ness School Finance Workshop, at the Europlace Institute of Finance Conference, at the ... informed traders and N realistic informed traders trade a risky asset.
OPTIMISTIC & PESSIMISTIC TRADING IN FINANCIAL MARKETS
Laurent Germain Toulouse Business School, SUPAERO and Europlace Institute of Finance Fabrice Rousseau Department of Economics National University of Ireland, Maynooth Anne Vanhems Toulouse Business School
PRELIMINARY VERSION
WORK IN PROGRESS
JULY 2005
0
We thank Hayne Leland, Jean Charles Rochet, Jean Claude Gabillon, Stefano Lovo, Denis Hilton, Elyes Jouini, Christophe Bisi`ere and the seminar participants at the Toulouse Business School Finance Workshop, at the Europlace Institute of Finance Conference, at the 3rd International Finance Conference, at the Global Finance Conference and at Maynooth.
1
Abstract
There is an abundant literature in finance on overconfidence, however there exists a different psychological trait well known to financial practitioners and psychologists [see Hilton at al. (2004)] which is optimism. This trait has received little attention. Our paper analyses the consequences of optimism and pessimism on financial markets. We develop a general model of optimism/pessimism where M unrealistic informed traders and N realistic informed traders trade a risky asset with competitive market makers. unrealistic traders can (i) misperceive the expected returns of the risky asset (scenario 1) or (ii) in addition to the previous can make a judgemental error on both the volatility of the asset returns and the variance of the noise in his/her private signal (scenario 2). We show, in scenario 1, that optimistic traders purchase more or sell smaller quantities whereas pessimistic traders sell more or purchase smaller quantities than if they were realistic. As market makers correctly anticipate that, we obtain that (i) the price level and the market depth are equal to the ones predicted by a standard model a la Kyle (1985) and (ii) unrealistic and realistic traders obtain the same expected profit. In scenario 2, we show that (i) unrealistic traders may earn negative, higher or lower expected profit than realistic traders, (ii) the expected profit of the realistic and the unrealistic trader is a non-monotonic function of the two degrees of error, and (iii) market depth is also a non-monotonic function of the error on the volatility of asset as well as the error on the variance of the noise. All of the above results are derived when market makers are realistic. We show that the results are not altered if market makers are themselves optimistic or pessimistic. The expected profit for the unrealistic market makers can either be positive or negative.
2
1
Introduction
Economic and financial theory have widely used the assumption that agents behave rationally. Such an assumption has led to the failure of explaining some properties observed in markets like (i) the low responsiveness or sometimes high responsiveness of the price to new information [Ritter (1991) and Womack (1996)], (ii) the excessive volume traded [Dow and Gorton (1997)], (iii) underreaction or overreaction of market participants [Debondt and Thaler (1985)], and (iv) the excessive volatility observed in financial markets [Shiller (1981, 1989)]. In order to try to explain these properties, financial economists have departed from the rationality assumption and have instead assumed that investors may have some psychological traits which would lead them to behave irrationally. An abundant literature in financial economics assumes that investors are overconfident, i.e. investors tend to overestimate the probability that their judgments are correct. This phenomenon has been referred to as “miscalibration”.1 In order to model that behavior, it is assumed that investors overestimate the precision of their private information. Most of that literature predicts that overconfident investors trade to their disadvantage. In other words, overconfident investors get lower expected profit than their rational counterpart [Odean (1998b), Gervais and Odean (1999), Caball´e and S´akovics (2003), Biais et al. (2004) among others]. However, Kyle and Wang (1997) and Benos (1998) find that overconfident traders may earn larger expected profit than rational ones. Moreover, a common finding to all these papers except Caball´e and S´akovics (2003) is that trading volume, price volatility as well as price efficiency increase with the level of overconfidence. There is a large body of evidence in the psychology literature that people, with good mental health, do not have accurate perception of themselves and their surrounding world. People’s perceptions have a tendency to be positively biased, i.e. people hold “positive illusions”. “Positive illusions” have been widely documented in psychology [Taylor and Brown (1988, 1994), Langer (1974) and Weinstein (1980), to name but a few]. Taylor and Brown (1988, 1994) analyze the “better than average” effect, Langer (1974) focuses on the illusion of control i.e. the individual’s tendency to overestimate the control they have over outcomes, whereas Weinstein (1980) looks at unrealistic optimism. Unrealistic optimism is defined as the people’s tendency to systematically overestimate the probability that good things will happen to them and, at the same time, to underestimate the probability that bad things will happen.2 One way psychologists test the presence of that trait is to ask their subjects how they behave in bad times. If in bad times they expect the best, they can be considered as optimistic. This trait seems to characterize some of the investors’ behavior in financial markets. Moreover, Hilton et al. (2004) show that overconfidence and “positive illusions” 1
See Hilton et al. (2001) and Biais et al. (2004). Harris and Middleton (1994), and Hoorens (2001) look at the “comparative optimism”. These two papers focus on the illusion of control and optimistim about health: people think on being less at risk but no more in control than others. 2
3
differ even if those two traits share some characteristics in both cases the agent over-value its ability. This optimistic psychological trait is already well known to financial practitioners. Indeed they interchangeably use optimistic market for bullish market and pessimistic market for bearish market. Another proof is the recent launch in October 1996 ,by the Union des Banques Suisses and Gallup Organization, of an index measuring the level of optimism in the American market, the Index of Investor Optimism.3 Our study belongs to the “positive illusions” strand of the literature where unrealistic and realistic traders disagree on both the mean and the variance of the returns of the risky asset. Indeed, optimistic (pessimistic) traders overestimate (underestimate) the mean and underestimate (overestimate) the variance of the returns of the risky asset. Being optimistic is believing that (i) the expected value of the asset is greater and less volatile than what would predict the analysis of its fundamental value and (ii) the precision of one’s own information is greater than it actually is. Few papers have looked at the effect of optimism. B´enabou and Tirole (2002) look at the value of self-confidence for rational agents and at their behavior used to enhance it. “Positive thinking” is found to improve welfare despite the fact that it can be self-defeating. In Brocas and Carrillo (2004), optimism about the chance of success of a project may lead entrepreneurs to invest in it without gathering further information. This entrepreneurial optimism leads to excessive investment. De Mezza and Southey (1996) find that entrepreneurs self select from the part of the population displaying an optimistic bias. Manove and Padilla (1999) look at the screening problem faced by bankers in order to separate optimistic entrepreneurs from realistic ones. Optimistic entrepreneurs have perceptions biased by wishful thinking. They show that, because of the existence of optimistic entrepreneurs, competitive banks may not be sufficiently conservative in their lendings. In Manove (2000), some entrepreneurs are unrealistically optimistic about their firms productivity. He shows that unrealistic entrepreneurs may earn more than realistic entrepreneurs and may even drive out of business all realistic entrepreneurs. This result is echoed by Heifetz and Spiegel (2004) in a different framework. In a game where agents meet per pair and then interact, they show that agents displaying optimism or pessimism will not disappear in the long run but will takeover the entire population. That result contradicts most of the result obtained in the overconfident literature. In Heifetz and Spiegel (2004), optimistic (pessimistic) agents over- (under-) estimates the impact of their actions. Our study differ from the previous studies on optimism as none of the above works look at the impact on financial markets of having optimistic (pessimistic) traders. Cornelli et al. (2005) is the only other work, we are aware of, looking at the possibility for a subset of investors to be optimistic or pessimistic. They establish the existence of unrealistic traders possibly optimistic or pessimistic in the grey market.(pre-IPO market). In our paper, we assume the existence of a subset of informed investors being optimistic and/or pessimistic 3
The methodology used www.ubs.com/investoroptimism.
