Manipulation, Panic Runs, and the Short Selling Ban∗
Pingyang Gao
Booth School of Business
The University of Chicago
Xu Jiang
Fuqua School of Business
Duke University
Jinzhi Lu
Booth School of Business
The University of Chicago
Very Preliminary and incomplete.
∗
Abstract
This paper identifies conditions under which a short-selling ban improves the ex-ante firm
value. Short selling improves price discovery and enables stakeholders to make better invest-
ment decisions. However, manipulative short selling can arise as a self-fulfilling equilibrium
and mislead the investment decisions. The adverse effect is amplified by the firm’s vulnera-
bility to panic runs. Overall, short selling reduces the ex-ante firm value if both manipulative
short selling is strong and the firm is very prone to runs. The results contribute to our
understanding of the function of short selling in the capital markets and to the controversy
around the regulations against short selling.
JEL classification:
Key words:
1 Introduction
This paper presents a model to evaluate the effi ciency consequences of banning short sales.
Ever since the first regulation against short selling was enacted by Amsterdam exchange in
1610, such regulations have been controversial (e.g., Bris et al. (2007)). A salient feature
of the restrictions on short selling is that they are often imposed on financial stocks in the
bad times. For example, in September 2008 the SEC banned short-sales of shares of 799
companies for two weeks and the U.K. and Japan declared a ban on short selling for “as
long as it takes” to stabilize the markets. Similarly, in August 2011, France, Spain, Italy,
and Belgium imposed temporary bans on short selling for some financial stocks during the
European sovereign debt crisis.
Proponents for short selling makes a straightforward argument. Like other selling and
buying, short selling allows investors to express their negative opinions through trading and
improves the informativeness of stock prices. The price discovery, in turn, leads to better
decisions and more effi cient allocation of capital in the economy.
In contrast, opponents of short selling have argued that short sellers may manipulate the
market through “bear raids.” Speculators with initial short positions may employ various
tactics, including spreading rumors, to drive down the share prices in order to close their
initial short positions at a lower cost. Goldstein and Guembel (2008) (hereafter referred to as
GG) presents a model in which manipulative short selling, whereby an uninformed speculator
nonetheless shorts a firm’s stock and earns a profit, can arise in a self-fulfilling equilibrium.
In this paper, we combine both arguments to evaluate the short-selling (SS) ban on the
ex-ante effi ciency. We augment a coordination game with the manipulative short selling
from GG. There is a speculator who receives information about the state with a known
probability. After privately learning about whether he has received an informative signal or
not, the speculator trades in a market in the style of Kyle (1985) in which the stock price
endogenously reflects some of the speculator’s information. A continuum of investors then
observe the stock price and make their decisions that collectively affect the firm’s cash flow.
Conditional on the stock price, the coordination subgame yields a unique equilibrium using
1
the global games methodology. We then solve for the speculator’s trading strategy, determine
the entire equilibrium, and compare the equilibrium outcomes under two regimes of allowing
and banning short selling.
Our main result is that the SS ban improves the ex-ante effi ciency if the firm’s vulnerability
to runs is suffi ciently high and the speculator’s information quality is not suffi ciently high.
Financial firms are more vulnerable to runs, while crises are often associated with high degree
of uncertainty in the market. As a result, our main result provide a possible justification for
the SS ban on the financial firms’stock during the financial crisis.
To see why the high vulnerability to run is necessary, it is useful to consider a benchmark
where the investment decision is made by representative investor, as studied in GG. In this
case, the SS ban always reduces the ex-ante effi ciency despite the equilibrium existence of
manipulative short selling. The intuition for this somewhat surprising result is compelling.
The investor can always ignore the stock price in making the decision and thus cannot be worse
off with the stock price. The implication is that a short selling ban that cannot discriminate
between informed and uninformed short selling always reduces the ex-ante effi ciency. The
existence of rational bear raids is not suffi cient to justify the ban on short selling.
The medium level of the speculator’s information quality is also necessary. To see this,
consider two extremes. At one extreme, if the market is populated mainly by informed
speculators, then manipulative SS arises but very infrequently. The informational benefit
from informed short selling dominates the cost of the infrequent manipulative SS. As a result,
the SS ban strictly reduces information and effi ciency. At the other extreme, if the market is
mainly populated by uninformed speculators, then based on the intuition in GG, manipulative
short selling does not arise in equilibrium. As a result, the SS ban removes the informed short
selling and thus reduces the effi ciency.
The intuition behind the main result is as follows. In a coordination game, the investors’
investment decisions are driven not only by information about the fundamental but also by
their concern about others’actions. Since investors use the stock price to update their beliefs
about both the state and others’decisions, the investors’responsiveness to the stock price
results from both motivations. The former improves while the latter reduces the effi ciency
2
of the investment decisions from the society’s perspective. We show that the SS ban essen-
tially reduces the stock price informativeness and makes investors less sensitive to the stock
price. Such a reduction in the sensitivity to the stock price reduces the use of information in
the investment decision but also mitigates the coordination failure. When the speculator’s
information is not suffi ciently good and when the coordination friction is suffi ciently high,
the informational loss is dominated by the improvement of coordination, and, as a result, the
effi ciency is improved.
Our paper is related to the literature on short sales. Short sales are a basic component
in modern finance theories of asset pricing and portfolio choice. Most theoretical studies
thus have viewed short sales as an institutional constraint and focused on identifying its
consequences (e.g., Miller (1977), Diamond and Verrecchia (1987), Duffi e et al. (2002), Abreu
and Brunnermeier (2003), Scheinkman and Xiong (2003)). In most of these studies, banning
short sales have an adverse effect on effi ciency.
A few papers have studied the ex-post consequences of short selling in the presence of
rigid frictions. The study that is most closely related to ours is GG. We built on the manip-
ulative short selling in GG and extend GG to model a coordination decision-making game.
This extension generates our main result that the SS ban can improve effi ciency, while in
GG the SS ban cannot improve effi ciency despite the manipulative short selling. Brunner-
meier and Oehmke (2013) show that short selling forces firms with market-based leverage
requirement to liquidate the illiquid assets. In their model there is no informational feedback
from the stock price to real decisions. The effect of stock price on the liquidation decision
is assumed. Liu (2014) also studies a coordination game with short selling. In his model,
investors in the coordination game receive private information and observes the stock price
as public information. Short selling is assumed to add noise into the stock price and makes
the public information noisier. In contrast, short selling in our model allows speculator’s
private information to be endogenously impounded into the stock price and makes the stock
price more informative. Liu (2016) studies the interaction between a investors’coordination
game with interbank market trading and focuses on the feedback loop between interbank
market rate and the coordination game. Interbank market serves as a provision of liquidity
3
and banks do not learn any information from the interbank market. In contrast, we focus on
the interaction between managers learning from price and the coordination game.
