theory and narrow framing. The momentum phenomenon found by Jegadeesh and Titman(1993)has been well analyzed in the behavioral finance literature.In the Barberis,Shleifer,and Vishny (1998,BSV henceforth) model,momentum can occur because of the investors'conservatism.Hong and Stein (1999) explicitly add momentum traders-traders buying stocks after a price increase-to their model. Many other researchers,including DSSW and Cutler,Summers,and Poterba (1990),have also investigated momentum trading or positive feedback trading.The simplest way of motivating positive feedback trading is extrapolative expectations.Namely,as investors form expectations by extrapolating trends,they buy into price trends.This can be due to some important psychological biases of investors,including representativeness and the law of small numbers (Barberis and Thaler,2003). Robert J.Shiller(2002,pl4)has the following vivid description on momentum trading or feedback trading: When speculative prices go up,creating successes of some investors,this may attract public attention,promote word-of-mouth enthusiasm,and heighten expectations for further price increases....This process in turn increases investor demand,and thus generates another round of price increases....The high prices are ultimately not sustainable,since they are high only because ofexpectations of further price increases.... We will demonstrate that we are able to generate price momentum in our model even without momentum traders.It is worth noting,although momentum trading/positive trading is not needed to obtain our major results,to make our model more realistic and more flexible,we do not rule out the possibility of momentum trading in the following analyses. Assumption 2.There is a manipulator in the market who is a large market player and is able to influence the asset price.In other words,the manipulator is a price-setter rather than a price-taker.He enters the market at time I without any initial endowment of the speculative asset. At each period of time tz1,the manipulator sets a price target for that period and then submits 10
10 theory and narrow framing. The momentum phenomenon found by Jegadeesh and Titman (1993) has been well analyzed in the behavioral finance literature. In the Barberis, Shleifer, and Vishny (1998, BSV henceforth) model, momentum can occur because of the investors’ conservatism. Hong and Stein (1999) explicitly add momentum traders—traders buying stocks after a price increase—to their model. Many other researchers, including DSSW and Cutler, Summers, and Poterba (1990), have also investigated momentum trading or positive feedback trading. The simplest way of motivating positive feedback trading is extrapolative expectations. Namely, as investors form expectations by extrapolating trends, they buy into price trends. This can be due to some important psychological biases of investors, including representativeness and the law of small numbers (Barberis and Thaler, 2003). Robert J. Shiller (2002, p14) has the following vivid description on momentum trading or feedback trading: When speculative prices go up, creating successes of some investors, this may attract public attention, promote word-of-mouth enthusiasm, and heighten expectations for further price increases. … This process in turn increases investor demand, and thus generates another round of price increases. … The high prices are ultimately not sustainable, since they are high only because of expectations of further price increases. … We will demonstrate that we are able to generate price momentum in our model even without momentum traders. It is worth noting, although momentum trading/positive trading is not needed to obtain our major results, to make our model more realistic and more flexible, we do not rule out the possibility of momentum trading in the following analyses. Assumption 2. There is a manipulator in the market who is a large market player and is able to influence the asset price. In other words, the manipulator is a price-setter rather than a price-taker. He enters the market at time 1 without any initial endowment of the speculative asset. At each period of time t ≥ 1, the manipulator sets a price target for that period and then submits