to
compute
4
that
index
can
be
found
at
and derive the equilibrium and its properties under that assumption. We study, in a Kyle (1985) framework, a financial market where optimistic (pessimistic) investors as well as realistic traders are present. In that respect, it is similar to the studies done in Benos (1998) and Kyle and Wang (1997) for overconfident traders. We model the optimistic behavior in two different ways. First, we measure the effect of additive misperception. Additive misperception refers to the misperception of the average return of the risky financial asset. The risky asset is normally distributed with mean 0 and variance σv2 , optimistic (pessimistic) traders perceive that the average is not 0 but a positive (negative) parameter a (additive misperception). We find that optimistic (pessimistic) traders purchase (sell) larger quantities or sell (purchase) smaller quantities. This additive part of the quantity is proportional to the misperception a. The price and market depth are equal to the one predicted by a standard model a la Kyle (1985). The market maker being rational, she knows that the information contained in the aggregate order flow is not affected by the presence of unrealistic traders. As a consequence, she rationally prices the aggregate order flow. In this case, the expected profits of the realistic and the unrealistic traders are the same. Second, we measure the effects of combining additive and multiplicative misperception. Multiplicative misperception is defined by two parameters κ1 and κ2 which altogether characterize the degree of misperception. Indeed, optimistic (pessimistic) traders under-scale (over-scale) the volatility of the risky asset σv2 by a parameter κ1 and under-scale (over-scale) the precision of their own signal σε2 by a parameter κ2 . As an overconfident trader, an optimistic trader over-value the precision of his own information. This is measured by the parameter κ2 . The lower is κ1 , the lower the variance of prior information anticipated by the unrealistic investors.4 We look at three different situations. In the first one, realistic market makers trade with realistic traders and with only one type of unrealistic traders. In the second one, realistic market makers trade with the realistic traders and with the two types of unrealistic (optimists and pessimists). The third situation studies the case where the market makers are themselves optimistic or pessimistic. We show that, for the three situations, an equilibrium does not always exist. The misperception of the variances lead both the unrealistic and the realistic traders (as a reaction to the former’s behavior) to distort their responsiveness to private information. Indeed, as a result to misperceiving variances unrealistic traders may over-trade too much on their private information leading the realistic traders to leave the market, implying its closure. We find that unrealistic traders may trade more or less on their private information than their realistic counterpart. If this overtrading is not too large, they benefit from it as their expected profit is larger than the one obtained by realistic traders. If that overtrading is too large, they may earn in expected term less than the realistic not ruling out 4
In Odean (1998b) those two parameters define the level of overconfidence. Nevertheless, the author does not study the impact of the variations of those parameters on the traders and Odean (1998b) assumes that κ1 is greater than one whereas we assume that for optimistic traders it is lower than one.
5
negative expected profit. We also give some comparative statics on the expected profit. We find that both expected profit for the realistic and unrealistic can be a non-monotonic function of both degree of errors made on the two variances. In the same vein, the liquidity parameter can also be non-monotonic with respect to κ1 and κ2 . In all three situations, the traders’ unrealism and the market makers’ unrealism have two distinct effects on the price function: (i) the misperception of the expectation implies a downward or an upward shift of the price function, (ii) the misperceptions of the variances affects the slope of the price function ,i.e. the market depth, as well as the size of the shift of the price function. The first effect can be explained as follows. Optimistic (pessimistic) traders increase (decrease) the size of their market orders due to the misperception of the expectation, i.e. if they received a positive signal they purchase a larger (smaller) quantity whereas if they received a negative signal they sell a smaller (larger) quantity. If the overall effect of the traders’ misperception is to increase the aggregate order flow, the rational market makers’ reaction is to decrease the price function by an amount proportional to this extra positive aggregate order flow as this order flow is not correlated with the value of the risky asset. If the market makers are themselves optimistic or pessimistic, their misperception of the expectation has the opposite effect to the effect of the traders’ misperception. An optimistic (pessimistic) market maker shifts up (down) the price function. The second effect si due to the fact that both the unrealistic traders and the realistic traders may trade more or less on their private information. Knowing that, a rational market maker adjusts the market depth accordingly. The market depth is also affected by the market makers’ misperception of the variance of prior information. Optimistic (pessimistic) market makers set a higher (smaller) market depth than realistic ones. Indeed, optimistic (pessimistic) market makers think that the prior information is more (less) precise than it is and therefore believes that the informed private information is less (more) substantial than it is in reality. As a consequence, she adjusts her price less (more) aggressively. This is done by reducing (increasing) the liquidity parameter, λ, and therefore by increasing (decreasing) market depth. As a response informed traders trade large quantities. Whenever, the market maker is unrealistic her expected profit may be positive or negative. Finally we show that for situation 1, the price efficiency improves with κ2 whereas it decreases with κ1 , whereas the volatility decreases with κ2 and increases with κ1 . In other models like Benos (1998) or Daniel, Hirshleifer and Subrahmanyam (2000) overconfidence is defined by κ2 , only. By allowing these two parameters to characterize the unrealistic behavior, our findings are more general than the ones obtained in Benos (1998), Kyle and Wang (1997), Daniel, Hirshleifer and Subrahmanyam (2000) to name but a few.5 Some of our results are qualitatively and quantitatively different from the ones obtained with overconfident traders. Since overconfident traders do not misperceive the expectation of the risky asset, 5
All these authors implicitly assume that κ1 = 1. Benos (1998) studies the extreme case where overconfident traders preceive their private information as being non noisy, that is κ1 = 1 and κ2 = 0.
6
the first effect is not present with overconfident traders. The second effect is not new, however the combination of the two misperceptions of the variances is not present when traders are overconfident. The paper unfolds as follows. In the next section, the general model is presented with the definition of an equilibrium in our model. In section 3, the model is solved for the additive misperception alone as well as for the case where the additive and the multiplicative misperception are combined for the case when the market makers are realistic. In that section we look at the two cases: (i) where only one type of unrealistic traders are present, (ii) where the two types of unrealistic traders are present. In section 4, we derive the equilibrium under the assumption that market makers are themselves optimistic or pessimistic and trade with only one type of unrealistic traders. The last section summarizes our results and concludes. All proofs are gathered in the appendix.
2
Model
We study a financial market where a market maker and several traders exchange a risky asset whose future value v˜ follows a gaussian distribution with zero mean and variance σv2 . Traders participating in that market can either be informed or uninformed. The uninformed traders are the so-called noise traders and submit a market order which is the realization of a normally random variable ue with zero mean and variance σu2 . The informed traders are risk neutral and can be one of two types: realistic or unrealistic (optimistic or pessimistic). N traders are realistic whereas M are unrealistic. Both type of traders observe a noisy signal of the future value of the risky asset ³
s˜k = v˜ + ε˜k , with ε˜k → N 0, σε2
´
∀k = 1, ..., N, N + 1, ..., N + M.