Our model makes a methodological contribution to the coordination with market ma-
nipulation. We use the global games methodology to obtain the unique equilibrium of the
coordination game to conduct welfare analysis. However, the market manipulation compo-
nent from GG employs two rounds of trading and is too complicated to be combined with
the coordination game. We use a one round trading setting to simplify the market manipu-
lation component and integrate it with the coordination game. This formulation of market
manipulation may be used in other settings.
Finally, our paper is related to the literature on the welfare effects of public information
in coordination games (Angeletos and Pavan (2007)). Morris and Shin (2002) is probably
the first to show that in settings with coordination motives, more precise public information
may decrease welfare when private information is suffi ciently precise. In our setting there
is no private information per se so even with coordination, more public information should
increase welfare. The detrimental effect of more public information comes from the interaction
of coordination with feedback effect.
The rest of the paper is organized as follows. Section 2 introduces the model set up.
Section 3 highlights the multiple equilibria problem of the model. In section 4, we pin-down
the unique equilibrium using the global game technique. In section 5, the main results,
specifically the result that short selling could be detrimental to bank value, are presented.
Section 6 concludes.
2 Model setup
Our model augments a coordination game with the manipulative short selling from GG. We
start with a coordination game. Consider a risk-neutral economy with no discounting and
four dates (t = 0, 1, 2, 3), one firm with an underlying project, and a continuum of investors.
The underlying state θ is either good or bad with equal probability, i.e., θ ∈ {H,L} with
Pr(θ = H) = 12 .
4
At t = 2, after observing a signal yet to be described below, each investor makes a binary
investment decision, i.e., ai ∈ {0, 1}. If the investor does not invest, she receives a payoff
normalized to 0. If she invests, then her payoff u is jointly determined by the state θ and the
aggregate investing population l ≡∫i∈[0,1]
aidi:
u = θ − δl.
The parameter δ ≥ 0 captures the degree of strategic complementarity among investors’
investment decisions. δ is often referred to as the project’s vulnerability to runs. The project’s
aggregate value is
v = (1− l) (θ − δl) .
So far we have a standard coordination game. The only information source for investors is
the stock price endogenously determined in a Kyle setting. Specifically, there are three types
of traders. The first is a speculator who learns perfectly about θ with probability α ∈ (0, 1]
and nothing with the complementary probability, that is, the speculator observes a signal
s ∈ {H,L, φ}. We call a speculator with s = H(L) as a positively (negatively) informed
speculator and a speculator with no information (s = φ) as an uninformed speculator. The
speculator chooses an order d(s) ∈ {−1, 0, 1}. The second group of traders are liquidity
traders who trade for reasons orthogonal to state θ. Their aggregate order is denoted as u,
which is normally distributed with mean zero and variance σ2n. Finally, the third group of
traders is the market maker who observes the total order flow q = d+ u and sets price equal
to the expected firm value:
P1 = E[v|q].
The stock price and the order flow have identical information content. For simplicity, we
assume that investors observe the order flow (instead of the stock price).
In sum, the timeline of the events is as follows.
At t = 0, the speculator’s information endowment s is realized.
At t = 1, the trading occurs. Both the stock price and the order flow are observed.
5
At t = 2, investors observe the order flow q and make decisions.
At t = 3, the firm’s terminal cash flows are realized.
As a benchmark, GG’s setting is special case of our model with δ = 0. [There is another
difference between our setting and that of GG is the assumption that noise trading is normally
distributed rather than discretely distributed. As will be discussed below, this assumption
allows us to solve the unique equilibrium with only one round of trading instead of two
rounds of trading as in GG, resulting in tractable analysis of welfare when we introduce the
coordination friction into the feedback model.]
We make a few assumptions before proceeding.
A1 : H > 0 > L
A2 : H + L > δ
A3 : cr > H + L
A1 states that it is socially optimal to invest in the good state and not to invest in
the bad state. A2 guarantees that in the absence of any information, the default choice is to
invest. It enhances the assumption in GG that rH +rL > 0 to accommodate the coordination
game represented by δ. Finally, A3 requires that the feedback effect is suffi ciently strong so
that the investors’decisions can be influenced by the stock price even in the absence of the
coordination failure (see Proposition 3 of GG). cr is defined by equation (14) in the Appendix.
To simplify the notation, we normalize H = 1 and L = −r with r ∈ (0, 1). r is a measure
of the strength of the informational feedback effect. A higher r indicates that the investment
decision is more sensitive to information.
In addition, we introduce a tier-breaker. If the negatively (positively) informed speculator
is indifferent among d ∈ {−1, 0, 1}, he always choose d = −1 (d = 1). This rules out a
degenerate equilibrium where no trading takes place.
The effi ciency is defined as the expected firm value that aggregates the payoffs to all
6
investors:
V ≡ E[(1− l)(rθ − δl)]. (1)
A perfect Bayesian equilibrium (PBE) of our model consists of the speculator’s trading
strategy d(s), each investor’s withdrawal strategy ai(q) and beliefs about the fundamental
θ such that (1) both the speculator and the investors maximize their respective objective
functions, given their beliefs and the strategies of others and (2) each investor uses Bayes’
Rule, if possible, to update beliefs about θ.
3 The equilibrium
In this section, we solve for the equilibrium using the backward induction.
3.1 The coordination subgame
We first solve for the subgame after the investors have observed the order flow q.We conjecture
and later verify that q satisfies maximum likelihood ratio property (MLRP), that is, a higher q
indicates that θ = H is more likely. Intuitively, positively informed speculator always chooses
d = 1 and negatively informed speculator always chooses d = −1. Given the conjectures,
regardless of the uninformed speculator’s choice of d, a higher order flow indicates that it
is more likely that the order flow comes from positively informed speculator and therefore a
higher probability of θ = H.
Now consider the strategy of investor i when observing order flow q. If she chooses to
withdraw, then she gets 0 for sure. If she chooses to continue, then her expected payoff
differential is
∆(q) = E[θ|q]− δE[l|q].
As is standard in coordination game, multiple equilibria arise if q is common knowledge,
making it diffi cult to conduct comparative statics. We apply the global games methodology
to obtain the unique equilibrium. Specifically, we assume that each investor observes the
order flow q with noise and focus on the equilibrium when the noise converges to 0. This
7
results in an unique equilibrium of the coordination game.
Lemma 1 The investors play a common threshold strategy. The common threshold q∗ is
determined by the following equation.