his order to clear the market at the target price. The assumption that the manipulator is a large trader is conventional in the literature on trade-based manipulation.In order to move the market with strategic trading,the manipulator must have the power to influence the price (see Jarrow(1992)and Allen and Gale (1992)).In Table 1,we provide many historical cases where many investors,such as wealthy individuals or a group of investors(e.g.,Vanderbilt during the Harlem Railroad corner (see Allen and Gale (1992)),large hedge funds,and trading pools in the US history)can be classified as large traders. These traders often not only have deep pockets,but also are influential in the securities markets. Assumption 3.There is also a continuum number of arbitrageurs,with measure 1,enters the market at time t=1.They are price-takers and trade shares of the speculative asset based on recent price movements.If the price moves up in the current period,they sell some shares to take profits.If the price goes down,they buy.Formally,they submit the following orders at time t: D.=-a(e-P)=-a(△P） 0 where a-0. Although the new trades of the arbitrageurs in each period only depend on short term price movements,the total position of the speculative asset held by the arbitrageurs,O,,is negatively proportional to price deviation from fundamentals.This is because the arbitrageurs have already heldf--al，化-P-）上-a-P）shres ofeivee t-1,if they buy additional -a(p-P)shares at time t,the total position of the speculative asset held by them will be ,=-a(P-P)shares.The arbitrageurs play two roles in our model.First,they provide necessary liquidity to the market so that trading can take place at equilibrium for each period.For instance,if the manipulator wants to move the asset price up by submitting a purchasing order,there must be some investors selling sufficient number of shares of the speculative asset.Because the behavior-driven investors in a sense are momentum 11
11 his order to clear the market at the target price. The assumption that the manipulator is a large trader is conventional in the literature on trade-based manipulation. In order to move the market with strategic trading, the manipulator must have the power to influence the price (see Jarrow (1992) and Allen and Gale (1992)). In Table 1, we provide many historical cases where many investors, such as wealthy individuals or a group of investors (e.g., Vanderbilt during the Harlem Railroad corner (see Allen and Gale (1992)), large hedge funds, and trading pools in the US history) can be classified as large traders. These traders often not only have deep pockets, but also are influential in the securities markets. Assumption 3. There is also a continuum number of arbitrageurs, with measure 1, enters the market at time t=1. They are price-takers and trade shares of the speculative asset based on recent price movements. If the price moves up in the current period, they sell some shares to take profits. If the price goes down, they buy. Formally, they submit the following orders at time t: ( ) ( ) Da,t = −α Pt − Pt−1 = −α ∆Pt (1) where α>0. Although the new trades of the arbitrageurs in each period only depend on short term price movements, the total position of the speculative asset held by the arbitrageurs, Qt , is negatively proportional to price deviation from fundamentals. This is because the arbitrageurs have already held a portfolio of ( ) ( ) 1 0 1 1 Q 1 P P 1 Pt P t j t = − j − j = − − − − = − ∑ α − α shares of the speculative asset at time t −1, if they buy additional ( ) −α Pt − Pt−1 shares at time t , the total position of the speculative asset held by them will be ( ) Qt = −α Pt − P0 shares. The arbitrageurs play two roles in our model. First, they provide necessary liquidity to the market so that trading can take place at equilibrium for each period. For instance, if the manipulator wants to move the asset price up by submitting a purchasing order, there must be some investors selling sufficient number of shares of the speculative asset. Because the behavior-driven investors in a sense are momentum
followers,a new class of investors is therefore needed in the model.Second,our model rules out fundamental risk.The arbitrageurs'trading strategy ensures that the price of speculative asset will not move away from fundamentals explosively.We call a the arbitrage parameter and will discuss its meaning and implication further in the next section. Assumption 4.Although the manipulator enters the market at time 1,the market already existed at time 0.The price of the speculative asset at time 0 was P,which was equal to the fundamental value of the asset.There were q behavior-driven investors who held one share of the speculative asset per person at the market close of day 0. The manipulator tends to move the asset price up by a fixed amount of >0 for t (t >1)consecutive periods from day I to day t.That is -P-=6>0,t=1,2,,1 2) By the close of day t,the manipulator has accumulated certain number of shares of the speculative asset.He starts liquidating his shares from day t+I and keeps doing so until he has sold all of his shares by time T-l for some T>t +1.We define t=T-1-t as the length of time the manipulator takes to liquidate his shares accumulated by time t. In order to ensure market equilibrium for each day,8 shall satisfy certain condition as discussed subsequently.Assumption 4 is not the only possible assumption that can make manipulation profitable,but is a simple one. Assumption 5.The manipulator leaves the market right after he has sold all his shares at T-1. The market ends at time T and by then investors receive a liguidating dividend of Po for each share of the speculative asset. Assumption 5 is not really needed for discussing the manipulation issue in the model.We 3
