G R E D E G Documents de travail

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1 G R E D E G Documents de travail WP n ASSET MISPRICING AND HETEROGENEOUS BELIEFS AMONG ARBITRAGEURS *** Sandrine Jacob Leal GREDEG Groupe de Recherche en Droit, Economie et Gestion 250 rue Albert Einstein Valbonne -

2 ASSET MISPRICING AND HETEROGENEOUS BELIEFS AMONG ARBITRAGEURS Sandrine Jacob Leal 12 1 st draft July 2008 This version October 2008 ABSTRACT In financial markets, some assets may be overpriced (underpriced), i.e. whose price is above (below) the asset fundamentals. While professional traders may observe that the asset price does not reflect its fundamental value, such asset mispricing may persist over a substantial period of time. The purpose of this paper is to investigate under which conditions such asset mispricing may persist in the presence of well-funded arbitrageurs and how heterogeneous beliefs among arbitrageurs may affect mispricing duration. The paper is based on a financial heterogeneous agent model with boundedly rational agents, namely: arbitrageurs and chartists, who follow simple decision rules. This paper uses a market maker based method of price formation in order to study the price dynamics induced by the interplay between arbitrageurs and chartists. The results indicate that whenever arbitrageurs have heterogeneous beliefs about the asset fundamental value, first even though chartists do not dominate the market, arbitrageurs trading activity may not bring asset price back towards the asset fundamentals and asset mispricing may persist. Second if chartists dominate the market, chartists trading activity may further price correction. JEL classification: G12, G14. Keywords: Price dynamics, Bounded rationality, Heterogeneous agents, Trading strategies. 1 Correspondence to: Sandrine Jacob Leal, University of Nice Sophia Antipolis, GREDEG CNRS, 250, Rue Albert Einstein Sophia Antipolis, F Valbonne Cedex, France. E mail: jacob@gredeg.cnrs.fr. 2 University of Siena, Italy. 1

3 1. INTRODUCTION In financial markets, some assets may be overpriced (underpriced), that is whose price may not be justified by the evolution of the fundamentals. While professional traders are present in the market and may observe that the asset price does not reflect its fundamental value, such asset mispricing may persist over a substantial period of time 3. This paper investigates under which conditions such asset mispricing may persist in the market when well-funded arbitrageurs and chartists coexist. In particular, conditions are given under which chartists trading activity may prevent arbitrageurs from correcting asset mispricing and lengthen mispricing correction. Many financial heterogeneous agent models have been proposed to explain the stylized facts observed in financial time series, including excess volatility (Lux, 1995), clustered volatility (Brock and LeBaron, 1996; Lux, 1997; Youssefmir and Huberman, ) and/or fat tails of returns distribution (Lux, 1998), which cannot be fully explained through a random walk model with a representative rational agent (e.g., Shiller, 1981; Cutler et al., 1989). These empirical observations have challenged the efficient market hypothesis which has been the dominating paradigm in finance in the 1970s and 1980s. In an efficient market, any predictable structure in asset returns cannot persist. However, from the late 1980s, the behavioral agent-based approach based on heterogeneous agent models and bounded rationality has become increasingly popular. Most works on heterogeneous agent models consider two main types of agents, namely: fundamentalists and chartists 5. Early models include Zeeman (1974), which proposed explaining the observed stylized fact of temporary bull and bear markets, and Frankel and Froot (1986). The latter develop a model for exchange rates with time varying weights of forecasting strategies. Simulations of the model reveal that the exchange rate may exhibit temporary bubbles. Subsequent research in the field has revealed that financial heterogeneous agent models may also yield more complex price dynamics. Many of these models have, more recently, been successful in explaining simultaneously the stylized facts observed in financial time series such as clustered volatility and fat tails of returns distribution including Lux (1995, 1998), Youssefmir and Huberman (1997), LeBaron et al. (1999), Lux and Marchesi (2000), Hommes (2001, 2002), Iori (2002), De Grauwe and Grimaldi (2005, 2006), De Grauwe and Kaltwasser (2006), Boswijk et al. (2007), Alfarano et al. (2008). In particular, clustered volatility arises through the interaction and switching between fundamentalists and chartists trading strategies. In most heterogeneous agent models with fundamentalists and chartists, the former act as a stabilizing force, bringing 3 An often cited example is the overpricing of Internet stocks during the 1990s, during which Internet stock prices reached much higher levels than their underlying fundamentals (Ofek and Richardson, 2001). Empirical evidence also suggests that stock prices often exhibit excess volatility, that is movements in stock prices may be much larger than movements in the underlying economic fundamentals (earlier discussions include LeRoy and Porter, 1981; Shiller, 1981). 4 This work applies to a wide range of systems, ranging from economies to computer networks and telecommunications. 5 Evidence from survey data on investors expectations (Shiller, 1987; Frankel and Froot, 1987a, 1987b, 1990a, 1990b; Allen and Taylor, 1990; Ito, 1990) has shown that financial practitioners use different trading and forecasting strategies. In addition to fundamental analysis, technical analysis and chartist strategies are extensively used among practitioners. More recent evidence from survey data includes Liu (1996), Menkhoff (1997), Lui and Mole (1998), Shiller (2000), Cheung et al. (2004), Maditinos et al. (2007). 2