E[r|q∗]− δ
2= 0. (2)
Lemma 1 characterizes the equilibrium common threshold in an intuitive manner. Collect-
ing these two expectations and imposing the equilibrium condition that the marginal investor
has to be indifferent between continuing and running at the threshold q∗, i.e. ∆(q∗) = 0,
we obtain equation 2 that uniquely determines the common threshold q∗. A standard result
in the global games methodology results in E[l|q∗] = 12 . An investor uses the signal q
∗ to
forecast other investors’signals and actions. At qi = q∗, she conjectures that exactly half of
the other investors will get a signal higher than q∗ and stay whereas the other half will get
a signal lower than q∗ and withdraw. Therefore, she expects that half of the investors with
stay: E[l|q∗] = 12 .
The key quantity left is thus E[θ|q∗], the marginal investor’s expectation of the funda-
mental. Since θ is binary, the expectation is fully characterized by the conditional probability
β(q∗j ) ≡ Pr(θ = H|q∗j ). A higher β means that the marginal investor has to be more optimistic
about the state to invest.
β(q∗j ) =δ2 + r
1 + r. (3)
β(q∗j ) is an equilibrium variable and depends on the trading strategy of the speculator.
to which we turn now.
3.2 The trading decision and the equilibrium
Anticipating the unique equilibrium for the subgame of coordination, the speculator decides
his trading strategy. The trading strategy and the investors’investment decisions are then
jointly determined by solving a fixed point problem.
When short selling is banned, it is straightforward to show that both the negatively
informed speculator and the uninformed speculator find it optimal not to trade. As GG
8
has shown, the manipulation is only one-sided. It is never optimal for the speculator to
buy when he is uninformed. It is also straightforward to show that the positively informed
speculator buys. Given the trading strategy, both the market maker and the investors make
their decisions accordingly.
Proposition 1 Suppose short sales are banned. Then the positively informed speculator buys
while other speculators do not trade, i.e. d∗(H) = 1 and d∗(L) = d∗(φ) = 0. The investment
threshold is q∗B.
When the short selling is allowed, the possibility of manipulative short selling compli-
cates the derivation of the equilibrium. Short-selling lowers the order flow and induces more
investors not to invest. This has two effects on the speculator’s profit. First, even in the
absence of changes to the investors’ investment decisions, the market maker interprets the
lower order flow as a bad signal about the state and lowers the price accordingly. Since
the market maker cannot distinguish whether the short-selling originates from the negatively
informed speculator or the uninformed speculator, the price is set as an weighted average
conditional on the speculator is negatively informed or uninformed. Thus, shorting is prof-
itable for the informed speculator but not profitable for the uninformed speculator. Second,
the lower price also has an informational feedback on the investment decisions. Agents use
the stock price to make inference about the state and about other investors’decisions. A
lower price is a bad signal about the state and thus discourages investors from investing.
The reduction in the investment indeed reduces the firm’s terminal cash flow and creates a
self-fulfilling equilibrium. Both the informed and uninformed speculator can profit from this
informational feedback effect. Therefore, while the negatively informed speculator always
shorts, the uninformed speculator shorts if and only if the profit from the informational feed-
back effect compensates the cost from his informational disadvantage, a condition satisfied
when the fraction of informed speculator is suffi ciently large.
Proposition 2 Suppose short sales are allowed. If α > α (δ) where α (δ) is defined in the
appendix, then
9
1. the informed speculator trades in the direction of his information: i.e., d∗(H) = 1 and
d∗(L) = −1;
2. the uninformed speculator short sells, i.e., d∗(φ) = −1;
3. the investment threshold is q∗A.
Proposition 2 generates a suffi cient condition for manipulative short selling to arise,
namely, α > α (δ). Setting δ = 0, Proposition 2 replicates the main result in GG that
manipulative short selling arises when the probability that the speculator is informed is suf-
ficiently high. We extend this result to our setting of a coordination game in which δ > 0.
4 The analysis
Having characterized the two equilibria, we are ready to present our main result about the
effi ciency of banning short selling.
We have defined the effi ciency as the ex-ante expected firm value V in equation (1). After
some algebra, for a given regime j ∈ {A,B}, we can write it generally as
V = V FB − 1
2(εH + rεL).
Note that V FB = 12rH is the effi ciency in the first-best case when θ is publicly known and
the investment decision is made by a single investor. In this case, the representative investor
invests in the good state and does not invest in the bad state. Relative to the first-best,
the effi ciency is ultimately reduced by two types of errors in the investment decisions, under-
investment in the good state or over-investment in the bad state. Denote their respective
probabilities as εH and εL. Underinvestment reduces 1 unit of effi ciency by foregoing the
payoff H = 1 in the good state, while overinvestment generates a loss of L = −r when the
state is bad.
The ban affects the speculator’s trading strategy, the information content of the order
flow, the investors’investment decisions and ultimately the effi ciency. We analyze each effect
in turn.
10
Table 1: Investment errors in various scenarios
j A B B −A B −A Prob*cost
εHIj Φ(q∗A−1σn
) Φ(q∗B−1σn
) Φ(q∗B−1σn
)− Φ(q∗A−1σn
) + α
εLIj 1− Φ(q∗A+1σn
) 1− Φ(q∗Bσn
) Φ(q∗A+1σn
)− Φ(q∗Bσn
) + αr
εHUj Φ(q∗A+1σn
) Φ(q∗Bσn
) Φ(q∗Bσn
)− Φ(q∗A+1σn
) − (1− α)
εLUj 1− Φ(q∗A+1σn
) 1− Φ(q∗Bσn
) Φ(q∗A+1σn
)− Φ(q∗Bσn
) + (1− α) r
First, the ban affects the speculator’s trading strategy and investors’investment decisions.
It does not affect the buy order from the positively informed speculator but replaces the sell
order from both the negatively informed speculator and the uninformed speculator with no
trading. Accordingly, the ban does not affect the order flow distribution when the speculator
is positively informed but moves the distribution to the right by one unit in other cases.
Rationally anticipating the consequences of the SS ban on the information content of the
stock price, the investors adjust their investment decisions accordingly. They increases the
order flow threshold above which they will invest by an amount smaller than 1.
Lemma 2 The SS ban changes the investment threshold as follows: q∗A < q∗B < q∗A + 1.
Given the order flow distribution and the investment threshold, the investment errors
in various scenarios are summarized in Table 1. We examine the effi ciency of the resulting
investment decisions. We break down the effi ciency effect further by the speculator’s type,
whether the speculator is informed (I) or uninformed (U). Note that the expected error εθj
satisfies
εθj = αεθIj + (1− α) εθUj .
Consider first the informed speculator who buys in the good state and shorts in the
bad state. In the good state, the order flow has a mean of 1 and variance of σ2n because
the positively informed speculator buys. Thus, the probability of underinvestment is εHIj =
Φ(q∗j−1
σn) for j ∈ {A,B}. This explain the second row in Table 1. Similarly, in the bad state,
the order flow has a mean of −1 if SS is allowed and 0 if SS is banned. The probability
of overinvestment is thus εLIj = 1 − Φ(q∗j−d∗Ij(L)
σn), where d∗Ij (L) is the negatively informed
11
speculator’s trading strategy in regime j ∈ {A,B}. This explains the third row in Table 1.
