depend on fundamental shocks, we abstain from this. We furthermore assume that there are traders in total. Let be the number N K 7 of technical traders. We are then able to define the weight of technical traders as NKW t C t /= . (5) Similarly, the weight of fundamental traders is given as NKNW t F t /)( −= . (6) Obviously, (5) and (6) imply that . Ct F t WW −=1 The number of technical and fundamental trades is determined as follows. As in Kirman (1991, 1993), we assume that two traders meet at random in each time step, and that the first trader will adopt the opinion of the other trader with a certain probability )1( δ− . In addition, there is a small probability ε that a trader will change his attitude independently. Contrary to Kirman’s approach, however, the probability that a trader converts another trader is asymmetric and depends on the current and past myopic profitability of the rules (indicated by the fitness variables CA and FA , which we define in the sequel). Suppose that technical trading rules have generated higher myopic profits than fundamental trading rules in the recent past. Then it is more likely that a technical trader will convince a fundamental trader than vice versa. Similarly, when fundamental trading rules are regarded as more profitable than technical trading rules, the chances are higher that a fundamental trader will successfully challenge a technical trader. Thus, we express the transition probability of as K ⎪ ⎪ ⎪ ⎩ ⎪ ⎪ ⎪ ⎨ ⎧ −− − − −+=− − −+ − =+ = − − + −− −→ − −− −− −→ − −+ −− 111 1 1 1 11 1 1 1 11 1 ) 1 )1((1 ) 1 )1((1 ttt tFC t t tt tCF t t tt t ppyprobabilitwithK N KN N KpyprobabilitwithK N K N KNpyprobabilitwithK K δε δε , (7) where the probability that a fundamental trader is converted into an technical trader is 8 ⎪⎩ ⎪ ⎨ ⎧ − >+ =− →− otherwise AAfor Ft C tCF t λ λδ 5.0 5.0)1( 1 (8) and the probability that a technical trader is converted into a fundamental trader is ⎪⎩ ⎪ ⎨ ⎧ + >− =− →− otherwise AAfor Ft C tFC t λ λδ 5.0 5.0)1( 1 , (9) respectively. Finally, we measure the fitness (attractiveness) of the trading rules as C t C ttt C t dADPPA 121])exp[](exp[ −−− +−= , (10) and F t F ttt F t dADPPA 121])exp[](exp[ −−− +−= , (11) respectively. Formulations (10) and (11) are as in Westerhoff and Dieci (2006) which, in turn, were inspired by Brock and Hommes (1998). Note that the fitness of a trading rule depends on two components. First, the agents take into account the most recent performance of the rules, indicated by the first terms of the right-hand side. The timing we assume is as follows. Orders submitted in period t-2 are executed at the price stated in period t-1. Whether or not these orders produce myopic profits then depends on the realized price in period t. Second, the agents have a memory. The memory parameter measures how quickly current myopic profits are discounted. For , agents obviously have no memory, while for 10 ≤≤ d 0=d 1=d they compute the fitness of a rule as the sum of all observed myopic profits. 3 Some simulation results The dynamics of international financial markets display certain stylized facts (Mantegna and Stanley 2000, Cont 2001, Lux and Ausloos 2002). These features include a random 9 In the last two panels, we plot the autocorrelation functions for raw returns and for absolute returns, respectively. Absence of significant autocorrelation between raw returns suggests that prices advance in a random walk-like manner. Despite the sporadic development of bubbles and crashes, it is thus hard to predict prices within our model. However, the autocorrelation coefficients for absolute returns are clearly significant and decay slowly. The autocorrelation coefficients are even positive for more than 100 lags. This is also in agreement with the second panel, and is a clear sign of volatility clustering, as observed in many real financial markets. ---------- Figure 1 ---------- From figure 1 we can also understand what is driving the dynamics of our model. Comparing the second and the third panel reveals that periods