These two types of traders differ in the beliefs they have about the expectation of the risky asset value or the expectation of prior information. realistic traders, correctly, believe that the expectation is zero. However, unrealistic traders believe that it is a. That expectation is positive (negative) if their are optimistic (pessimistic). The term unrealistic trader refers to a trader who uses the wrong distribution for the asset return. However that trader rationally anticipates the behavior of both the market maker and the realistic trader. The strategy of each realistic trader i is a Lesbegue measurable function, Xi : < → 1, κ2 > 1 for pessimistic traders. Whenever κ1 = κ2 = 1, the unrealistic trader does not misperceive both variances. We study three different cases. In the first one, only one type of unrealistic along with realistic traders are trading with realistic market makers. In the second one, we remove the assumption of only one type of unrealistic traders and allow pessimistic and optimistic traders. In the last one, we remove the rationality assumption of the market makers from the first case. 3.2.1
One Type of unrealistic Traders
We now characterize the equilibrium. Proposition 3 Whenever ³
´
³
´
κ21 M (1 − τ ) (1 + 2τ )2 + N (1 + τ ) + 2κ1 κ2 τ M (1 + 2τ )2 + 2N (1 + τ ) +4κ22 τ 2
(1 + τ ) N ≥ 0.
there exists an equilibrium of the following form: xri = αr + β r si , ∀i = 1, ..., N, = αun + β un sj , ∀j = 1, ..., M, xun j p = µ + λy = µ + λ
Ã
N P
i=1
The coefficients are such that:
11
xri
+
M P
j=1
xun j
!
+u .
(1)
for the optimistic/pessimistic traders αun = β un =
2 (1 + 2τ ) κ2 τ σu
q
2
σv Mκ1 (1 + 2τ ) [κ1 (1 − τ ) + 2κ2 τ ] + N (κ1 + 2κ2 τ )2 (1 + τ ) κ1 (1 + 2τ ) σu q
σv Mκ1 (1 + 2τ )2 [κ1 (1 − τ ) + 2κ2 τ ] + N (κ1 + 2κ2 τ )2 (1 + τ )
a ;
for the realistic traders αr = 0, βr =
(κ1 + 2κ2 τ ) σu
q
σv Mκ1 (1 + 2τ )2 [κ1 (1 − τ ) + 2κ2 τ ] + N (κ1 + 2κ2 τ )2 (1 + τ )
;
for the market maker µ = −
2 (1 + 2τ ) κ2 τ a, (2τ + N + 1) (κ1 + 2κ2 τ ) + Mκ1 (2τ + 1) q
σv Mκ1 (1 + 2τ )2 [κ1 (1 − τ ) + 2κ2 τ ] + N (κ1 + 2κ2 τ )2 (1 + τ ) . λ = σu [(2τ + N + 1) (κ1 + 2κ2 τ ) + Mκ1 (2τ + 1)] Proof. See Appendix. The existence of the equilibrium is subject to condition (1). This condition guarantees that there is enough trading from the part of the realistic relatively to the amount traded by unrealistic. When τ < 1 and when τ > 1 with M ≤ N 1+τ , the condition is satisfied for any combination of κ and κ . Whenever 1 2 (τ −1)(1+2τ )2 1+τ > (τ −1)(1+2τ the condition is restrictive and limits the possible τ > 1 and M N )2 combinations of κ1 and κ2 . If τ > 1, the larger the number of unrealistic traders, M, and/or the larger κ1 the more restrictive the condition. The number of realistic as well as κ2 affect the condition in the opposite way. The smaller κ2 , the more unrealistic traders respond to their private information, as they believe that their information is less noisy than it is. As M grows the proportion of the order flow due to the unrealistic traders grow. As a response to both, realistic traders scale down their quantities to reduce their impact on the price. Everything else being equal, as M becomes extremely large and κ2 very small, realistic traders leave the market, implying its closure. When κ1 = κ2 , the equilibrium described above is identical to the one found in proposition 1. As long as the unrealistic trader misperceives both variances by the same amount, this misperception has no effect on his behavior. This can be understood by looking at the way the unrealistic trader updates his beliefs concerning the future value of the asset after having observed his signal. The conditional expectation for any unrealistic trader j is given by
Eir [ v˜| s = sj ] =
κ1 κ2 sj + a. κ1 + κ2 τ κ1 + κ2 τ 12
When κ1 = κ2 , the above conditional expectation is equal to the conditional expectation for the unrealistic misperceiving the expectation only. From proposition 1, we know that unrealistic and realistic traders trade with the same intensity on private information. Therefore, only the misperception of the expectation matters. This implies that the liquidity is not affected by κ1 or κ2 . In all other cases when κ1 6= κ2 , both misperceptions affect the variables of the model. The unrealistic trading intensity on private information increases with κ1 and decreases with κ2 . As prior information is misbelieved to be noisier and/or the private information is perceived to be more precise, the unrealistic trades more on private information. The comparative statics concerning the realistic’s trading intensity are not as straightforward. If τ < 12 , i.e. prior information is relatively less precise than the noise in private information, the realistic response is as expected, he decreases his trading intensity with κ1 and increases it with κ2 in order to reduce the impact of his market order onto the price. As prior information becomes less precise relative to the noise in the private signal, the trading intensity for the realistic may increase with κ1 and decrease with κ2 , following the unrealistic’s behavior. When κ1 > κ2 (κ1 < κ2 ), the unrealistic trader’s responsiveness to private information is higher (lower) than the realistic one. The following figures show how κ1 and κ2 influence the values of the liquidity parameter, λ, and the parameter µ. [Insert [Insert [Insert [Insert
Figure Figure Figure Figure
2] 3] 4] 5]
The comparative statics regarding µ depend on whether the unrealistic traders are optimistic or pessimistic. For the case where unrealistic traders are optimistic (pessimistic), the parameter µ increases (decreases) with κ1 whereas it decreases (increases) with κ2 . The comparative statics on the liquidity parameter, λ, are not affected by whether unrealistic traders are optimistic or pessimistic. We, then, obtain that - for small κ2 , the liquidity parameter increases and then decreases with κ1 , for large κ2 , the liquidity parameter increases with κ1 , - for small κ1 , the liquidity parameter decreases with κ2 , for large κ1 , the liquidity parameter increases and then decreases with κ2 . Proposition 4 Provided the equilibrium exists, the expected profits are given by for the unrealistic traders E (Πun ) =
σv2 (1 + 2τ )2 κ1 [κ1 (1 − τ ) + 2κ2 τ ] , λ ((2τ + N + 1) (κ1 + 2κ2 τ ) + Mκ1 (2τ + 1))2 13
for the realistic traders E (Πr ) =
σv2 (1 + τ ) [κ1 + 2κ2 τ ]2 . λ ((2τ + N + 1) (κ1 + 2κ2 τ ) + Mκ1 (2τ + 1))2
Whenever κ1 < κ2 , the realistic traders earn strictly larger expected profit than unrealistic traders. Provided the equilibrium exists and that κ2 < κ1 , ³
´
• the unrealistic might earn negative expected profit κ2 τ2τ < κ1 , −1 • the unrealistic profit however lower than the ³ might earn positive expected ´ ) 2τ realistic one κ2 2τ2τ(1+τ < κ < κ , 1 2 τ −1 2 −1 • the unrealistic might earn expected profit larger than the realistic one ³
´
) κ1 < κ2 2τ2τ(1+τ . 2 −1
Proof. See Appendix. The expected profits are computed under the true distributions of v˜ and ε˜. We look at four different situations depending on the value of both τ and the ratio M for each case (optimistic or pessimistic): situation 1 is for τ ≤ N 1+τ √1 , situation 2 for √1 < τ ≤ 1, situation 3 for 1 < τ and M ≤ , N 2 2 (1−τ )(1+2τ )2
< M . In the following graphs, the and situation 4 for 1 < τ and (1−τ 1+τ N )(1+2τ )2 lower (upper) dashed square (rectangle) corresponds to the scenario where the unrealistic traders are optimistic (pessimistic). κ1
Optimism/Pessimism (situation 1) E(Πun) >E(Πr)
E(Πr) >E(Πun) 1
κ1 = κ 2
E(Πun) >E(Πr)
E(Πr) >E(Πun)
κ2
1
Figure 6: Equilibrium regions for a low τ (τ ≤ √12 ) whith optimistic or pessimistic traders only.