4 prices back to the fundamentals, while the latter act as a destabilizing force, pushing prices away from the fundamentals. A common result of these works is that whenever the proportion of chartists, mainly trend followers, exceeds some threshold value, the price trend tends to be reinforced and prices deviate further away from fundamentals. The purpose of this paper is precisely to investigate under which conditions this holds and whether the fact that chartists dominate the market is a necessary condition in order to explain mispricing persistence. In line with the new behavioral approach, the present paper develops a simple financial heterogeneous agent model, with boundedly rational agents, namely: arbitrageurs (or fundamentalists) and chartists, and a stylized representation of the market institution by a market maker 6. However while previous works mainly focus on the interaction between fundamentalists and chartists, I account for arbitrageurs with heterogeneous beliefs about the asset fundamentals realistic versus unrealistic beliefs and different chartist trading strategies trend followers versus contrarian traders 7. Simulations of the model reveal that whenever arbitrageurs have heterogeneous beliefs about the asset fundamental value, first even though chartists do not dominate the market, arbitrageurs trading activity may not bring the asset price back towards the asset fundamental value. Consequently asset mispricing may markedly persist in the market. Second when chartists dominate the market, chartists trading activity may further price correction. In order to investigate under which conditions chartists trading activity may prevent arbitrageurs from correcting asset mispricing, one needs to define mispricing persistence. A preliminary step of this work consists in exploring a simple setting (benchmark case) in which arbitrageurs have homogeneous beliefs. Then I introduce arbitrageurs with heterogeneous beliefs realistic versus unrealistic beliefs and analyze the price dynamics induced by the interplay between arbitrageurs and chartists. The paper is organized as follows. Section 2 presents first the simple version of the model (benchmark case) and subsequently the core model along with the main results of the simulations. Section 3 summarizes the main findings of this work and points to future extensions of the present model. 2. THE MODEL In the model discussed here, two types of boundedly rational agents, namely: arbitrageurs (or fundamentalists) and chartists follow simple decision rules and trade a single risky asset. I use a market maker based method of price formation in order to study the price dynamics induced by each trading strategy as well as by the interplay between arbitrageurs and chartists. 6 Earlier models using a representation of market institution by a market maker include Beja and Goldman (1980), Day and Huang (1990), Chiarella (1992), and recently Lux and Marchesi (2000), Farmer (2002), Farmer and Joshi (2002). 7 Lo and MacKinlay (1990), Jegadeesh and Titman (1995), as well as more recently Galariotis et al. (2007) suggest that the profitability of contrarian traders is economically significant. 3

5 Model framework In time t, the asset price is denoted P t and the asset fundamental value is denoted V t. The latter is assumed to be constant over time 8. Arbitrageurs observe the asset price which prevails in the market and base their trading strategy upon any differential between the asset price and its perceived fundamental value 9. Chartists do not account for the asset fundamental value; rather they base their trading strategies upon observed historical patterns in past prices in order to forecast future movements in prices. Chartists are of two types, namely: trend followers and contrarian traders. Trend followers believe that past price movements tend to repeat in the future. Contrarian traders rather believe that past price movements tend to revert in the future. In each period, arbitrageurs and chartists can place buy or sell orders. Since arbitrageurs and chartists orders typically do not match, excess aggregate demand is taken up by a market maker, who sets the new price based upon the previous price and the level of excess demand. At some point in time (t 0 ), there is a positive 10 shock on the asset price which is not related to the asset fundamental value. The asset price then departs from the asset fundamental value. The shock on the asset price, as well as the evolution of the asset fundamental value, is illustrated in Fig. 1. P t, V t Shock on P t P t V t P t = V t 0 t 0 t Fig. 1: Evolution of the asset price and fundamental value. In order to investigate under which conditions asset mispricing may persist while arbitrageurs with heterogeneous beliefs and chartists coexist in the market, one has first to define mispricing persistence. Consequently first I implement some benchmark cases in which all arbitrageurs have realistic beliefs about future payoffs. These benchmarks include first a setting in which there are only arbitrageurs (benchmark 1), in order to determine when arbitrageurs trading activity should bring the asset price back to the fundamentals. Second a setting in which I introduce chartists though their trading activity does not further affect the price dynamics 8 I consider a simple case in which no new information (neither positive nor negative) alters the intrinsic value of the asset over time. Consequently, future cash flows of the asset are constant. 9 The perceived fundamental value of the asset depends on arbitrageurs beliefs about the asset future payoffs. Beliefs about asset future payoffs are based on the prospect of future cash flows. Rather beliefs about future prices independent of cash flows are precluded in this work. 10 Similar results of the model may be derived if a negative shock on the asset price is considered. 4

6 (benchmark 2). In this setting, I focus on the effect of arbitrageurs trading activity on the price dynamics, while chartists are present in the market. Finally a setting in which chartists trading activity further affects the price dynamics (benchmark 3), in order to investigate whether mispricing may persist while well-funded arbitrageurs are present in the market. Second I extend these benchmark cases and introduce heterogeneous realistic as well as unrealistic beliefs among arbitrageurs in order to investigate whether the fact that chartists dominate the market is still a necessary condition for destabilizing asset price Price dynamics and homogeneous beliefs among arbitrageurs. In this setting, well-funded arbitrageurs have all the same realistic beliefs about the asset future payoffs 11. In each period, arbitrageurs know the true asset fundamental value (V t ). All arbitrageurs are thus immediately aware that the shock on the asset price is not related to the evolution of the asset fundamentals. Agents behavior First before the shock on the asset price, arbitrageurs have no reasons to trade the asset since its price reflects its fundamentals. As long as there is no asset mispricing, arbitrageurs expect that in the next period the price will still reflect the asset fundamental value. They remain inactive. After the positive (negative) shock on the asset price however, the asset is overvalued (undervalued). Arbitrageurs expect that the price will go back to the asset fundamental value in the future. Since any differential between P t and V t represents a potential gain for arbitrageurs, they sell (buy) the asset as long as its price is above (below) the asset fundamental value 12. Arbitrageurs orders are a function of the difference between the asset fundamental value and its current price: x = t+ 1 0 if P t = V t x = t 1 β(v + t P t) 13 if P t V t Second there are two types of chartists, namely: trend followers and contrarian traders. Trend followers trade the asset based upon the belief that any past price movement tends to repeat in the future. Contrarian traders trade the asset based upon the belief that any past price movement rather tends to revert in the future. Following Brock and Hommes (1998), chartists expectations about the asset price in the next period are: 11 This may be due to the fact that information about the asset fundamental value diffuses quickly and correctly. In these settings, arbitrageurs perceived asset fundamental value coincides with the true asset fundamental value. 12 To a certain degree, this hypothesis justifies the use, along this work, of the term arbitrageurs rather than fundamentalists. While fundamentalists tend to buy (sell) overvalued (undervalued) stocks, arbitrageurs tend to exploit more aggressively such potential gains. Besides, this hypothesis holds since I assume that arbitrageurs are well funded, i.e., do not face any financial constraints. 13 In Farmer and Joshi (2002), such rule defines order based strategies, which implies that agents continue to buy (sell) as long as there is asset mispricing. 5