where technical analysis is rather popular are associated with higher volatility. Also, bubbles may be triggered in these periods. The trend-extrapolating (and highly noisy) nature of technical analysis has obviously a destabilizing impact on the dynamics. Note that technical analysis is quite profitable during the course of a bubble. As a result, more traders learn about this due to their interactions with other traders. Since technical analysis consequently gains in popularity, bubbles may possess some kind of momentum. A major shift from technical to fundamental analysis may be witnessed when a bubble collapses. A dominance of fundamental analysis then leads to a period where prices are closer towards fundamental values and where volatility is less dramatic.2 2 Why do the weights of technical and fundamental analysis vary so erratically? Since prices fluctuate randomly it is hard for traders to make systematic profits, i.e. the difference in the fitness of the rules is (usually) rather limited, which, in turn, enables “spontaneous” swings in opinion. Put differently, if one of the rules outperformed the other one, it would also dominate the market. In addition, traders may change their opinion independently of market circumstances. 12 3.2 Setting 2: 100=N Now we turn to the case with traders. Figure 2 may be directly compared with figure 1, since it is based on the same simulation design. The only difference is that the number of traders is quadrupled. As indicated by the third panel, the popularity of the trading strategies now varies only very slowly over time. Therefore, there are extremely long periods where one or the other trading strategy dominates the market, which has some obvious consequences for the dynamics. For instance, between time steps 1500 and 2700 the majority of traders rely on fundamental analysis, and hence we find a period where prices are more or less in line with fundamental values and where absolute returns are rather low. Afterwards, technical analysis gains in strength and for the next 2000 time steps volatility is elevated. Since the model is calibrated to daily data, 2000 time steps correspond to a time span of about 8 years. Although some stylized facts may still be replicated for agents, the dynamics of our model appears less convincing than before. 100=N 100=N ---------- Figure 2 ---------- Apparently, to generate realistic dynamics, the popularity of technical and fundamental trading rules has to vary more quickly, at least from a technical point of view. If there are only 25 traders, it may – in an extreme scenario – only take 25 time steps to accomplish a regime change from pure technical to pure fundamental analysis (or vice versa). An increase in the number of agents naturally increases the duration of such a complete regime switch. As seen in figure 2, regime changes may take a very long time if the number of agents is equal to 100 (of course, internal and external factors delay regime changes). In the next section, we try to show that this is not directly a “problem” of setting the number of agents too high. To achieve a reasonable fit of actual 13 market dynamics with our model, the relation between the number of agents and the number of direct interactions between them per trading time step has to be within certain limits. 3.3 Setting 3: 500=N Let us increase the number of agents up to 500=N . In addition, let us assume that there is not only one direct interaction between the agents per trading time step but that there are 20 contacts. Clearly, we now always run the interaction part of the model 20 times before we iterate the trading part of the model. As a result, the whole system may then again complete a full regime turn from pure fundamental to pure technical analysis (or the other way around) within 25 trading time steps. Figure 3 presents the results. The qualitative similarities between figure 1 and figure 3 are striking. We recover bubbles and crashes, excess volatility, fat tails for the distribution of the returns, absence of