Therefore, when the speculator is informed, the ban on the quality of the investment
decision is captured by
∆εθI = εθIB − εθIA.
Now consider the uninformed speculator who shorts when SS is allowed and does not trade
when SS is banned. The order flow has a mean of d∗Uj (θ) in regime j and state θ, and the
investment errors can be expressed as in the fourth and fifth row of TABLE Y. Therefore,
when the speculator is uninformed, the effect of the ban on the investment effi ciency is
captured by
∆εθU = εθUB − εθUA.
We can characterize the ban’s effect on investment effi ciencies as follows.
Lemma 3 The SS ban affects the accuracy of the investment decisions as follows.
1. when the speculator is informed, it reduces the investment accuracy, that is, ∆εθI > 0
for any θ;
2. when the speculator is not informed, the ban increase investment accuracy in the good
state and reduces investment accuracy. That is, ∆εHU < 0 and ∆εLU > 0.
3. ∆εLI = ∆εLU = −∆εHU = ∆ε0.
Lemma 3 is intuitive. First, the ban suppresses the information from the negatively in-
formed speculator and increases the overinvestment in the bad state, despite the rational
adjustment by investors. The ban removes the sell order from the negatively informed spec-
ulator and thus degrades the informational value of the order flow. Hence, ∆εLI > 0. Second,
the ban also increases the investment error when the speculator is positively informed, that
is, ∆εHI > 0. Even though the ban does not affect the buy strategy of the positively informed
speculator, it changes the strategy of other types of speculators whose order flow cannot be
distinguished from the positively informed speculator. In particular, the suppression of the
12
sell orders dilutes the information content of the positively informed speculator’s buy order,
which adversely affects the investors’use of information in the investment decisions. Collec-
tively, these two channels explain Part 1 of Lemma 3 and capture the conventional wisdom
that the SS ban reduces the effi ciency by degrading the information value of the stock price.
When the speculator engages in manipulative short selling, that is, shorting when he
is uninformed, how the SS ban affects the investment accuracy depends on the state. In
the good state, the ban reduces investment and leads to more underinvestment, that is,
∆εHU > 0. In the bad state, the ban also reduces investment, but the reduction leads to less
underinvestment, that is, ∆εLU < 0. This explains Part 2 of Lemma 3.
Finally, Part 3 of Lemma 3 shows an articulate relationship among the investment errors.
First, since the speculator is not informed, the investment error is symmetric. When the
ban reduces the underinvestment in the good state, it increases the overinvestment in the
bad state by the same amount. that is, ∆εHU = −∆εLU . Second, in the bad state, both the
uninformed and informed speculators use the same trading strategy across the two regimes.
Thus, the ban has the same effect on the investment accuracy, that is, ∆εLI = ∆εLU .
In sum, when the speculator is informed, the ban reduces effi ciency ∆εHI > 0 and ∆εLI =
∆ε0 > 0. When the speculator is uninformed, the ban reduces effi ciency in the bad state
but increases effi ciency in the good state, that is, ∆εLU = −∆εHU = ∆ε0 < 0. There is a
trade-off. Collecting these errors and weighting them by their probabilities and associated
consequences, we can write out the effi ciency difference across the two regimes as
∆V ≡ 2(VBan − VAllowed)
= −α∆εHI − (αr − (1− α)(1− r))∆ε0. (4)
The next proposition presents our main result.
Proposition 3 Banning short-selling improves the effi ciency if and only if α ∈ (α(δ), α∗∗(δ)),
where α∗∗(δ) is defined in the appendix. This set is empty at δ = 0 and nonempty when δ is
suffi ciently large.
13
We illustrate Proposition 3 with two special cases, one in the absence of coordination
friction δ = 0 and the other with extreme coordination friction δ− > 1− r.
Consider first the case with δ = 0, resulting in a single-person decision making setting.
In this case, the argument from GG shows that the ban reduces the effi ciency. Since the
single investor can always ignore a signal, she cannot be worse off from having additional
information. Since the ban reduces the stock price informativeness, it reduces the effi ciency.
Alternatively, we can also use the expression of ∆V to analyze the ban’s consequences.
In particular, there is an endogenous connection between δ and the possible value of α for
manipulative short-selling to be optimal, which is best illustrated by focusing on the extreme
values of δ. Note that the first term of ∆V is always negative, as discussed above. Thus, the
sign of ∆V crucially depends on the sign of the second term.
When δ → 0, manipulative short-selling to be optimal (i.e. q∗A > −∞), it is necessary that
α > 1−r. Otherwise, if α ≤ 1−r, then the stock price cannot be informative enough to change
the investor’s decision. In other words, short sales can come from both the negatively informed
speculator and the uninformed speculator. The cost of suppressing the former dominates the
benefit of suppressing the latter because α has to be suffi ciently large. α > 1 − r results in
αr− (1−α)(1−r) being positive, i.e. conditional on the state being bad, the beneficial effect
from preventing manipulative short-selling of the uninformed speculator is dominated by the
cost of preventing informed short-selling. This explains why the set is empty when δ = 0.
Before explaining the case when δ approaches 1 − r, it is useful to present the following
Lemma.
Lemma 4 α(δ) is decreasing in δ.
Lemma 4 is intuitive. As the coordination concern becomes severe, investors are more
pessimistic and they are only willing to stay if the probability of the good state is suffi ciently
high. Anticipating this fragility, manipulative short selling is more likely. In fact, as δ ap-
proaches 1−r, an uninformed investor short sells even as the fraction of informed speculators
approaches 0, i.e. α(δ)→ 0, a case which we now turn to.
14
Now consider another extreme case in which δ approaches 1 − r.In this case, α (δ) →
0.When α → 0, clearly αr − (1− α)(1− r) < 0, i.e. conditional on the state being bad, the
beneficial effect from preventing manipulative short-selling of the uninformed speculator is
dominated by the cost of preventing informed short-selling. since the likelihood of informed
short-selling becomes suffi ciently small. The second term of ∆V therefore becomes positive.
The first term, while still being negative, decreases in magnitude when α becomes smaller
as the benefit from informed short-selling decreases regardless of whether the state is good
or bad when the probability of being informed decreases. In fact, when δ → 1 − r, we can
calculate that
∆V → (1− r
2− α)[Φ(
1
σn)− Φ(
1
2σn)]
Note that Φ( 1σn
)− Φ( 12σn
) > 0, that is, the ban reduces the sensitivity of investors’deci-
sions to the order flow. In addition, 1−r2 and α represent the respective effects of coordination
and information on investors’ investment decisions. When 1−r2 > α, then the coordination
effect dominates the information effect. In this case, the ban, by mitigating the investors’
response to the order flow, improves the effi ciency. Otherwise, the ban reduces the effi ciency.