7 e t+ 1 = t 1 + t 1 t 2 P P δ(p P ) 14 with δ > 0 for trend following strategy and δ < 0 for contrarian strategy. Trend followers buy (sell) the asset when its price has increased (decreased) in previous periods. Contrarian traders buy (sell) the asset when its price has decreased (increased) in previous periods. However if observed prices were constant, chartists both trend followers and contrarian traders would expect that the asset price will be unchanged and remain inactive. Chartists orders are thus: x = t+ 1 0 if Pt+ 1 = P t 1 e e xt+ 1 = φ Pt 1 P t 2 if Pt+ 1 < P t 1 xt+ 1 = φ Pt 1 P t 2 if Pt+ 1 > Pt 1 e Price formation Price adjustments and transactions are assumed to occur simultaneously. As, for instance, in Day and Huang (1990) and more recently in Farmer and Joshi (2002), because agents orders typically do no match, a market maker 15 is assumed to match demand and supply in each period and modify the asset price according to aggregate excess demand. The price in the next period (P t+1 ) is computed as follows: P = P + μ( x ) t+ 1 t t+ 1 If agents net orders are negative (positive), the market maker must buy (sell) some shares of the asset and decrease (increase) P t+1. Transactions are then executed at this new price. a. Price dynamics with arbitrageurs only. In this section, I investigate the price dynamics within a simple setting in which there are arbitrageurs only and the latter have realistic beliefs about the asset future payoffs (benchmark 1). They know V t in each period. If there are only arbitrageurs in the market, the asset price would immediately go back to the asset fundamental value owing to arbitrageurs trading activity, i.e., arbitrage works. The parameter values of the simulation are given in Table Chartists do not account for the current price because it is assumed that they do not observe it. 15 The representation of market institution by a market maker allows investigating independently the effect of each trading strategy on price dynamics. In this work, chartists and arbitrageurs with heterogeneous beliefs are successively introduced. 6

8 Table 1: Parameters for the simulation with arbitrageurs only - Benchmark 1. Description of parameter (symbol) Value Description of parameter (symbol) Value Number of periods (t) 50 Date of the shock (t 0 ) 10 Number of agents (N) 1000 Proportion of arbitrageurs (aa) 1 Arbitrageurs orders parameter (β) 1 16 Chartists expectation parameter (δ) 1 Chartists orders parameter (ϕ) 1 17 Market adjustment parameter (μ) Initial price (P 0 ) 100 Initial fundamental value (V 0 ) 100 Size of the shock (b) 0.1 As expected, if there are only arbitrageurs in the market, the asset price immediately goes back to the asset fundamental value after the positive (negative) shock on the asset price. This is illustrated in Fig. 2. Since after the shock, all arbitrageurs realize that the price does not reflect the asset fundamental value, they sell (buy) at the same time. Arbitrage works. Fig. 2: Price dynamics when there are only arbitrageurs in the market (i.e., aa = 1). The asset fundamental value (asset price) is represented by the blue (red) line. b. Price dynamics with arbitrageurs and chartists. In this setting, I introduce chartists, trend followers as well as contrarian traders in order to investigate whether asset mispricing may persist while chartists are present in the market. As a first step in the analysis of the interplay between arbitrageurs and chartists, I 16 A more general setting may be explored so that arbitrageurs orders are rather proportional to the difference between V t and P t (as for instance, in Day and Huang (1990), Farmer (2002), Farmer and Joshi (2002)), such that β 1. However in the present paper, β = 1 is a convenient assumption in order to derive the expected results and settle the appropriate benchmark case for the discussion of more complex situations presented in section 2.2. This parameter value holds all along this work. 17 As for arbitrageurs, I assume that chartists orders in each period is a function of the difference between P t 1 and P t 2, i.e., δ = 1. Besides, the same parameter value holds for both types of chartists in order to ensure that chartists trading activity does not further affect price dynamics when chartists are split into equal shares. This parameter value holds all along this work. 18 The market adjustment parameter (μ) depends on the number of agents in the market. In order to neutralize the effect of the market maker s activity, and avoid an extra bias on the evolution of the asset price, I set μ = 1/N, with N is the number of agents in the market. This parameter value holds all along this work. 7