autocorrelation for raw returns, and volatility clustering, i.e. our model again mimics key stylized facts of financial markets quite well. ---------- Figure 3 ---------- Two further comments are required. Note first that periods of high volatility may or may not be associated with bubbles and crashes. It may thus happen that prices fluctuate wildly around fundamental values. We consider it interesting that there is not a strict one-to-one relation between high volatility and bubble periods.3 Finally, although the model once again generates a distribution which deviates from the normal distribution, in the sense that there is more probability mass in its tails, the fat-tailedness 3 This implies that technical analysis may also outperform fundamental analysis in a non-bubble period; otherwise its weight – which is driven by the agent’s learning behavior – would not have increased. 14 importance of the trading rules is not flexible enough – due to the assumed tandem recruitment process. Of course, one could also consider increasing the number of agents further, say, to 5000 traders. Interesting dynamics may still be recovered as long as the number of contacts between the agents per trading time step is appropriately adjusted. One interesting extension of the current setup may be to consider that (also) the probability that an agent changes his opinion independently from social interactions is state dependent. One could, for instance, assume that the probability to switch from a technical to a fundamental attitude is relatively high if fundamental analysis outperforms technical analysis. In this sense, the agents would then (also) display some kind of individual economic reasoning behavior. Finally, we would like to point out that, with a bit of experience, it is quite simple to program our model. It should therefore be possible, even for interested laymen, to reproduce the dynamics of our model. From a scientific point of view, replication of results is important. Everything required for such an exercise is given in our paper. 17 References Alfarano, S., Lux, T. and Wagner, F. (2005): Estimation of agent-based models: the case of an asymmetric herding model. Computational Economics, 26, 19-49. 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(1990): Bulls, bears and market sheep. Journal of Economic Behavior and Organization, 14, 299-329. De Grauwe, P., Dewachter, H. and Embrechts, M. (1993): Exchange rate theory – chaotic models of foreign exchange markets. Blackwell: Oxford. De Grauwe, P. and Grimaldi, M. (2006): Heterogeneity of agents, transactions costs and the exchange rate. Journal of Economic Dynamics and Control, 29, 691-719. 18 Farmer, D. and Joshi, S. (2002): The price dynamics of common trading strategies. Journal of Economic Behavior and Organization, 49, 149-171. Frankel, J. and Froot, K. (1986): Understanding the U.S. dollar in the eighties: the expectations of chartists and fundamentalists. Economic Record, 62, 24-38. Graham, B. and Dodd, D. (1951): Security analysis. New York. Hommes, C. (2006): Heterogeneous agent models in economics and finance. In: Tesfatsion, L. and Judd, K. (eds.): Handbook of computational economics Vol. 2: Agent-based computational economics. North-Holland, Amsterdam, 1107-1186. Kirman, A. (1991): Epidemics of opinion and speculative bubbles in financial markets. In: Taylor, M. (ed.): Money and financial markets. Blackwell: Oxford, 354-368. Kirman, A. (1993): Ants, rationality, and recruitment. Quarterly Journal of Economics, 108, 137-156. LeBaron, B. (2006): Agent-based computational finance. In: Tesfatsion, L. and Judd, K. (eds.): Handbook of computational economics Vol. 2: Agent-based computational economics. North-Holland, Amsterdam, 1187-1233. Li, H. and Rosser, B. (2001): Emergent volatility in asset markets with heterogeneous agents. Discrete Dynamics in Nature and Society, 6, 171-180. Li, H. and Rosser, B. (2004): Market dynamics and stock price volatility. European Physical Journal B, 39, 409-413. Lux, T. (1995): Herd behavior, bubbles