By continuity, we can prove the more general result that the short selling ban improves the
effi ciency if and only if α is not suffi ciently high and δ is suffi ciently high, i.e. the set is
non-empty when δ is suffi ciently large.
To better understand Proposition 3, we provide a numerical example. For ease of nota-
tion we write ∆V as ∆V (α, δ). We assume that rH = 1, σn = 12 , rL = −0.5. Note that in
this case δ = 0.5.When δ = 0, we can numerically calculate that α(0) ' 0.91. In addition,
q∗SA(α(0), 0) ' −0.2 ∈ (−12 , 0) and ∆V (α(0), 0) ' −0.0875 < 0, i.e. banning short-selling de-
creases firm value. When we increase δ and keep α = α(0), short-selling is clearly still optimal
for the uninformed speculator. When δ = 0.48, ∆V (α(0), 0.45) ' −0.0882 < ∆V (α(0), 0),
i.e. banning short-selling is even worse. However, when δ = 0.48, α(0.4) decreases to around
0.03, which enlarges the range for short-selling to be optimal for the uninformed speculator.
For example, when α = 0.2 > α(0.48), ∆V (0.2, 0.48) = 0.0145 > 0. In addition, note that α
cannot be too large. For example, when α = 0.4, ∆V (0.4, 0.48) = −0.0168 < 0, i.e. banning
15
short-selling is bad when α = 0.4.
5 Empirical and policy implications
TBC
6 Conclusions
TBC
7 Appendix: Proofs
7.1 Proof of Lemma 1:
Proof. We first prove that the investors play a common threshold strategy. It is proved in
two steps. In the first step, we show that all investors use the same strategy. In the second
step, we show that the equilibrium strategy must be a single threshold. Note that the proof
assumes that it is optimal for the positively informed speculator to choose d = 1 and the
negatively informed speculator to choose d = −1. This will be proved at the very end after
step 2.
First, suppose that investor i chooses to withdraw if and only if qi ∈ Si, and investor j
chooses to run if and only if qj ∈ Sj , where Si and Sj are subsets of the real line. Suppose
that Si 6= Sj . This implies that at least at least one of the sets of Si or Sj must be non-empty.
Without loss of generality suppose that Si is not empty. This implies that there exists a q0
such that q0 ∈ Si but q0 /∈ Sj . This implies that upon observing qi = q0, investor i stays
but investor j withdraws, i.e. ∆(qi = q0) > 0 ≥ ∆(qj = q0), which is a contradiction as
∆(qi = q0) = ∆(qj = q0) as ∆(q) only depends on q for any fixed speculators’ strategies.
Therefore all investors use the same strategy.
Second, upon observing qi, investor i’s expected payoff of staying relative to withdrawing
16
is
∆(qi)
= Pr(θ = H|qi)(H − δE[l|qi]) + Pr(θ = L|qi)(L− δE[l|qi])
= L+ Pr(θ = H|qi)(H − L)− δE[l|qi]
Assume now that positively informed speculator will choose d = +1 and negatively in-
formed speculator will choose d = −1. Denote the uninformed speculator’s strategy by
du ∈ [−1, 1]. Denote the density of q conditional on r as g. Then
Pr(θ = H|qi)
=g(qi|θ = H)
g(qi|θ = H) + g(qi|θ = L)
=
g(qi|θ=H)g(qi|θ=L)
g(qi|θ=H)g(qi|θ=L) + 1
Note that
g(qi|θ = H)
g(qi|θ = L)
=αφ( qi−1√
σ2n+σ2ε) + (1− α)φ( qi−du√
σ2n+σ2ε)
αφ( qi+1√σ2n+σ2ε
) + (1− α)φ( qi−du√σ2n+σ2ε
)
=
αφ(
qi−1√σ2n+σ
2ε
)
φ(qi−du√σ2n+σ
2ε
)+ (1− α)
αφ(
qi+1√σ2n+σ
2ε
)
φ(qi−du√σ2n+σ
2ε
)+ (1− α)
=αe
2(1−du)qi+d2u−1
2(σ2n+σ2ε) + (1− α)
αe−2(1+du)qi+d2u−1
2(σ2n+σ2ε) + (1− α)
Thus, for any du ∈ (−1, 1), limqi→−∞
Pr(θ = H|qi) = 0 and limqi→+∞
Pr(θ = H|qi) = 1. Therefore
by continuity, both upper dominance region and lower dominance region exists. Therefore
there exists finite q and q such that ∆(qi) < 0 if q < q and ∆(qi) > 0 if q < q. When du = −1,
17
limqi→−∞
Pr(θ = H|qi) = 1−α2−α and lim
qi→+∞Pr(θ = H|qi) = 1. Therefore the upper dominance
region exists and q exists. If 1−α2−αH + 1
2−α(−L) < 0, then the lower dominance region exists
and thus q exists. If 1−α2−αH + 1
2−α(−L) ≥ 0, then the lower dominance region does not exist.
When du = +1, limqi→−∞
Pr(θ = H|qi) = 0 and limqi→+∞
Pr(θ = H|qi) = 12−α . Therefore
the lower dominance region exists and thus q exists. If 12−αH + 1−α
2−α(−L) > δ, then the
upper dominance region exists and thus q exists. If 12−αH + 1−α
2−α(−L) ≤ δ, then the upper
dominance region does not exist.
As a summary, in equilibrium we either have 1) ∆(qi) < 0 if q < q and ∆(qi) > 0 if q > q
or 2) ∆(qi) < 0 if q < q or 3) ∆(qi) > 0 if q > q. We now show that under either of the three
scenarios, common threshold strategy is the equilibrium.
For case 1) and 2), denote qB = sup{qi : ∆(qi) < 0}, i.e. the highest signal below which a
investor prefers to withdraw. Note that it is possible for qB = +∞, which then implies that
∆(qB) ≤ 0. If all investors use threshold strategy then investor i will withdraw when qi < qB,
for any i. Suppose in equilibrium they do not use threshold strategy, then for investor i,
there exists signals smaller than qB such that a investor observing qi will stay. Denote qA to
be the largest of them, i.e. qA = sup{qi < qB : ∆(qi) ≥ 0}. q < qA < qB and thus is finite.