9 implement a situation in which chartists trading activity does not further affect the evolution of the asset price (benchmark 2). This was done by matching the population of trend followers and contrarian traders, such that the positive autocorrelation in returns induced by trend followers is cancelled by the negative autocorrelation in returns induced by contrarian traders 19. I thus focus on the effect of arbitrageurs trading activity on the price dynamics by varying the proportion of arbitrageurs in the market. This setting allows investigating whether arbitrageurs are able to bring the asset price back towards the asset fundamentals while chartists are present in the market. The parameter values of the simulation are given in Table 2. Table 2: Parameters for the simulation with arbitrageurs and chartists split into equal shares Benchmark 2. Description of parameter (symbol) Value Description of parameter (symbol) Value Number of periods (t) 50 Date of the shock (t 0 ) 10 Number of agents (N) 1000 Proportion of arbitrageurs (aa) from 0 to 1 Proportion of trend followers (cc) 0.5 Proportion of contrarian traders (1-cc) 0.5 Arbitrageurs orders parameter (β) 1 Chartists expectation parameter (δ) 1 Chartists orders parameter (ϕ) 1 Market adjustment parameter (μ) Initial price (P 0 ) 100 Initial fundamental value (V 0 ) 100 Size of the shock (b) 0.1 As emphasized in benchmark 1, if there are only arbitrageurs in the market (i.e., aa = 1), price correction occurs immediately after the shock on the asset price. For intermediate values of the proportion of arbitrageurs (aa varying from 0.9 to 0.1), first the asset price eventually corrects. Second, mispricing duration is decreasing in the proportion of arbitrageurs. The larger the proportion of arbitrageurs is, the shorter mispricing duration. Third, price correction is markedly delayed only if chartists dominate (i.e., aa 0.2). However even though chartists trading activity does not further affect the evolution of the asset price, mispricing may persist for some period of time. Finally, if there are no arbitrageurs in the market, chartists trading activity supports the asset price at its post-shock level. These findings are illustrated in Fig This implies that chartists are split into equal shares, i.e., same proportion of trend followers and contrarian traders (i.e., cc = 0.5). 8

10 Fig. 3: Price dynamics when chartists are split into equal shares (i.e., cc = 0.5) and the proportion of arbitrageurs (aa) varies from 1 to 0. The second step of this analysis is to investigate the price dynamics when chartists trading activity further affects the evolution of the asset price (benchmark 3). This was done by varying the proportion of trend followers (cc varying from 0 to 1). This setting allows investigating the effect of chartists trading activity on the price dynamics and studying whether chartists may prevent arbitrageurs from correcting asset mispricing. In this setting, it is confirmed that if arbitrageurs dominate (aa varying from 1 to 0.6), the price correction occurs almost immediately after the shock, whatever the proportion of trend followers (or contrarian traders). For extreme values of proportion of trend followers (cc close to 0 or 1), mispricing duration is lengthened but arbitrageurs trading activity still brings the asset price back towards the asset fundamental value almost immediately after the shock. Mispricing duration is thus increasing in the proportion of trend followers (or contrarian traders). These findings are illustrated in Fig. 4 (with aa = 0.6) 20. Fig. 4: Price dynamics when proportion of arbitrageurs = 0.6 and the proportion of trend followers (cc) varies from 0 to See Fig. 1 3 in Annexes for aa varying from 0.9 to 0.7, respectively. 9

11 If arbitrageurs do not dominate (i.e., aa varying from 0.5 to 0.3) however, price eventually corrects but later than in benchmark 2 (see Fig.3). This finding is illustrated in Fig. 5 (with aa = 0.4) 21. Fig. 5: Price dynamics when the proportion of arbitrageurs = 0.4 and the proportion of trend followers (cc) varies from 0 to 1. The results of these simulations confirm that as long as arbitrageurs dominate (aa [1; 0.3]), whatever the proportion of trend followers in the market, price correction eventually occurs. Besides, mispricing duration is increasing in the proportion of trend followers (or contrarian traders). However when chartists dominate (aa = 0.2 and aa = 0.1), price correction is markedly delayed 22 (with respect to benchmark 2, i.e., cc = 0.5). As emphasized by previous works (e.g., Lux, 1995, 1998; Lux and Marchesi, 2000; De Grauwe and Grimaldi, 2005, 2006), if chartists dominate, chartists trading activity may destabilize asset prices. Furthermore, when chartists are mainly trend followers (i.e., cc > 0.8), the asset price overreacts after the shock 23, which is consistent with earlier works on overreaction in asset markets (e.g., Farmer, 2002; Farmer and Joshi, 2002). These works actually suggest that trend following strategies tend to induce short-run positive autocorrelation in returns and amplify trends in price, thus providing an explanation for stocks prices overreaction. These findings are illustrated in Fig See Fig. 4 5 in Annexes for aa = 0.5 and aa = 0.3, respectively. 22 When aa = 0.2, mispricing duration is lengthen by more than 35% with respect to benchmark 2 (i.e., cc = 0.5) if chartists are mainly contrarian traders (i.e., cc = 0.1). If chartists are mainly trend followers (i.e., cc = 0.9 and cc = 1), mispricing duration is even larger (increased by 85% and 150%, respectively). When aa = 0.1, mispricing duration is lengthen by 30% if chartists are mainly trend followers (i.e., cc = 0.9). For extreme values of cc (i.e., cc = 0, 0.1 and 1), asset mispricing never corrects. 23 Asset price overreacts at most by 6.5% when chartists are only trend followers (i.e., cc = 1). 10

12 Fig. 6: Price dynamics when the proportion of arbitrageurs = 0.2 and the proportion of trend followers varies (cc) from 0 to 1. Fig. 7: Price dynamics when the proportion of arbitrageurs = 0.1 and proportion of trend followers varies (cc) from 0 to 1. Finally, and not surprisingly, when there are no arbitrageurs in the market the asset price never goes back to the asset fundamental value. First when trend followers dominate contrarian traders (i.e., cc > 0.5), the asset price further departs from the fundamentals. As mentioned above this is consistent with earlier works (e.g., Farmer, 2002; Farmer and Joshi, 2002), trend followers trading activity amplifies trends in the asset price and induces positive short-term autocorrelation. Second when contrarian traders dominate trend followers (i.e., cc < 0.5), the asset price tends to fluctuate around the asset fundamental value. The initial shock on the asset price is followed by an infinite sequence of endogenous fluctuations of the asset price between its post-shock level and its fundamental value. These endogenous fluctuations are due to contrarian traders trading activity (i.e., contrarian traders buy (sell) when the price has decreased (increased)). These findings are illustrated in Fig. 8. Fig. 8: Price dynamics when there are no arbitrageurs (i.e., aa = 0) and the proportion of trend followers (cc) varies from 0 to 1. 11