14
Optimism/Pessimism (situation 2)
κ1 =
2τ (1 + τ ) 2τ 2 − 1
E( Πr )> E(
κ1 = κ 2
Π un )
κ1
r
un
Π E(
2τ (1 + τ ) 2τ 2 − 1
Π E( )>
) un )
r)
Π E(
(Π >E
E( Πr )> E( Π un )
1 r
un
Π E(
Π E( )>
)
κ1 = κ 2 un )
r)
Π E(
(Π >E
2τ 2 − 1 2τ (1 + τ )
κ2
1
Figure 7: Equilibrium regions for an intermediate τ ( √12 < τ ≤ 1) whith optimistic or pessimistic traders only. Optimism/Pessimism (situation 3)
2τ (1 + τ ) κ1 = κ 2 2τ 2 − 1
E( Πr )>
0> E( Π un E( ) Πr )> E( E Π un (Π u ) > n) > E( 0 Πr )
1
un )
Π E(
E( Πr )
E( Π un )>
E( Π un )>
0> E( Πr )>
2τ τ −1
E( Πr )>
κ1 = κ 2
0
E( Πu
n
)
κ1
un )
Π E( r)> Π ( E
>0
κ1 = κ 2
>0
r)>
Π E(
κ2
1
Figure 8: Equilibrium regions for both a large τ (1 < τ ) and a relatively low number of unrealistic traders in the market (M ≤ (1−τ 1+τ ), whith optimistic or pessimistic traders only. N )(1+2τ )2 Optimism (situation 4) κ1 = κ 2
E(Π r )> 0> E( E(Π u Π n r n ) ) ) >E > (Π u E( n Πr )> )> 0 0
E( Πu
N o eq
uilibr ium
κ1
κ1 = κ 2
2τ (1 + τ ) 2τ 2 − 1
κ1 = κ 2
un )
r)>
2τ τ −1
>
0
Π E(
Π E( 1 1
2τ 2 − 1 2τ (1 + τ )
κ2
Figure 9: Equilibrium regions for both a large τ (1 < τ ) and a relatively high number of unrealistic traders in the market 15
( (1−τ 1+τ < )(1+2τ )2
M ) N
with optimistic traders.
um
Pessimism (situation 4) E( 0> Πr ) > E( Πu Π un E( n Π un ) ) > ) E( >0 Πr )> 0
κ1 = κ 2 κ1 = κ 2
E(
E( Πr )>
No e
qui lib ri
κ1
2τ (1 + τ ) 2τ 2 − 1
κ1 = κ 2
un )
r)>
2τ τ −1
>
0
Π E(
Π E( 1 1
2τ 2 − 1 2τ (1 + τ )
κ2
Figure 10: Equilibrium regions for both a large τ (1 < τ ) and a relatively high number of unrealistic traders in the market ( (1−τ 1+τ κ1 , it is always the case that unrealistic traders earn less expected profit than realistic ones, however they do earn nonnegative profit in expected term In that case we have β r > β un . For κ1 > κ2 , depending on the relative value of κ1 with respect to κ2 and on the value of the other parameters (i.e. depending in which situation of the different ones cited above we are) the realistic trader may earn more expected profit than the unrealistic with the possibility for the latter to earn negative expected profit (situations 3 and 4), or the unrealistic trader may earn on average larger profits than the realistic. For all situations but situation 4, the existence condition does not depend on κ1 and κ2 as the condition (1) is always verified for either a small τ (τ ≤ 1) or for a relative low number of unrealistic traders present in the market (M ≤ (1−τ 1+τ ). N )(1+2τ )2 When κ1 (1 − τ ) + 2κ2 τ < 0, the unrealistic trader earns negative unconditional expected payoff. When the actual variance of the noise is large relative to the ex-ante volatility, a large error concerning σε2 compared to the one on σv2 is costly for the unrealistic trader. He then earns negative expected profit. This corresponds to the situation where σε2 > σv2 , however the unrealistic trader believes that the noise in the private information is smaller than the ex-ante volatility of the asset. In that case, the unrealistic trader trades too much on his private information. Whenever the previous condition is reversed, the unrealistic trader can earn, on average, the same as the realistic trader, more or less than the realistic trader. For a small τ , the expected profit of the unrealistic trader is greater than the one of the realistic whenever the unrealistic trading intensity on private information is greater than the realistic one. As a consequence of a large quantity traded by the unrealistic trader, the realistic trader scales down his quantity resulting in an 16
overall reduction in their expected profit. As τ increases, a small relative error on σε2 compared to one made on σv2 is beneficial for the unrealistic implying higher expected profit for the unrealistic than for the realistic. As the relative error made on σε2 increases, the unrealistic over-trades too much on his private information leading to expected profit lower than the realistic one. As the unrealistic increases his over-trading, he might even earn negative expected profit and might lead the market to close down. For both cases (optimistic and pessimistic traders), as τ increases, the slope ) of the two lines (κ2 τ2τ and κ2 2τ2τ(1+τ 2 −1 ) becomes flatter implying that the region −1 where the unrealistic trader earns more than the realistic trader shrinks. Ultimately, for an infinite τ and for the parameters where the equilibrium exists, when κ2 < κ1 , the unrealistic trader earns negative expected profit and the realistic positive, whereas when κ1 < κ2 the unrealistic trader earns positive expected profit although lower than the realistic one. When τ is infinite, the unrealistic trader never earns profit, in expected terms, higher than the realistic trader. We now look more closely at the expected profit functions for both the realistic and the unrealistic. [Insert [Insert [Insert [Insert
Figure Figure Figure Figure
11] 12] 13] 14]
One can see the following comparative statics from the figures above For the optimist (Figure 11) we obtain that - for small κ2 , the expected profit increases and then decreases with κ1 , for large κ2 , the expected profit increases with κ1 , - for small κ1 , the expected profit decreases and then increases with κ2 , for large κ1 , the expected profit increases with κ2 . For the pessimist (Figure 12) we obtain that - the impact of κ1 on the expected profit is the same as the one obtained above for the optimistic, - for small κ1 , the expected profit decreases with κ2 , whereas for large κ1 , the expected profit decreases and then increases with κ2 . When optimistic or pessimistic traders are present (Figure 13 and 14) we obtain for the realistic that - for small κ2 , the expected profit decreases and then increases with κ1 , for large κ2 , the expected profit decreases with κ1 , - for small κ1 , the expected profit increases with κ2 , whereas for large κ1 , the expected profit decreases and then increases with κ2 . Figure 11 illustrates the possibility for optimistic traders to obtain negative expected profit. As κ2 increases, for the optimistic case, the unrealistic trader decreases β un , which reduces his impact on prices and overall increases his profit. 17
For the pessimistic, this is reversed as an increase in κ2 leads to a more pessimistic trader. The comparative statics are then reversed. For the realistic trader, the comparative statics do not depend on whether the unrealistic traders are optimistic or pessimistic. Price efficiency is equal to var (v| p) =
σv2 (κ1 + 2τ κ2 ) (1 + 2τ ) . (N + 2τ + 1) (κ1 + 2τ κ2 ) + Mκ1 (2τ + 1)
We find that price efficiency increases with κ2 whereas it decreases with κ1 . The ex-ante volatility is equal to var (p) =
σv2 (N (κ1 + 2τ κ2 ) + Mκ1 (2τ + 1)) . (2τ + N + 1)2 {(N + 2τ + 1) (κ1 + 2τ κ2 ) + Mκ1 (2τ + 1)}