12 followers, a new class of investors is therefore needed in the model. Second, our model rules out fundamental risk. The arbitrageurs’ trading strategy ensures that the price of speculative asset will not move away from fundamentals explosively. We call α the arbitrage parameter and will discuss its meaning and implication further in the next section. Assumption 4. Although the manipulator enters the market at time 1, the market already existed at time 0. The price of the speculative asset at time 0 was P , which was equal to the 0 fundamental value of the asset. There were 1 q behavior-driven investors who held one share of the speculative asset per person at the market close of day 0. The manipulator tends to move the asset price up by a fixed amount of δ > 0 for ut (t ) consecutive periods from day 1 to day u > 1 ut . That is 1 0, 1,2,..., . PP t t tt u δ − => = − (2) By the close of day ut , the manipulator has accumulated certain number of shares of the speculative asset. He starts liquidating his shares from day ut +1 and keeps doing so until he has sold all of his shares by time T-1 for some T . We define > tu +1 d u t = T −1− t as the length of time the manipulator takes to liquidate his shares accumulated by time ut . In order to ensure market equilibrium for each day, δ shall satisfy certain condition as discussed subsequently. Assumption 4 is not the only possible assumption that can make manipulation profitable, but is a simple one. Assumption 5. The manipulator leaves the market right after he has sold all his shares at T-1. The market ends at time T and by then investors receive a liquidating dividend of P for each 0 share of the speculative asset. Assumption 5 is not really needed for discussing the manipulation issue in the model. We
make this assumption here following the convention in the literature and the widespread belief that in the long run,fundamental rules.The assumption is useful in discussing certain asset price anomalies such as long-term reversal.It is easy to see from assumptions 3 and 5 that the net purchases of arbitrageurs are zero over the whole time periods. Here,we assume the speculative asset has no fundamental risk.We also assume that there is no heterogeneous information.This does not mean that fundamental risks and information asymmetry are not important in the real market or in market manipulation.Rather,we use this simplified setup to highlight the manipulator's trading strategies when the market is not fully rational.With this simple framework,we demonstrate that manipulation is possible even if there is no information asymmetry on asset fundamentals. 3.Results and Interpretations To solve the model,we first find out the accumulated holding of the speculative asset by the manipulator at the market close by time t.We have the following proposition. Proposition 1:By the market close at day t,the manipulator has accumulated N=a.t. shares of the speculative asset with an average cost of P per share. Proof:By Assumptions I to 3,it follows immediately that for each period t,such as 1sts the manipulator shall buy a.8 shares at a price of P+to.A simple calculation yields the statement in Proposition 1. Proposition 1 highlights the important impact of arbitrage on the manipulator's trading strategy.To move the price of the speculative asset by an amount of the manipulator must purchase a.8 shares of the asset.If a is sufficiently large,the manipulator must have a very deep pocket to move the market.Put another way,when there is no limit of arbitrage,namely, a->o,it is almost impossible for the manipulator to "pump-and-dump"the speculative asset. 13