and crashes. Economic Journal, 105, 881-896. Lux, T. (1998): The socio-economic dynamics of speculative markets: interacting agents, chaos, and the fat tails of return distributions. Journal of Economic Behavior and Organization, 33, 143-165. Lux, T. (2009a): Financial power laws: empirical evidence, models and mechanisms. In: 19 Legends for figures 1-6 Figure 1: The first three panels show the evolution of log prices, the returns, and the weights of technical trading rules, respectively. The left-hand panel in the fourth line depicts the distribution of the returns (the dotted line gives the corresponding normal distribution), whereas the left-hand panel presents estimates of the tail index. The bottom two panels depict the autocorrelation coefficients of raw and absolute returns, respectively. The simulation is based on 5000 time steps (omitting a longer transient period) and traders. The remaining parameters are specified in section 3. 25=N Figure 2: The same simulation design as in figure 1, except that we now consider agents. 100=N Figure 3: The same simulation design as in figure 1, except that we now consider agents and 20 direct interactions per trading time step. 500=N Figure 4: Four repetitions of figure 1 using different random number streams. Figure 5: Four repetitions of figure 2 using different random number streams. Figure 6: Four repetitions of figure 3 using different random number streams. 22 0 1250 2500 3750 5000 0.0 0.5 0.5 time lo g pr ic e 0 1250 2500 3750 5000 0.00 0.06 0.06 time re tu rn 0 1250 2500 3750 5000 0.50 0.90 0.10 time w ei gh ts 3 0 3 0.00 0.05 0.10 return pr ob de ns ity 0 5 10 5.00 2.00 8.00 largest observations ta il in de x 1 25 50 75 100 0.0 0.2 0.2 lag ac f r 1 25 50 75 100 0.00 0.20 0.40 lag ac f r Figure 1 23 0 1250 2500 3750 5000 0.0 0.5 0.5 time lo g pr ic e 0 1250 2500 3750 5000 0.00 0.06 0.06 time re tu rn 0 1250 2500 3750 5000 0.50 0.90 0.10 time w ei gh ts 3 0 3 0.00 0.05 0.10 return pr ob de ns ity 0 5 10 5.00 2.00 8.00 largest observations ta il in de x 1 25 50 75 100 0.0 0.2 0.2 lag ac f r 1 25 50 75 100 0.00 0.20 0.40 lag ac f r Figure 2 24 0 1250 2500 3750 5000 0.0 0.5 0.5 time lo g pr ic e 0 1250 2500 3750 5000 0.00 0.06 0.06 time re tu rn 0 1250 2500 3750 5000 0.50 0.90 0.10 time w ei gh ts 0 1250 2500 3750 5000 0.0 0.5 0.5 time lo g pr ic e 0 1250 2500 3750 5000 0.00 0.06 0.06 time re tu rn 0 1250 2500 3750 5000 0.50 0.90 0.10 time w ei gh ts 0 1250 2500 3750 5000 0.0 0.5 0.5 time lo g pr ic e 0 1250 2500 3750 5000 0.00 0.06 0.06 time re tu rn 0 1250 2500 3750 5000 0.50 0.90 0.10 time w ei gh ts 0 1250 2500 3750 5000 0.0 0.5 0.5 time lo g pr ic e 0 1250 2500 3750 5000 0.00 0.06 0.06 time re tu rn 0 1250 2500 3750 5000 0.50 0.90 0.10 time w ei gh ts Figure 5 27 0 1250 2500 3750 5000 0.0 0.5 0.5 time lo g pr ic e 0 1250 2500 3750 5000 0.00 0.06 0.06 time re tu rn 0 1250 2500 3750 5000 0.50 0.90 0.10 time w ei gh ts 0 1250 2500 3750 5000 0.0 0.5 0.5 time lo g pr ic e 0 1250 2500 3750 5000 0.00 0.06 0.06 time re tu rn 0 1250 2500 3750 5000 0.50 0.90 0.10 time w ei gh ts 0 1250 2500 3750 5000 0.0 0.5 0.5 time lo g pr ic e 0 1250 2500 3750 5000 0.00 0.06 0.06 time re tu rn 0 1250 2500 3750 5000 0.50 0.90 0.10 time w ei gh ts 0 1250 2500 3750 5000 0.0 0.5 0.5 time lo g pr ic e 0 1250 2500 3750 5000 0.00 0.06 0.06 time re tu rn 0 1250 2500 3750 5000 0.50 0.90 0.10 time w ei gh ts Figure 6 28 BERG Working Paper Series on Government and Growth 1 Mikko Puhakka and Jennifer P. 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December 1995 12 Heinz-Dieter Wenzel and Bernd Hayo, Are the fiscal Flows of the European Union Budget explainable by Distributional Criteria? June 1996 13 Natascha Kuhn, Finanzausgleich in Estland: Analyse der bestehenden Struktur und Ü- berlegungen für eine Reform, June 1996 14 Heinz-Dieter Wenzel, Wirtschaftliche Entwicklungsperspektiven Turkmenistans, July 1996 15 Matthias Wrede, Öffentliche Verschuldung in einem föderalen Staat; Stabilität, vertikale Zuweisungen und Verschuldungsgrenzen, August 1996