13 c. Comparison of settings when arbitrageurs have realistic beliefs about the asset fundamental value. In order to investigate under which conditions asset mispricing may persist in the presence of well-funded arbitrageurs, I explored several situations. This work has first allowed analyzing independently the effect of each trading strategy on price dynamics. Second it has allowed investigating the price dynamics induced by the interplay between arbitrageurs and chartists within a simple framework in which arbitrageurs have realistic beliefs about the asset future payoffs. In the first setting (benchmark 1), there are arbitrageurs only in the market. I then introduced chartists, both trend followers and contrarian traders. In the second setting (benchmark 2), chartists trading activity does not further affect the price dynamics. This has allowed focusing on the effect of arbitrageurs trading activity on price dynamics. Finally and most interestingly, in the third setting (benchmark 3), chartists trading activity may further affect the evolution of the asset price. The main findings of each benchmark are summarized in Table 3. Table 3: Main findings of each benchmark when arbitrageurs have homogeneous beliefs about the asset fundamental value. Main features Main findings Benchmark 1 Benchmark 2 Arbitrageurs only Arbitrageurs and chartists Chartists trading activity does not further affect the evolution of the asset price (i.e., cc = 0.5). Price correction occurs immediately after the shock on the asset price. Mispricing duration is decreasing in the proportion of arbitrageurs in the market; Even if chartists trading activity does not further affect the evolution of the asset price, mispricing may persist; Price correction is markedly delayed only if chartists dominate. Benchmark 3 Arbitrageurs and chartists Chartists trading activity may further affect the evolution of the asset price (i.e., cc [0, 1]). Mispricing duration is increasing in the proportion of trend followers (or contrarian traders) in the market; If chartists dominate, asset mispricing may persist not only if the chartists are trend followers, but also if the chartists are contrarian traders. 12

14 The simulations presented above revealed that if chartists affect the evolution of the asset price (benchmark 3), chartists trading activity may explain mispricing persistence (asset mispricing persists longer than in benchmark 2). As suggested by earlier works (e.g., De Grauwe and Grimaldi, 2005, 2006), when chartists mainly trend followers dominate, they may destabilizes asset prices. Besides, while those works mainly focus on the distinction between fundamentalists and trend followers, I have investigated a more general setting with trend followers as well as contrarian traders. Similar conclusions may be drawn for contrarian traders. Contrarian traders trading activity may also prevent arbitrageurs from correcting the asset mispricing and destabilize stocks price. These preliminary results are useful benchmarks in order to explore a more complex framework in which arbitrageurs have heterogeneous beliefs and investigate under which conditions chartists destabilize asset prices. This is discussed in the following section Price dynamics and heterogeneous beliefs among arbitrageurs 24. In the previous section, benchmark cases allowed deriving well-known results as well as defining mispricing persistence from simple settings namely: (i) if there are only well-funded arbitrageurs in the market, arbitrage works asset mispricing is immediately corrected, (ii) mispricing duration is decreasing in the proportion of arbitrageurs in the market and (iii) chartists trading activity may destabilize asset prices. In this setting, I introduce arbitrageurs with heterogeneous beliefs about the asset fundamentals. The main difference with the setting discussed above is that all arbitrageurs do not immediately realize that the shock on the asset price is not related to the evolution of the fundamentals. Rather some arbitrageurs with unrealistic beliefs believe so 25 thus do not immediately identify that there is asset mispricing. However they become sequentially aware of the true asset fundamental value 26. For a while, arbitrageurs with realistic and unrealistic beliefs about the asset fundamental value coexist in the market. First, as described in the previous setting, arbitrageurs with realistic beliefs know the true asset fundamental value and immediately identify asset mispricing. Accordingly, as soon as the asset price does not reflect the asset fundamentals, they expect that the price will go back to the asset fundamental value. These arbitrageurs orders (X t+1 ), as described previously, are: xt+ 1 = β(vt P t) 24 In this setting, chartists behaviour, as well as the price formation mechanism, is the same as described in the previous setting. 25 This setting allows considering situations in which information about the asset s fundamental value diffuses slowly, there is asymmetric information among arbitrageurs or differential interpretation of such information. 26 This was done by using a simple uniform rule. In each period, an additional fraction of arbitrageurs become informed of the true asset s fundamental value. 13

15 If the price is above (below) the asset fundamental value, arbitrageurs sell (buy) the asset, since they identify that the asset is overvalued (undervalued). Second arbitrageurs with unrealistic beliefs about the asset fundamental value expect that in the future the asset price will remain at its post-shock level, reflecting the perceived new asset fundamental value ( V t+1 ) 27. Since after the shock, they do not identify that there is asset mispricing, they remain inactive 28. Immediately after the shock, these arbitrageurs orders (X t+1 ) are: x t+ 1 = 0. In subsequent periods however, the asset price may further depart from the perceived new fundamental value ( V t ), due to other agents trading activity. Arbitrageurs with unrealistic beliefs thus expect that the asset price will go back to V t, rather than the true fundamental value (V t ). Accordingly, they trade the asset in order to exploit any differential between P t and V t. These arbitrageurs orders (X t+1 ) are: xt+ 1 = β(v t P t) If the asset price which prevails in the market is above (below) V t, arbitrageurs with unrealistic beliefs sell (buy) the asset, since they believe that the asset is overvalued (undervalued). a. Price dynamics with arbitrageurs only. In this section, as in the previous section, I investigate the price dynamics within a simple setting in which there are arbitrageurs only. However for some period of time, arbitrageurs with realistic as well as unrealistic beliefs coexist in the market. If there are only arbitrageurs in the market (benchmark 4), the price would go back to the asset fundamental value as soon as all arbitrageurs know the true asset fundamental value. The parameter values of the simulation are given in Table This perceived fundamental value does not coincide with the true asset fundamental value. 28 This holds as long as P=V. t t 14