The ex-ante volatility increases with κ1 and decreases with κ2 . The effect of κ1 onto both volatility and price efficiency accords to intuition. However the effect of κ2 deserves more attention. As κ2 increases, unrealistic trade less on their private information which leads the realistic to trade more on their private information. The effect on the realistic dominates the effect on the unrealistic leading to an increase in price efficiency. For the effect on the volatility the effect on the unrealistic dominates the effect on the realistic implying a reduction in volatility. 3.2.2
Two types of Unrealistic Traders: Optimists and Pessimists
We now look at the case where the two types of unrealistic traders, namely optimistic and pessimistic traders, together with realistic traders are present in the market. Let us assume that Θ denotes the total number of optimistic traders present in the market, P the total number of pessimistic traders and finally N the total number of realistic traders. Let xo , xp , and xr be the quantity submitted by the optimistic, pessimistic, and realistic traders respectively. Let κl1 σv2 and κl2 σε2 be the perceived variances for the unrealistic trader l with l = o, p where o stands for optimistic and p for pessimistic. We assume that the perceived expectation for the optimistic traders is Eo [˜ v] = a > 0 whereas it is Ep [˜ v] = −a0 < 0 for the pessimistic traders. The following result can be obtained. Proposition 5 Whenever h
d1 = (1 + 2τ )2 Θκo1 (κp1 + 2τ κp2 )2 (κo1 (1 − τ ) + 2τ κo2 ) +P κp1 (κo1 +
2τ κo2 )2
i
(κp1 (1 − τ ) + 2τ κp2 ) + N (κo1 +
(2) 2τ κo2 )2
(κp1 +
there exists an equilibrium of the following form:
18
2τ κp2 )2
(1 + τ ) ≥ 0.
xoj = αo + β o sj , ∀j = 1, ..., Θ, xpk = αp + β p sk , ∀k = 1, ..., P, xri = αr + β r si , ∀i = 1, ..., N, p = µ + λy = µ + λ
Ã
Θ P
j=1
The coefficient are such that for the optimistic traders
xoj
+
P P
k=1
xpk
+
N P
i=1
xri
!
+u .
2a o κ τ (1 + 2τ ) (κp1 + 2τ κp2 ) , λd 2 κo1 (1 + 2τ ) (κp1 + 2τ κp2 ) , = λd
αo = βo
for the pessimistic traders 2a0 p κ τ (1 + 2τ ) (κo1 + 2τ κo2 ) , λd 2 κp (1 + 2τ ) (κo1 + 2τ κo2 ) , = 1 λd
αp = − βp
for the realistic traders αr = 0, (κo1 + 2τ κo2 ) (κp1 + 2τ κp2 ) , βr = λd for the price 2 (1 + 2τ ) τ 0 p o (a P κ2 (κ1 + 2τ κo2 ) − aΘκo2 (κp1 + 2τ κp2 )) , √d σv d1 , λ = σu d
µ =
where d = (1 + 2τ ) [Θκo1 (κp1 + 2τ κp2 ) + P κp1 (κo1 + 2τ κo2 )]+(N + 2τ + 1) (κo1 + 2τ κo2 ) (κp1 + 2τ κp2 ) . Proof. See Appendix. The form of the equilibrium is similar to the one obtained for the simpler case where only unrealistic traders of one type were present in the market. As in the previous case, the equilibrium exists only if the unrealistic traders do not overtrade too much on their private information due to their misperceptions of the variances. κp κo Let us define ρ = κ2p and θ = κ2o . We now analyze the degree by which traders 1 1 trade on their private information.
19
When ρ = θ = 1, all the results are independent of κ1 and κ2 . The traders’ responsiveness to private information as well as the liquidity parameter are identical to the one predicted by a model a la Kyle (1985) with Θ + P + N realistic traders. The size and sign of the shift for the price function depends on the misperceptions of the expectations. If the aggregate misperception of the optimists, Θα, is greater than the pessimists’ one, the price shift is negative. For any other cases, and from the expression of the βs, one can see that when ρ ≥ θ (θ > ρ), the optimist trades more (less) intensely on private information than the pessimist. Whenever one of these ratio is less than one, the particular unrealistic trader trades more intensely on private information than his realistic counterpart. In these cases the sign and the size of µ depends on the size of both the misperception of the asset expectation by both type of unrealistic traders, and the misperception of both variances by the unrealistic trader, as well as the number of optimistic and the number of pessimistic. Obviously depending on these parameters, µ can either be positive of negative. The following proposition focuses on the expected profits for each type of traders. Proposition 6 Provided the equilibrium exists, the unconditional expected profits are given by for the optimistic traders E [Πo ] =
σv2 (1 + 2τ )2 κo1 (κp1 + 2τ κp2 )2 [κo1 (1 − τ ) + 2κo2 τ ] , λd2
for the pessimistic traders E [Πp ] =
σv2 (1 + 2τ )2 κp1 (κo1 + 2τ κo2 )2 [κp1 (1 − τ ) + 2κp2 τ ] , λd2
for the realistic traders E [Πr ] =
σv2 (1 + τ ) (κo1 + 2τ κo2 )2 (κp1 + 2τ κp2 )2 , λd2
where d is defined as before. Proof. See Appendix. As before, the unrealistic traders can suffer expected losses. This situation −1 for the optimistic traders and when happens when τ > 1 and when θ ≥ τ2τ −1 ρ ≥ τ2τ for the pessimistic. However, the realistic trader always earns a positive expected profit. Given the expression of the expected profit for the three types of traders, the exact different cases as in proposition 4 are obtained with 3 different types of traders. Therefore some parameter configurations imply that unrealistic traders obtain more expected profit than realistic traders, with a possibility for either the optimists or the pessimists to earn the most expected profit. 20
4
Unrealistic Market Makers
We now look at the case where the market makers are unrealistic as well as M traders among the M + N traders. The unrealistic traders misperceive the distributions of both v˜ and ε˜j as before. Given the fact that the market maker has no access to any private signal, she misperceives the expectation and variance of the distribution of prior information. The market makers believe that the distribution of the asset is such that ³
´
v˜ → N a, κ1 σv2 . Proposition 7 Whenever d01 = κ1 (2τ + 1) (κ1 + 2κ2 τ ) (Mκ1 (2τ + 1) + N (κ1 + 2κ2 τ )) ³
´
−τ Mκ21 (2τ + 1)2 + N (κ1 + 2κ2 τ )2 ≥ 0, there exists a unique linear equilibrium. It is characterized by the following parameters, for the optimistic/pessimistic traders αun =
2τ + 1
β un =
κ1 (2τ + 1)
q
σv d01 q
(2κ2 τ a − a (κ1 + 2κ2 τ )) σu ,
σv d01
σu ,
for the realistic traders αr = − βr
a
q
(κ1 + 2κ2 τ ) (2τ + 1) σu , σv d01 κ1 + 2κ2 τ q = σu , σv d01
for the market maker (2τ + 1) [a (κ1 + 2κ2 τ ) (M + N + 1) − 2Mκ2 τ a] , 0 dq σv λ = 0 d0 , d σu 1
µ =
where d0 = Mκ1 (2τ + 1) + N (κ1 + 2κ2 τ ) + (2τ + 1) (κ1 + 2κ2 τ ). Proof. See Appendix. In the following discussion, we only look at the effect of an optimistic market maker as the pessimistic case is symmetric. Through the misperceptions of both the expectation and the variance, the market maker’s optimism affects the price 21