13 make this assumption here following the convention in the literature and the widespread belief that in the long run, fundamental rules. The assumption is useful in discussing certain asset price anomalies such as long-term reversal. It is easy to see from assumptions 3 and 5 that the net purchases of arbitrageurs are zero over the whole time periods. Here, we assume the speculative asset has no fundamental risk. We also assume that there is no heterogeneous information. This does not mean that fundamental risks and information asymmetry are not important in the real market or in market manipulation. Rather, we use this simplified setup to highlight the manipulator’s trading strategies when the market is not fully rational. With this simple framework, we demonstrate that manipulation is possible even if there is no information asymmetry on asset fundamentals. 3. Results and Interpretations To solve the model, we first find out the accumulated holding of the speculative asset by the manipulator at the market close by time ut . We have the following proposition. Proposition 1: By the market close at day ut , the manipulator has accumulated N = α ⋅tu ⋅δ shares of the speculative asset with an average cost of δ + + 2 1 0 ut P per share. Proof: By Assumptions 1 to 3, it follows immediately that for each period t, such as u 1 , ≤ t ≤ t the manipulator shall buy α ⋅δ shares at a price of P0 + tδ . A simple calculation yields the statement in Proposition 1. ■ Proposition 1 highlights the important impact of arbitrage on the manipulator’s trading strategy. To move the price of the speculative asset by an amount of δ , the manipulator must purchase α ⋅δ shares of the asset. If α is sufficiently large, the manipulator must have a very deep pocket to move the market. Put another way, when there is no limit of arbitrage, namely, α → ∞ , it is almost impossible for the manipulator to “pump-and-dump” the speculative asset
Therefore,the assumption of the "limits of arbitrage"is essential for the manipulator's trading strategy to work.Proposition 1 also suggests that,the higher the original price of the asset,the more money the manipulator needs to put up for purchasing the shares.This implies,ceteris paribus,small cap stocks are more likely to be subject to price manipulation We first consider a simple but interesting case in which behavior-driven investors are extremely unwilling to take losses,namely,q3 =0.This is a strong implication of the dispositional effect that has been supported by several empirical studies,such as Odean(1998) and Grinblatt and Han (2001) Proposition 2:If q3 =0,then the manipulator can sell his shares at a high price P=P+t 8 from time t=t +1 through time t=T-1 by appropriately choosing a positive 8.By doing so, the manipulator's total profit is x()-ao The trading volume stays at a+g shares per period from time t=1 to time t=t.From time t=t +1 to time t=T-1,trading volume per period is q2 shares--the manipulator sells q2 shares to new behavior-driven investors each period. Proof:Set Because0 behavior-driven investors will not sell their shares ta without a profit.The manipulator is able to sell g2 shares to the new behavior-driven investors each period from day t=t+1 through time t=T-1 by maintaining the equilibrium price at P.=P+1.The average selling price is P per share.As a result,the manipulator's total profit is =-+-a小 (3)
14 Therefore, the assumption of the “limits of arbitrage” is essential for the manipulator’s trading strategy to work. Proposition 1 also suggests that, the higher the original price of the asset, the more money the manipulator needs to put up for purchasing the shares. This implies, ceteris paribus, small cap stocks are more likely to be subject to price manipulation. We first consider a simple but interesting case in which behavior-driven investors are extremely unwilling to take losses, namely, 0 q3 = . This is a strong implication of the dispositional effect that has been supported by several empirical studies, such as Odean (1998) and Grinblatt and Han (2001) Proposition 2: If q , then the manipulator can se 3 = 0 ll his shares at a high price P3 = P0 + tuδ from time t through time = tu +1 t = T −1 by appropriately choosing a positive δ. By doing so, the manipulator’s total profit is 2 2 1 2 1 δ α ⋅δ − = ⋅ − ⋅ u u u t t t N . The trading volume stays at 1 αδ + q shares per period from time t = 1 to time u t = t . From time t to time = tu +1 t = T −1, trading volume per period is 2 q shares--the manipulator sells 2 q shares to new behavior-driven investors each period. Proof: Set α δ ⋅ ⋅ = u d t t q2 . Because 0 q3 = , behavior-driven investors will not sell their shares without a profit. The manipulator is able to sell 2 q shares to the new behavior-driven investors each period from day 1 t = tu + through time t = T −1 by maintaining the equilibrium price at Pt P tuδ u = 0 + . The average selling price is u Pt per share. As a result, the manipulator’s total profit is 2 0 2 1 2 1 π δ α ⋅δ − = ⋅ + = ⋅ − + u u u t t t t N P P u (3)