16 Table 4: Parameters for the simulation with only arbitrageurs (realistic vs. unrealistic beliefs) Benchmark 4. Description of parameter (symbol) Value Description of parameter (symbol) Value Number of periods (t) 50 Date of the shock (t 0 ) 10 Date at which all arbitrageurs are informed (t i ) 20 Number of agents (N) 1000 Additional number of informed arbitrageurs in each period (n) 111 Proportion of arbitrageurs (aa) 1 Arbitrageurs orders parameter (β) 1 Chartists expectation parameter (δ) 1 Chartists orders parameter (ϕ) 1 Market adjustment parameter (μ) Initial price (P 0 ) 100 Initial fundamental value (V 0 ) 100 Size of the shock (b) 0.1 As expected, if there are only arbitrageurs in the market, since arbitrageurs have heterogeneous beliefs about the asset fundamental value, after the shock, the asset price does not immediately go back to the asset fundamental value. Rather asset mispricing persists until all arbitrageurs know the asset true fundamental value, although there are no chartists in the market. The trading activity of arbitrageurs with unrealistic beliefs supports asset mispricing, since they do not know the true asset fundamentals. This is illustrated in Fig. 9. Fig. 9: Price dynamics when there are only arbitrageurs in the market (i.e., aa = 1). The asset fundamental value (asset price) is represented by the blue (red) line. b. Price dynamics with arbitrageurs and chartists. In this setting, I introduce chartists, trend followers as well as contrarian traders. As in the previous section, as a first step in the analysis of the interplay between arbitrageurs and chartists, I implement a setting in which chartists trading activity does not further affect the evolution of the asset price (benchmark 5). I thus focus first on the effect of arbitrageurs trading activity on price dynamics by varying the proportion of arbitrageurs in the market. The parameter values of the simulation are given in Table 5. 15

17 Table 5: Parameters for the simulation with arbitrageurs and chartists split into equal shares - Benchmark 5. Description of parameter (symbol) Value Description of parameter (symbol) Value Number of periods (t) 50 Date of the shock (t 0 ) 10 Date at which all arbitrageurs are informed (t i ) 20 Number of agents (N) 1000 Additional number of informed arbitrageurs in each period (n) 111 Proportion of arbitrageurs (aa) from 0 to 1 Proportion of trend followers (cc) 0.5 Proportion of contrarian traders (1-cc) 0.5 Arbitrageurs orders parameter (β) 1 Chartists expectation parameter (δ) 1 Chartists orders parameter (ϕ) 1 Market adjustment parameter (μ) Initial price (P 0 ) 100 Initial fundamental value (V 0 ) 100 Size of the shock (b) 0.1 In this setting, simulations reveal that first, as expected, the price correction occurs as soon as all arbitrageurs know the true asset fundamental value even though chartists are present in the market (aa varying from 1 to 0.8). Second while chartists may dominate (aa varying from 0.7 to 0.3), arbitrageurs trading activity still brings the asset price back towards the asset fundamental value. Price correction occurs almost as soon as all arbitrageurs know the true asset fundamental value. Besides, for proportion of arbitrageurs varying from 1 to 0.3, mispricing duration is decreasing in the proportion of arbitrageurs. Third asset mispricing markedly persists only for extremely high proportion of chartists in the market (aa varying from 0.2 to 0.1). Chartists trading activity may thus lengthen mispricing duration if chartists widely dominate 29. Finally, and not surprisingly, if there are no arbitrageurs in the market, the asset price stays at its post-shock level. These findings are illustrated in Fig. 10. Fig. 10: Price dynamics when chartists are split into equal shares (i.e., cc = 0.5) and the proportion of arbitrageurs (aa) varies from 1 to However with respect to benchmark 2 (i.e., arbitrageurs with realistic beliefs and same proportion of trend followers and contrarian traders), the effect of chartist s trading activity on mispricing duration is weaker in this case. While in benchmark 2, the mispricing duration increases by 50% and 80% with respect to benchmark 1 for aa [0.2; 0.1], respectively. In benchmark 5 (i.e., arbitrageurs with realistic as well as unrealistic beliefs and same proportion of trend followers and contrarian traders), the increase in mispricing duration with respect to benchmark 4 is lower than before (i.e., 30% and 40% for aa [0.2; 0.1], respectively). 16