function in two opposite ways. On the one hand, the market depth decreases with the misperception of the variance, i.e. the higher the misperception the higher the market depth. An optimistic market maker thinks that the prior information is more precise than it is and therefore believes that the informed private information is less substantial than it is in reality. As a consequence, she adjusts her price less aggressively. This is done by reducing the liquidity parameter, λ, and therefore by increasing market depth. As a response, informed traders trade more intensely. On the other hand, the overall level of price is shifted up due to the misperception of the expectation. Indeed, an optimistic market maker wrongly believes that the expectation of the risky asset is higher than it is and therefore increases the overall level of prices. However that effect may be mitigated by the effect of the trader’s misperception of the expectation as seen in the equation defining µ. When the situation is symmetric (both the market makers and the unrealistic traders are optimistic or pessimistic), the level of price can either be reduced (µ < 0) or increased (µ > 0). Whenever the situation is asymmetric the shift is positive (negative) with an optimistic (pessimistic) market 2τ maker. For any positive shift (¯ a ≥ (κ1 +2κ2Mκ a) or extreme negative shift 2 τ )(M+N+1) 2τ (¯a < (κ2Mκ a) of the price function, the unrealistic trader, irrespective of being 1 +2κ2 τ ) optimistic or pessimistic, decreases his market order (αun < 0) or increases it (αun > 0), respectively. This implies that an optimistic trader finding the level of price too high decreases his market order whereas a pessimistic trader finding the level of price too low increases it. For intermediate negative shift of the price 2τ 2τ ( (κ1 +2κ2Mκ a>a ¯ > (κ2Mκ a), the unrealistic trader always decreases his 2 τ )(M+N+1) 1 +2κ2 τ ) market order. This effect induces that both types of informed traders scale down their market order if the market maker is optimistic. The unconditional expected profits of each traders are given in the following proposition. Proposition 8 Provided the equilibrium exists, the expected profits are given by for the unrealistic traders i (2τ + 1)2 h 2 σ κ (κ (1 − τ ) + 2κ τ ) − a (κ + 2κ τ ) (2κ τ a − (κ + 2κ τ ) a) , E [Π ] = 1 1 2 1 2 2 1 2 v λd02 un
for the realistic traders i (κ1 + 2κ2 τ )2 h 2 2 2 σ (τ + 1) + a (2τ + 1) , E [Π ] = v λd02 r
for the market maker h
E ΠMM
i
=
(2τ + 1) (κ1 + 2κ2 τ ) h 2 σv (¯ κ1 − 1) (Mκ1 (2τ + 1) + N (κ1 + 2κ2 τ )) λd02 +a (2τ + 1) (2Mκ2 τ a − a (κ1 + 2κ2 τ ) (M + N))] .
22
Proof. See Appendix. The expected profit of both types of traders is affected by the market marker’s misperception on both the expected return of the asset and the variance. The lower the perceived variance (the lower κ ¯ 1 ), the higher the market depth, and therefore the higher the trader’s unconditional expected profit. The misperception of the expectation affects differently the two types of traders. The realistic trader’s unconditional expected profit increases with it. The analysis for the unrealistic’s unconditional expected profit is not as straightforward. Whenever αun and a¯ have different sign, the unrealistic trader’s expected profit is larger. This happens with an asymmetric situation or may happen with a symmetric one. As before, the unrealistic trader may obtain greater, equal or smaller profit that the realistic one with the possibility for him to have negative expected profit. The market maker, when realistic (¯ κ1 = 1, a = 0), obtains zero expected profit. However, if she is unrealistic, either optimistic or pessimistic, she may obtain expected profit different from zero. When the market maker is pessimistic (optimistic), an increase ( a decrease) of κ ¯ 1 , increases (decreases) the expected profit through the reduction (increase) in market depth. The effect of the additive misperception is not as straightforward. Indeed, it depends on the sign of the traders’ additive misperception. If the situation is asymmetric (market maker is optimistic and traders are pessimistic, or the converse), the second term is always negative. If the situation is symmetric (both pessimistic or both optimistic), the sign of the second term can either be positive or negative.
5
Conclusion
We developed here a model of optimism and pessimism in financial markets. We argue that our unrealistic traders as well as misperceiving the expected returns of the asset, can misperceive the variance of both the volatility of the asset returns and the noise in the private signal. An optimistic (pessimistic) trader over-estimate (under-estimate) the expected returns of the asset and can underestimate (over-estimate) both variances. We study two scenarios, in the first one the unrealistic trader only misperceives the expected returns whereas in the second one, we allow the unrealistic trader to misperceive the expected returns and both variances. In scenario 1, we find that an optimistic (pessimistic) trader purchases (sells) a larger quantity or sells (purchases) a smaller one. It implies that the price function is below (above) with optimistic (pessimistic) traders than the one predicted by a standard model a la Kyle (1985). We show that the liquidity is not affected by the misperception of the expected returns. The expected profit for the unrealistic trader and for the realistic trader are shown to be equal. In scenario 2, we show that an equilibrium does not always exist. Indeed, the unrealistic trader may trade too much on his private information leading the realistic trader to leave the market, implying its closure. We find that the unrealistic investor may trade more on his private information than his 23
realistic counterpart. If this overtrading is not too large, he benefits from it as his expected profit is larger than the one obtained by his realistic counterpart. If that overtrading is too large, he may earn in expected term less than the realistic, he may even earn negative expected profit. We find that both expected profit for the realistic and unrealistic can be a non-monotonic function of the degree of errors made on both variances. In the same vein, the liquidity parameter can also be non-monotonic with respect to κ1 and κ2 . Finally we show that the price efficiency improves with κ2 whereas it decreases with κ1 , whereas the volatility decreases with κ2 and increases with κ1 . We also look at the case where the two types of unrealistic traders trade in the market and at the case where the market maker is unrealistic. In the latter case we show that even if optimistic (pessimistic) traders are present the price level may not be below (above) the price level obtained in a model a la Kyle (1985). If the market maker’s misperception of the expectation of the asset is relatively large compared to the unrealistic one, a situation where both the market maker and the unrealistic are optimistic leads to a higher level of price. We show that when the market maker is unrealistic, her expected profit may be either positive or negative. An interesting extension of the model would be to look at how the results obtained in the present model are modified in a dynamic setting. This is left for future research.