This implies that investors in the range [qA, qB] and (−∞, q) will withdraw for sure, while
investors in the range of [q, qA) may choose to stay or withdraw and denote their strategies
by n(qi) ∈ [0, 1]. Since investors are indifferent upon observing qA, we have
∆(qA) = 0 ≥ ∆(qB)
On the other hand, note that
E[l|qA]
= Φ(q − qA√
2σε) +
∫ qA
qn(qj)
1√2σε
φ(qj − qA√
2σε)dqj + Φ(
qB − qA√2σε
)− 1
2
18
and
E[l|qB]
= Φ(q − qB√
2σε) +
∫ qA
qn(qj)
1√2σε
φ(qj − qB√
2σε)dqj +
1
2− Φ(
qA − qB√2σε
)
= Φ(q − qB√
2σε) +
∫ qA
qn(qj)
1√2σε
φ(qj − qB√
2σε)dqj + Φ(
qB − qA√2σε
)− 1
2
where we used Φ(−x) = 1− Φ(x) to arrive at the last equality.
Thus
E[l|qB]− E[l|qA]
= Φ(q − qB√
2σε)− Φ(
q − qA√2σε
) +
∫ qA
qn(qj)
1√2σε
[φ(qj − qB√
2σε)− φ(
qj − qA√2σε
)]dqj
Since qB > qA, φ(qj−qB√
2σε)− φ(
qj−qA√2σε
) < 0 for any qj ∈ [q, qA]. Therefore
E[l|qB]− E[l|qA]
≤ Φ(q − qB√
2σε)− Φ(
q − qA√2σε
) < 0
In addition, Pr(θ = H|qi) is increasing in qi, which results in Pr(θ = H|qB) > Pr(θ =
H|qA). Correspondingly, ∆(qB) > ∆(qA) and thus, the contradiction. Therefore all investors
use the same threshold strategy in equilibrium.
For case 1) and 3), denote qC = inf{qi : ∆(qi) > 0}, i.e. the lowest signal above which a
investor will stay. Note that it is possible for qC = −∞, which then implies that∆(qC) ≥ 0. If
all investors use threshold strategy then investor i will stay when qi > qC , for any i. Suppose
in equilibrium they do not use threshold strategy, then for investor i, there exists signals
larger than qC such that a investor observing qi will withdraw. Denote qD to be the smallest
of them, i.e. qD = sup{qi > qC : ∆(qi) ≤ 0}. qC < qD < q and thus is finite. This implies
that investors in the range [qC , qD] and (q,+∞) will stay for sure, investors in the range of
(−∞, qC) will withdraw for sure, while investors in the range of [qD, q) may choose to stay
or withdraw and denote their strategies by n(qi) ∈ [0, 1]. Since investors are indifferent upon
19
observing qD, we have
∆(qD) = 0 ≤ ∆(qC)
Using similar techniques as above we can show that this inequality cannot hold. Therefore
all investors use the same threshold strategy in equilibrium.
We now prove the optimal strategies for the informed speculator.
First, consider the strategy of the negatively informed speculator. Since the speculator
knows that θ = L, he knows that equity value is non-positive. Thus, when he takes action d,
his profit will be π(d) = −dE[P |d]. Since E[P |d] ≥ 0, π(−1) ≥ π(0) ≥ π(1). Assumption A2
then implies that d = −1 for the negatively informed speculator.
Next, consider the strategy of the positively informed speculator. Since the speculator
knows that θ = H, he knows that equity value is (1 − l)(H − δl). On the other hand,
conditional on total order flow q, stock price
P (q) = Pr(θ = H|q)[1− l(q)][H − δl(q)] + Pr(θ = L|q)[1− l(q)][L− δl(q)]
= [1− l(q)][Pr(θ = H|q)(H − δl(q)) + Pr(θ = L|q)(L− δl(q))]
Therefore, when the speculator takes action d, the speculator’s profit will be
π(d) = d(E[V |d]− E[P |d])
= d(E[(1− l)(H − δl)|d]− E[(1− l)(Pr(θ = H|q)(H − δl) + Pr(θ = L|q)(L− δl))|d])
= dE[(1− l) Pr(θ = L|q)(H − L)|d]
Since E[(1 − l) Pr(θ = L|q)(1 − L)|d] ≥ 0, π(1) ≥ π(0) ≥ π(−1). Assumption A2 then
implies that d = 1 for the positively informed speculator.
Given that each investor uses a common threshold strategy, we now solve for the common
threshold. We assume that each investor will choose to run if and only if the order flow
20
qi ≤ q∗ for some threshold q∗ where q∗ is determined by the indifferent condition.
∆(q∗)
= Pr(θ = H|q∗)(H − δE[l|q∗]) + Pr(θ = L|q∗)(rL − δE[l|q∗])
= 0
When q = q∗,
E[l|q∗]
= Pr(qj ≤ q∗|qi = q∗)
=1
2
, which results in
Pr(θ = H|q∗) =δ2 − rLH − rL
(5)
As shown in the proof of Lemma 1, Pr(θ = H|q∗) is strictly increasing in q∗. Therefore,
equation (5) has at most one solution. The expression of Pr(θ = H|q∗), however, depends on
q∗ and therefore the uninformed speculators’strategies (as the informed speculator always
buy when observing θ = H and sell when observing θ = L), which we turn to now. Again
denote the density of q conditional on r as g. From Bayes’rule and given the speculators’
strategies that the uninformed speculator short-sells,
Pr(θ = H|q∗A)
=g(q∗A|θ = H)
g(q∗A|θ = H) + g(q∗A|θ = L)
=αφ(
q∗A−1√σ2n+σ2ε
) + (1− α)φ(q∗A+1√σ2n+σ2ε
)
αφ(q∗A−1√σ2n+σ2ε
) + (2− α)φ(q∗A+1√σ2n+σ2ε
)
21
Insert into equation (5) results in
αφ(q∗A−1√σ2n+σ2ε
) + (1− α)φ(q∗A+1√σ2n+σ2ε
)
αφ(q∗A−1√σ2n+σ2ε
) + (2− α)φ(q∗A+1√σ2n+σ2ε
)=
δ2 − LH − L
, which can be rearranged as
αφ(
q∗A−1√σ2n+σ2ε
)
φ(q∗A+1√σ2n+σ2ε
)+ 1− α =
δ2 − LH − δ
2
(6)
First note that equation (6) has a solution α ∈ (0, 1) only if 1−α <δ2−L
H− δ2
, or, equivalently,
α > 1−δ2−L
H− δ2
. When α ≤ 1−δ2−L
H− δ2
, then
αφ(q∗A−1√σ2n+σ2ε
) + (1− α)φ(q∗A+1√σ2n+σ2ε
)
αφ(q∗A−1√σ2n+σ2ε
) + (2− α)φ(q∗A+1√σ2n+σ2ε
)≥
δ2 − rLH − rL
, implying that it is always optimal for the investors to stay, or, equivalently, q∗A = −∞.