18 The second step of this analysis is to investigate the price dynamics when chartists trading activity further affects the evolution of the asset price (benchmark 6). This is done by varying the proportion of trend followers (cc varying from 0 to 1) as well as the proportion of arbitrageurs in the market. This setting allows investigating whether chartists trading activity may prevent arbitrageurs from correcting asset mispricing, while chartists do not dominate. In this setting, it is confirmed that as long as arbitrageurs widely dominate (with aa varying from 1 to 0.7), whatever the proportion of trend followers (cc varying from 0 to 1), price correction occurs almost as soon as all arbitrageurs know the true asset fundamental value. Besides, mispricing duration is increasing in the proportion of trend followers (or contrarian traders) 30. These findings are illustrated in Fig. 11 (with aa = 0.7) 31. Fig. 11: Price dynamics when the proportion of arbitrageurs = 0.7 and the proportion of trend followers (cc) varies from 0 to 1. However, contrary to existing results (e.g., Lux, 1995, 1998; Farmer and Joshi, 2002; De Grauwe and Grimaldi, 2005, 2006), the simulation reveals that while arbitrageurs still dominate (with aa = 0.6), price correction starts to be markedly delayed, with respect to benchmark 5 (i.e., cc = 0.5), if chartists are trend followers only (i.e., cc = 1) 32 or contrarian traders only 33 (i.e., cc = 0). Moreover immediately after the shock on the asset price, while arbitrageurs still dominate, asset price starts to overreact 34 when chartists are mainly trend followers (i.e., cc > 0.5). These findings are illustrated in Fig Moreover from aa = 0.7, first mispricing duration is more lengthen if contrarian traders dominate trend followers (i.e., cc < 0.5) than if trend followers dominate contrarian traders (i.e., cc > 0.5). Second there is a nonlinear relation between mispricing duration and proportion of trend followers (or contrarian traders). 31 See Fig. 6 7 in Annexes for aa varying from 0.8 and 0.9, respectively. 32 If chartists are trend followers only, mispricing duration is lengthen by 43% with respect to benchmark If chartists are contrarian traders only, mispricing duration is lengthen by 52% with respect to benchmark The price overreaction may consist in a rise of about 2% (with cc = 0.7) due to trend followers trading activity. While this figure may seem small, price overreaction is much larger for lower proportion of arbitrageurs in the market. Most importantly, price overreaction occurs for lower values of cc (i.e., cc = 0.7) with respect to benchmark 3. 17

19 Fig. 12: Price dynamics when the proportion of arbitrageurs = 0.6 and the proportion of trend followers (cc) varies from 0 to 1. Simulations reveal that as long as arbitrageurs dominate (aa varying from 1 to 0.6), arbitrageurs trading activity still brings the asset price back towards the asset fundamental value. However first mispricing duration is increasing in the proportion of trend followers (or contrarian traders). Second while arbitrageurs still dominate, for extreme values of the proportion of trend followers (with cc = 1, i.e., chartists are trend followers only, or cc = 0, i.e., chartists are contrarian traders only), price correction may be markedly delayed. Besides, the asset price starts to overreact due to trend followers trading activity. Furthermore while chartists still do not dominate (i.e., aa = 0.5) 35, price correction may never occur if chartists are contrarian traders only. The initial shock on the asset price is followed by an infinite sequence of endogenous fluctuations of the asset price around the asset fundamental value. Contrarian traders may also destabilize asset price and prevent arbitrageurs from correcting asset mispricing. Finally price overreaction, after the shock, is even more pronounced when agents in the market arbitrageurs and chartists are split into equal shares 36 (i.e., aa = 0.5). These findings are illustrated in Fig Fig. 13: Price dynamics when the proportion of arbitrageurs = 0.5 and the proportion of trend followers (cc) varies from 0 to These results hold for lower values of proportion of arbitrageurs (i.e., aa [0.5; 0.2]). 36 After the shock, the asset price may overreaction by 3% (with cc = 1). For extreme values of proportion of arbitrageurs (i.e., aa = 0.1), price overreaction may reach 8%. 37 See Fig in Annexes for aa varying from 0.4 to 0.2, respectively. 18

20 On the contrary, when chartists mainly trend followers (with cc = 0.7) widely dominate (i.e., aa 0.2), mispricing duration may be shortened with respect to benchmark 5 (i.e., chartists are split into equal shares) 38. Trend followers trading activity may further price correction as trend followers amplify the trend in prices initiated by arbitrageurs. These findings are illustrated in Fig. 14 and Fig. 15. Fig. 14: Price dynamics when the proportion of arbitrageurs = 0.1 and the proportion of trend followers = 0.5. Price correction occurs in t = 43. Fig. 15: Price dynamics when the proportion of arbitrageurs = 0.1 and the proportion of trend followers = 0.7. Price correction occurs in t = 33. The asset fundamental value (asset price) is represented by the blue (red) line. Finally, and not surprisingly, if there are no arbitrageurs in the market, it is confirmed that asset mispricing never corrects. As in the settings where arbitrageurs have realistic beliefs about the asset fundamental value, while if chartists are contrarian traders only, the asset price fluctuates between the asset price post-shock level and the asset fundamental value, if there are trend followers only, the asset price departs even further from the asset fundamental value 39. This finding is illustrated in Fig For aa = 0.2 and cc [0.6, 0.7], mispricing duration is shortened by 10%, while for lower values (i.e., aa = 0.1) and cc = 0.7, mispricing duration is shortened by more than 20%, with respect to benchmark 5, due to trend followers trading activity. 39 If chartists are equally divided between trend followers and contrarian traders, the asset s price remains at its post shock level. 19

21 Fig. 16: Price dynamics when there are no arbitrageurs in the market (i.e., aa = 0) and the proportion of trend followers (cc) varies from 0 to 1. c. Comparison of settings when arbitrageurs have heterogeneous beliefs about the asset fundamental value. In order to investigate under which conditions arbitrageurs trading activity brings the asset price back towards the asset fundamentals, while they may have heterogeneous beliefs about the asset fundamental value, I explored several situations. These settings have allowed investigating independently the effect of each trading strategy on the price dynamics and eventually, starting from documented results (found out in benchmarks 4 and 5), the price dynamics induced by the interplay between arbitrageurs and chartists. In the first setting (benchmark 4), there are arbitrageurs only with realistic as well as unrealistic beliefs about the asset fundamentals. In the second setting, I introduced chartists, both trend followers and contrarian traders. The setting where chartists trading activity does not further affect the price dynamics (benchmark 5) allowed focusing on the effect of arbitrageurs trading activity on the evolution of the asset price. Finally, in the third setting, chartists trading activity may further affect the evolution of the asset price (benchmark 6). I have eventually studied the price dynamics induced by the interplay between arbitrageurs and chartists, while arbitrageurs have heterogeneous beliefs about the asset fundamentals. The main findings of each benchmark are summarized in Table 6. 20