6
Bibliography
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Kyle, A. S., and F. A. Wang, 1997, “Speculation Duopoly with Agreement to Disagree: Can Overconfidence Survive the Market Test,” Journal of Finance, Vol. 52, No. 5, pp 2073-2090. Manove, M., and J. Padilla, 1999, “Banking (Conservatively) with Optimists,” RAND Journal of Economics, Vol. 30, No. 2, pp 324-350. Manove, M., 2000, “Entrepreneurs, Optimism and the Competitive Edge,” UFAE and IAE Working Papers 296.95. Odean, T., 1998a, “Are Investors Reluctant to Realize their Losses?,” Journal of Finance, Vol. 53, No. 5, pp 1775-1798. Odean, T., 1998b, “Volume, Volatility, Price, and Profit When all Traders are Above Average,” Journal of Finance, Vol. 53, No. 6, pp 1887-1924. Odean, T., 1999, “Do Investors Trade Too Much,” American Economic Review, Vol. 89, No. 5, pp 1279-1298. R´egner, I., D. Hilton, L. Cabantous, and S. Vautier, 2004, “Judgmental Overconfidence: One Positive Illusion or Many?,” Mimeo. Ritter, J., 1991, “The Long-Run Performance of Initial Public Offerings,” Journal of Finance, Vol. 46, No. 1, pp 3-27. Scheinkman, J. A., and W. Xiong, 2003, “Overconfidence and Speculative Bubbles,” Journal of Political Economy, Vol. 111, No. 6, pp 1183-1219. Shiller, R., 1981, “Do Stock Prices Move Too Much to Be Justified by Subsequent Changes in Dividends?,” American Economic Review, Vol. 71, pp 421-435. Shiller, R., 1989, “Comovements in Stock Prices and Comovements in Dividends,” Journal of Finance, Vol. 44, No. 3, pp 719-729. Weinstein, N., 1980, “Unrealistic Optimism About Future Life Events,” Journal of Personality and Social Psychology, Vol. 39, pp 806-820. Womack, K., 1996, “Do Brokerage Analysts’ Recommendations Have Investment Values,” Journal of Finance, Vol. 51, No. 1, pp 137-167.
7 7.1
Appendix Proofs
Proof of Proposition 1 (Equilibrium for Additive misperception) Take the results obtained in proposition 3 for the expression of the parameters αun , β un , αr , β r , µ, and λ and set κ1 = 1 and κ2 = 1. Proof Proposition 2 (Expected Profits for Additive misperception) Take the results obtained in proposition 4 for the expression of both profits and set κ1 = 1 and κ2 = 1. Proof of Proposition 3 (Equilibrium for Additive and Multiplicative misperception) The proof follows the same steps as the proof of proposition 5. As only one type of unrealistic trader is present, set M = Θ + P , αo = αp = αun , and β o = β p = β un where the superscript un is for the unrealistic traders. The
26
aggregate order flow is then given by y=
N P
i=1
xri +
M P
j=1
r un r xun j +u = (Nβ + Mβ ) v+β
N P
i=1
εi +β un
M P
j=1
εj +Nαr +Mαun +u.
Proof of Proposition 4 (Expected Profits for Additive and Multiplicative misperception) The computations of the expected profits follows the same steps as in the proof of proposition 6. As only one type of unrealistic traders is present set M = Θ + P , αo = αp = αun , β o = β p = β un and κo1 = κp1 = κ1 and κo2 = κp2 = κ2 . By doing so the following is obtained σv2 (1 + 2τ )2 κ1 [κ1 (1 − τ ) + 2κ2 τ ] , λ ((2τ + N + 1) (κ1 + 2κ2 τ ) + Mκ1 (2τ + 1))2 σv2 (1 + τ ) [κ1 + 2κ2 τ ]2 E (Πr ) = . λ ((2τ + N + 1) (κ1 + 2κ2 τ ) + Mκ1 (2τ + 1))2
E (Πun ) =
Having the expression of the expected profit for unrealistic traders and for realistic traders, we can compare them to each other. We compute the difference in expected profits, after some rearrangements that leads to E (Πun )−E (Πr ) =
´ κ1 (1 + 2τ ) σv2 ³ 2 2 (1 + 2τ ) κ [κ (1 − τ ) + 2κ τ ] − (1 + τ ) [κ + 2κ τ ] . 1 1 2 1 2 λD2
Finding the sign of E (Πun ) − E (Πr ) is equivalent to find the sign of (1 + 2τ )2 κ1 [κ1 (1 − τ ) + 2κ2 τ ] − (1 + τ ) [κ1 + 2κ2 τ ]2 . It is straightforward to prove that the above expression is equal to h
³
´
i
2τ (κ1 − κ2 ) κ1 1 − 2τ 2 + 2κ2 (1 + τ ) τ .
(3)
Whenever τ ≤ √12 , the expression (3) is of the sign of κ1 − κ2 and when κ1 − κ2 > 0 (< 0), we have E (Πr ) < E (Πun ) (E (Πun ) < E (Πr )). Whenever )τ √1 < τ , (3) has two positive roots κ1 = κ2 and κ1 = 2κ2 (1+τ κ2 . One can 2τ 2 −1 2 prove that the hlatter is always greater than the former. For any κ2 and κ1 i 2κ2 (1+τ )τ r un in the interval κ2 , 2τ 2 −1 κ2 we have E (Π ) < E (Π ), for any κ2 and κ1 outside the interval we obtain that E (Πun ) < E (Πr ). Given the expression of the unrealistic’s expected profit, one can see that if τ > 1 and κ1 > τ2τ κ, −1 2 unrealistic traders earn negative expected profits. Proof of Proposition 5 (Equilibrium for Additive and Multiplicative misperception with both types of Unrealistic Traders) Given the expressions of the market orders submitted by the optimistic traders, xo , by the pessimistic traders, xp , and finally by the realistic traders, xr , the aggregate order flow is equal to y = (Nβ r + Θβ o + P β p ) v +β r
N P
i=1
εi +β o
Θ P
j=1
27
εj +β p
P P
k=1
εk +Nαr +Mαo +P αp +u.
The optimistic trader maximizes his conditional expected profit Eo ((v − p) xoi | s = sj ) . max o xj
Substituting the form of the price as well as the market orders form for the N realistic traders, the P pessimistic and the Θ − 1 optimistic in the above expression, computing the first order condition and solving it for the market order, we obtain xoj =
1 [Eo (v| s = sj ) (1 − (Θ − 1) λβ o − P λβ p − Nλβ r ) 2λ −µ − (Θ − 1) λαo − P λαp − Nλαr ] .
(4)
We now need to compute Eo (v| s = sj ). On one side we have that Eo (v| s = sj ) = o (v,sj ) γ (sj − Eo (v)) + Eo (v) with γ = cov given the normality of the random varivaro [sj ]
ables. Given that Eo (v) = a and τ = Eo (v| s = sj ) =
σε2 , σv2
we obtain
κo1 κo2 τ + a s . j κo1 + κo2 τ κo1 + κo2 τ
Replacing the expression of the conditional expectation into the form of the order (4) and identifying the parameters we have κo1 (1 − λNβ r − λP β p ) , λ ((Θ + 1) κo1 + 2κo2 τ ) " 1 κo τ = a o 2 o (1 − (Θ − 1) λβ o − λP β p − Nλβ r ) λ (Θ + 1) κ1 + κ2 τ −µ − λ (Nαr + P αp )] .