Second, when α > 1−δ2−L
H− δ2
, then when q∗A = 0, equation (6) becomes
1 =δ2 − LH − δ
2
,or, equivalently,
δ = δ = H + L = 1− r
Thus, when α > 1−δ2−L
H− δ2
, one can solve for a close-form solution of q∗A to be
q∗A =σ2n + σ2
ε
2[ln(α− (1−
δ2 − LH − δ
2
))− lnα]
→ σ2n
2[ln(α− (1−
δ2 − LH − δ
2
))− lnα]
when σ2ε → 0
22
When α ≤ 1−δ2−L
H− δ2
,
q∗A = −∞
When the uninformed speculator does not trade, we can similarly calculate that
Pr(θ = H|q∗NT )
=g(q∗NT |θ = H)
g(q∗NT |θ = H) + g(q∗NT |θ = L)
=αφ(
q∗NT−1√σ2n+σ2ε
) + (1− α)φ(q∗NT√σ2n+σ2ε
)
αφ(q∗NT−1√σ2n+σ2ε
) + αφ(q∗NT+1√σ2n+σ2ε
) + 2(1− α)φ(q∗NT√σ2n+σ2ε
)
Insert into equation (5) and rearranging terms results in
αφ(
q∗NT+1√σ2n+σ
2ε
)
φ(q∗NT√σ2n+σ
2ε
)+ 1− α
αφ(
q∗NT−1√
σ2n+σ2ε
)
φ(q∗NT√σ2n+σ
2ε
)+ 1− α
=H − δ
2δ2 − L
(7)
The solution of equation (7) is unique as the left hand side of equation (7) is decreasing in
q∗NT . This is because the numerator is decreasing in q∗NT and the denominator is increasing
in q∗NT . When q∗NT → −∞, the left hand side approaches +∞ which is clearly larger than
the right hand side. In addition, when q∗NT → +∞, the right hand side approaches 0 which
is clearly smaller than the right hand side.
When short-sell is banned, then both uninformed and negatively informed speculator do
not trade. we can similarly calculate that
23
Pr(θ = H|q∗B)
=g(q∗B|θ = H)
g(q∗B|θ = H) + g(q∗B|θ = rL)
=αφ(
q∗B−1√σ2n+σ2ε
) + (1− α)φ(q∗B√σ2n+σ2ε
)
αφ(q∗B−1√σ2n+σ2ε
) + (2− α)φ(q∗B√σ2n+σ2ε
)
Insert into equation (5) and rearranging terms results in
αφ(
q∗B−1√σ2n+σ2ε
)
φ(q∗B√σ2n+σ2ε
)+ 1− α =
δ2 − LH − δ
2
(8)
Note that the left hand side of equation (8) is increasing in q∗B. When q∗B → +∞, the left
hand side becomes +∞, which is clearly larger than the right hand side. When q∗B → −∞,
the left hand side becomes 1−α. Thus, equation (8) has a solution if and only if 1−α <δ2−L
H− δ2
,
or α > 1−δ2−L
H− δ2
. Solving equation (8) results in
q∗B = 2q∗A +1
2= (σ2
n + σ2ε) ln
α− [1− (δ2−L
H− δ2
)]
α+
1
2
When α ≤ 1−δ2−L
H− δ2
, then q∗B = −∞.
7.2 Proof of Proposition 2:
Proof. The proposition is proved in three steps. In step 1, we show that it is never optimal
for the uninformed speculator to buy. In step 2, we derive the conditions where it is optimal
for the uninformed speculator to choose d = −1 under the conjecture that the uninformed
speculator will choose d = −1. In step 3, we derive the conditions where it is optimal for
the uninformed speculator to deviate to d = −1 under the conjecture that the uninformed
speculator will choose d = 0. Combining the conditions in steps 2 and 3 arrives in the
conditions as stated in Proposition 2.
24
Step 1: It is never optimal for the uninformed speculator to choose d = 1.
If the uninformed speculator chooses to buy, then from his perspective q ∼ N(1, σ2n) and
qi ∼ N(1, σ2n + σ2
ε).
Suppose the market conjectures that the uninformed speculator chooses d = 1, from
Bayes’rule and given the speculators’strategies,
Pr(θ = H|q)
=g(q|θ = H)
g(q|θ = H) + g(q|θ = L)
=φ( q−1√
σ2n+σ2ε)
αφ( q+1√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)
In addition,
l(q)
= Pr(qi ≤ q∗B|q)
= Φ(q∗B − qσε
)
where we denote the threshold when uninformed speculator chooses to buy by q∗B and q∗B > 0.
Thus
P (q) = Pr(θ = H|q)[1− l(q)][H − δl(q)] + Pr(θ = L|q)[1− l(q)][L− δl(q)]
=φ( q−1√
σ2n+σ2ε)
αφ( q+1√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)[1− Φ(
q∗B − qσε
)][H − δΦ(q∗B − qσε
)]
+αφ( q+1√
σ2n+σ2ε) + (1− α)φ( q−1√
σ2n+σ2ε)
αφ( q+1√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)[1− Φ(
q∗B − qσε
)][L− δΦ(q∗B − qσε
)]
25
Therefore, if the speculator chooses to buy, he expects to pay
E[P (q)]
=
∫ +∞
−∞
φ( q−1√σ2n+σ2ε
)
αφ( q+1√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)[1− Φ(
q∗B − qσε
)][H − δΦ(q∗B − qσε
)]1
σnφ(q − 1
σn)dq
+
∫ +∞
−∞
φ( q−1√σ2n+σ2ε
)
αφ( q+1√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)[1− Φ(
q∗B − qσε
)][L− δΦ(q∗B − qσε
)]1
σnφ(q − 1
σn)dq
whereas he expects the security to pay off
E[V ]
=
∫ +∞
−∞[1− Φ(
q∗B − qσε
)][H + L
2− δΦ(
q∗A − qσε
)]1
σnφ(q − 1
σn)dq
Therefore, the speculator’s payoff from short-selling is
E[V ]− E[P (q)]
=
∫ +∞
−∞[1
2−
φ( q−1√σ2n+σ2ε
)
αφ( q+1√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)][1− Φ(
q∗B − qσε
)](H − L)
× 1
σnφ(q − 1
σn)dq
=
∫ q∗B
−∞[1
2−
φ( q−1√σ2n+σ2ε
)
αφ( q+1√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)][1− Φ(
q∗B − qσε
)](H − L)
× 1
σnφ(q − 1
σn)dq
+
∫ +∞
q∗B
[1
2−
φ( q−1√σ2n+σ2ε
)
αφ( q+1√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)][1− Φ(
q∗B − qσε
)](H − L)
× 1
σnφ(q − 1
σn)dq
26
When σε → 0,
E[V ]− E[P (q)]
→∫ +∞
q∗B
[1
2−
φ( q−1σn
)
αφ( q+1σn
) + (2− α)φ( q−1σn
)]
1
σn(H − L)φ(
Q− 1
σn)dQ
= −H − L2
∫ +∞
q∗B
α(1− e−2q
σ2n )
[2− α(1− e−2q
σ2n )]
1
σnφ(q − 1
σn)dq
=H − L
2
∫ −∞−q∗B
α(1− e2q
σ2n )
[2− α(1− e2q
σ2n )]
1
σnφ(−q − 1
σn)dQ
=H − L
2
∫ −q∗B−∞
α(e2q
σ2n − 1)
[2 + α(e2q
σ2n − 1)]
1
σnφ(q + 1
σn)dq
=H − L
2
∫ −q∗B−∞
f(α, q, σn)dq
≤ H − L2
∫ +∞
−∞f(α, q, σn)dq < 0
where we used change of variables in arriving at the third inequality. So it is suboptimal
for the uninformed speculator to buy when the market’s conjecture is that the uninformed
speculator will buy.