22 Table 6: Main findings of each benchmark when arbitrageurs have heterogeneous beliefs about the asset fundamental value. Main features Main findings 40 Benchmark 4 Arbitrageurs only Mispricing correction occurs as soon as all arbitrageurs are informed of the true asset fundamental value. Benchmark 5 Arbitrageurs and chartists Chartists trading activity does not further affect the evolution of the asset price (i.e., cc = 0.5). Mispricing duration is decreasing in the proportion of arbitrageurs in the market; Even if chartists trading activity does not further affect the evolution of the asset price, asset mispricing may persist; Mispricing correction is markedly delayed only if chartists dominate. Benchmark 6 Arbitrageurs and chartists Chartists trading activity may further affect the evolution of the asset price (i.e., cc [0, 1]). Mispricing duration is increasing in the proportion of trend followers (or contrarian traders) in the market; While arbitrageurs dominate the market, mispricing correction may be markedly delayed (if chartists are trend followers only or contrarian traders only); If chartists dominate: 1. Trend followers trading activity may reduce mispricing duration. 2. Trend followers trading activity may induce price overreaction after the shock on the asset price. The simulations presented above have revealed that while chartists do not dominate, chartists trading activity may prevent arbitrageurs from correcting asset mispricing whenever 40 In bold the main differences with respect to the benchmarks in which all arbitrageurs have realistic beliefs about the asset s fundamental value (i.e., benchmarks 1 3). 21

23 arbitrageurs have heterogeneous beliefs about the asset fundamental value. Chartists trading activity may markedly lengthen mispricing duration and explain mispricing persistence, while arbitrageurs are aware of asset mispricing. However, and more surprisingly, when chartists dominate, trend followers trading activity may further price correction. 3. SUMMARY FINDINGS AND CONCLUDING REMARKS In this paper, I have presented and discussed a simple financial heterogeneous agent model with two types of boundedly rational agents, namely: arbitrageurs (or fundamentalists) and chartists. The purpose of the paper was to investigate under which conditions chartists trading activity may prevent arbitrageurs from correcting asset mispricing. The results of the simulations have illustrated how chartists trading activity may explain mispricing persistence, while arbitrageurs are present in the market. Our study confirmed that first when arbitrageurs have realistic beliefs about the asset fundamentals, when chartists dominate, chartists trading activity may destabilize asset price and prevent arbitrageurs from bringing the asset price towards the asset fundamentals. Besides, investigating a more general setting with contrarian traders as well, earlier results may be extended to the effect of contrarian traders trading activity on the price dynamics. Contrarian traders may destabilize asset prices and prevent arbitrageurs from bringing the asset price towards the asset fundamentals. However, when arbitrageurs have heterogeneous beliefs about the asset fundamental value, the simulations reveal that chartists trading activity may destabilize the asset price even when chartists do not dominate. Besides, after the shock on the asset price, the latter markedly overreacts when trend followers dominate contrarian traders. On the contrary, when chartists dominate, trend followers trading activity may reduce mispricing duration. Trend following strategies tend to amplify price trends induced by arbitrageurs trading activity. With some basic mechanisms now clear, I believe that extensions of the present model can incorporate other features and provide a useful framework for more formal tests. A key element missing from chartists behavior studied here is strategy switching mechanism. This would have very profound effects on price dynamics. It would produce complex dynamics and allow generating stylized facts observed in financial time series such as excess volatility, excess kurtosis and/or clustered volatility. In the future paper, I will present some results that include a switching mechanism between trend followers and contrarian traders. The aim of this work is twofold. First this will allow investigating whether chartists trading activity, unpredictable by arbitrageurs, may prevent arbitrageurs from getting to know the fundamentals. Second this will imply a more sophisticated learning process among arbitrageurs, through which information diffusion would be affected by chartists trading activity. 22

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27 Figure 1 Price dynamics when the proportion of arbitrageurs = 0.9 and the proportion of trend followers (cc) varies from 0 to 1. Figure 2 Price dynamics when the proportion of arbitrageurs = 0.8 and the proportion of trend followers (cc) varies from 0 to 1. Figure 3 Price dynamics when proportion of arbitrageurs = 0.7 and the proportion of trend followers (cc) varies from 0 to 1. Figure 4 Price dynamics when the proportion of arbitrageurs = 0.5 and the proportion of trend followers (cc) varies from 0 to 1. Figure 5 Price dynamics when the proportion of arbitrageurs = 0.3 and the proportion of trend followers (cc) varies from 0 to 1. 26

28 Figure 6 Price dynamics when the proportion of arbitrageurs = 0.8 and the proportion of trend followers (cc) varies from 0 to 1. Figure 7 Price dynamics when the proportion of arbitrageurs = 0.9 and the proportion of trend followers (cc) varies from 0 to 1. Figure 8 Price dynamics when the proportion of arbitrageurs = 0.4 and the proportion of trend followers (cc) varies from 0 to 1. Figure 9 Price dynamics when the proportion of arbitrageurs = 0.3 and the proportion of trend followers (cc) varies from 0 to 1. Figure 10 Price dynamics when the proportion of arbitrageurs = 0.2 and the proportion of trend followers (cc) varies from 0 to 1. 27