βo =
(5)
αo
(6) (7)
The second order condition is satisfied. Proceeding in the same way for the pessimistic trader, and assuming that Ep (v) = −a0 , we get the following system of equations κp1 (1 − λNβ r − λΘβ o ) , λ ((P + 1) κp1 + 2κp2 τ ) " κp2 τ 1 0 o p r = −a p p (1 − Θλβ − λ (P − 1) β − Nλβ ) λ (P + 1) κ1 + κ2 τ r −µ − λ (Nα + Θαo )] .
βp =
(8)
αp
(9) (10)
The second order condition is verified. Finally, the realistic maximizes his conditional expected profit. Given his si first order condition and the fact that E (v| s = si ) = 1+τ , the parameters for the realistic’s market order are such that (1 − λΘβ o − P λβ p ) βr = , (11) λ (N + 1 + 2τ ) 1 [µ + λΘαo + P λαp ] . (12) αr = − λ (N + 1) 28
The second order condition is satisfied. The market maker behaves competitively and sets a price such that p = E [v| y] = 0 +
cov (v, y) (y − E (y)) . var [y]
Given the expression of the aggregate order flow, the parameters of the price schedule are given by (Nβ r + Θβ o + P β p ) ³
λ =
´
(Nβ r + Θβ o + P β p )2 + N (β r )2 + Θ (β o )2 + P (β p )2 τ +
µ = −λ (Nαr + Θαo + P αp ) .
2 σu σv2
, (13) (14)
Solving the above system of six equations and six unknowns leads to the result of proposition 5. Proof of proposition 6 (Expected Profits) The expected profit of any trader, h, can be written as ³
´
³
´
³
³
E Πh = E (v − p) xh = E (v − µ − λy) β h (v + εh ) + αh
´´
.
Given the expression of y and cancelling some of the terms, the expected profit is equal to ³
h
E Π
´
= E
"Ã
Ã
r
o
p
v − λ (Nβ + Θβ + P β ) v + β
³
× β h (v + εh ) + αh
´i
.
r
N P
i=1
εi + β
o
Θ P
j=1
εj + β
p
P P
!!
εk + u
k=1
All random variables are independent and have a zero mean, we therefore get ³
h
E Π ³
E Πh
´ ´
Ã
2 h
r
o
p
h h
= E v β (1 − λ (Nβ + Θβ + P β )) − λβ ε h
i
= β h σv2 (1 − λ (Nβ r + Θβ o + P β p )) − λβ h σε2 .
Ã
β
r
N P
i=1
εi + β
o
Θ P
j=1
εj + β
Using the expressions of β r , β o , and β p , and after some simplifications we get ³
´
E Πh = β h σv2
"
#
(1 + 2τ ) (κo1 + 2τ κo2 ) (κp1 + 2τ κp2 ) − λβ h τ , d
where d is defines as above. Set h = o, p, or r and plug in the relevant value of β in the above expression, after some computations we obtain the desired result for each type of traders: κo1 (2τ + 1)2 (κp1 + 2τ κp2 )2 o [κ1 + 2τ κo2 − κo1 τ ] , 2 λd 2 p κ (2τ + 1) (κo1 + 2τ κo2 )2 p [κ1 + 2τ κp2 − κp1 τ ] , E (Πp ) = σv2 1 λd2 (2τ + 1)2 (κo1 + 2τ κo2 )2 (κp1 + 2τ κp2 )2 . E (Πr ) = σv2 λd2 E (Πo ) = σv2
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o
P P
k=1
εk
!!
,
Proof of Proposition 7 (Unrealistic Market Makers) After maximizing the traders expected utility we get for the different parameters α
un
=
β un = αr = βr =
·
¸
κ2τ 1 − (1 − λ (M − 1) β un − λNβ r ) , λ (M + 1) κ1 + κ2 τ κ1 (1 − λNβ r ) λ ((M + 1) κ1 + 2τ κ2 ) 1 [µ + λMαun ] − λ (N + 1) 1 − λMβ un . λ (N + 1 + 2τ )
(15)
(16)
The market maker sets a price, p, such that ´ cov (˜ v, y) ³ ¯ (y) , ¯ [ v˜| y] = E¯ [˜ y−E p=E v] + var (y)
where the upper bar denotes that the expectation, covariance and variance are computed given the wrong beliefs of the market maker. Given the market maker’s additive misperception we obtain λ =
(Mβ un + Nβ r ) κ1 (Mβ un + Nβ r )2 κ1 + (Mβ un2 + Nβ r2 ) τ +
2 σu σv2
µ = (1 − λMβ un − λNβ r ) a − λMαun − λNαr .
,
(17) (18)
Solving the above system of six equations with six unknowns leads to the desired result. Proof of Proposition 8 (Expected Profit) Follow the same steps as in proposition 6 for the expected profits of the traders. The market maker’s expected profit are equal to h
i
³
´
E ΠMM = −NE (Πr ) − ME (Πun ) + E ΠLiq . ³It
is ´straightforward to show that the expected profit of the liquidity traders, , are equal to λσu2 . Plug the expressions found for the two types of E Π traders and for the liquidity traders into the expression above and after some manipulations, the desired result is found. Liq
7.2
Figures
The simulations for figures 2, 3, 4, 5, 11, 12, 13, 14, are made for M = 3, N = 10, σv = 1, σu = 1, σε = 2.
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λ 0.35
1
0.3
0.8
0.25
0.6 0.2
κ2
0.4 0.4 0.2
0.6
κ1
0.8 1
Figure 2: Liquidity parameter as a function of κ1 and κ2 whith optimistic traders.
µ
0 -0.1
1
-0.2 -0.3 -0.4
0.8 0.6 0.2
0.4 0.4
κ1
κ2
0.2
0.6 0.8 1
Figure 3: µ parameter as a function of κ1 and κ2 whith optimistic traders.
λ 0.35
3
0.3
2.5
0.25 2 1.5
κ1
κ2
1.5
2 2.5 3
Figure 4: Liquidity parameter as a function of κ1 and κ2 whith optimistic traders.
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µ
0.4
3
0.35 0.3
2.5
0.25 2
κ2
1.5 1.5
2 2.5
κ1 Figure 5: µ parameter as a function of κ1 and κ2 whith pessimistic traders. 3
E (Π un )0.02 1
0
0.8
-0.02
0.6 0.2
0.4 0.4
κ1
κ2
0.2
0.6 0.8 1
Figure 11: Expected Profit of the optimistic trader as a function of κ1 and κ2 .
E (Π un ) 0.025
3
0.02 2.5
0.015 2 1.5
κ1
κ2
1.5
2 2.5 3
Figure 12: Expected Profit of the pessimistic trader as a function of κ1 and κ2 .
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E (Π r ) 0.045 0.04
1
0.035
0.8
0.03 0.6 0.2
0.4 0.4
κ2
0.2
0.6
κ1 Figure 13: Expected Profit of the realistic tarders as a function of κ1 and κ2 when optimistic traders are present in the market. 0.8
1
E (Π r ) 0.032 3
0.03 2.5
0.028 2 1.5
κ1
1.5
2
κ2
2.5 3
Figure 14: Expected Profit of the realistic traders as a function of κ1 and κ2 when pessimistic traders are present in the market.
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