We now show that when the market’s conjecture is that the speculator will not trade, it
is still suboptimal for the uninformed speculator to buy.
If the speculator chooses to buy, he expects to pay
E[P (q)]
=
∫ +∞
−∞
αφ( q−1√σ2n+σ2ε
) + (1− α)φ( q√σ2n+σ2ε
)
αφ( q−1√σ2n+σ2ε
) + αφ( q+1√σ2n+σ2ε
) + 2(1− α)φ( q√σ2n+σ2ε
)[1− Φ(
q∗NT − qσε
)][H − δΦ(q∗NT − qσε
)]
× 1
σnφ(q − 1
σn)dq
+
∫ +∞
−∞
αφ( q+1√σ2n+σ2ε
) + (1− α)φ( q√σ2n+σ2ε
)
αφ( q−1√σ2n+σ2ε
) + αφ( q+1√σ2n+σ2ε
) + 2(1− α)φ( q√σ2n+σ2ε
)[1− Φ(
q∗NT − qσε
)][L− δΦ(q∗NT − qσε
)]
× 1
σnφ(q − 1
σn)dq
27
whereas he expects the security to pay off
E[V ]
=
∫ +∞
−∞[1− Φ(
q∗NT − qσε
)][H + L
2− δΦ(
q∗NT − qσε
)]1
σnφ(q − 1
σn)dq
Therefore, the speculator’s payoff from short-selling is
E[V ]− E[P (q)]
=
∫ +∞
−∞[
αφ( q−1√σ2n+σ2ε
) + (1− α)φ( q√σ2n+σ2ε
)
αφ( q−1√σ2n+σ2ε
) + αφ( q+1√σ2n+σ2ε
) + 2(1− α)φ( q√σ2n+σ2ε
)− 1
2](H − L)[1− Φ(
q∗NT − qσε
)]
× 1
σnφ(q − 1
σn)dq
=
∫ q∗NT
−∞[
αφ( q−1√σ2n+σ2ε
) + (1− α)φ( q√σ2n+σ2ε
)
αφ( q−1√σ2n+σ2ε
) + αφ( q+1√σ2n+σ2ε
) + 2(1− α)φ( q√σ2n+σ2ε
)− 1
2](H − L)[1− Φ(
q∗NT − qσε
)]
× 1
σnφ(q − 1
σn)dq
+
∫ +∞
q∗NT
[αφ( q−1√
σ2n+σ2ε) + (1− α)φ( q√
σ2n+σ2ε)
αφ( q−1√σ2n+σ2ε
) + αφ( q+1√σ2n+σ2ε
) + 2(1− α)φ( q√σ2n+σ2ε
)− 1
2](H − L)[1− Φ(
q∗NT − qσε
)]
× 1
σnφ(q − 1
σn)dq
When σε → 0,
E[P (q)]− E[V ]
→ (H − L)
∫ +∞
q∗NT
[αφ( q−1√
σ2n+σ2ε) + (1− α)φ( q√
σ2n+σ2ε)
αφ( q−1√σ2n+σ2ε
) + αφ( q+1√σ2n+σ2ε
) + 2(1− α)φ( q√σ2n+σ2ε
)− 1
2][1− Φ(
q∗NT − qσε
)]
× 1
σnφ(q − 1
σn)dq
=1
2(H − L)
∫ +∞
q∗NT
αφ( q−1σn
)− αφ( q+1σn
)
αφ( q−1σn
) + αφ( q+1σn
) + 2(1− α)φ( qσn
)
1
σnφ(q − 1
σn)dq
=1
2(H − L)
∫ +∞
q∗NT
αφ( q−1σn
)
αφ( q−1σn
) + αφ( q+1σn
) + 2(1− α)φ( qσn
)
1
σn[φ(
q − 1
σn)− φ(
q + 1
σn)]dq
28
Thus
sgn{E[P (q)]− E[V ]}
= sgn(
∫ +∞
q∗NT
φ( q−1σn
)
αφ( q−1σn
) + αφ( q+1σn
) + 2(1− α)φ( qσn
)
1
σn[φ(
q − 1
σn)− φ(
q + 1
σn)]dq)
= sgn(
∫ +∞
q∗NT
1
α+ αe− 2q
σ2n + 2(1− α)e− 2q−1
2σ2n
1
σn[φ(
q − 1
σn)− φ(
q + 1
σn)]dq) < 0
To see why the sign is negative, note that using integration by parts,
∫ +∞
q∗NT
1
α+ αe− 2q
σ2n + 2(1− α)e− 2q−1
2σ2n
1
σn[φ(
q − 1
σn)− φ(
q + 1
σn)]dq
=1
α+ αe− 2q
σ2n + 2(1− α)e− 2q−1
2σ2n
[Φ(q − 1
σn)− Φ(
q + 1
σn)]|+∞q∗NT
−∫ +∞
q∗NT
∂
∂q[
1
α+ αe− 2q
σ2n + 2(1− α)e− 2q−1
2σ2n
][Φ(q − 1
σn)− Φ(
q + 1
σn)]dq
=1
α+ αe−2q∗NTσ2n + 2(1− α)e
−2q∗NT−1
2σ2n
[Φ(q∗NT − 1
σn)− Φ(
q∗NT + 1
σn)]
−∫ +∞
q∗NT
∂
∂q[
1
α+ αe− 2q
σ2n + 2(1− α)e− 2q−1
2σ2n
][Φ(q − 1
σn)− Φ(
q + 1
σn)]dq
Both terms are negative. The first term is negative because Φ(q∗NT−1σn
) − Φ(q∗NT+1σn
) < 0.
The second term is negative because ∂∂q [ 1
α+αe− 2q
σ2n +2(1−α)e− 2q−12σ2n
] < 0 and Φ( q−1σn
)− Φ( q+1σn
) <
0. Therefore given the conjecture is that the uninformed speculator will not trade, the
uninformed trader will not buy as well. We can similarly show that when the conjecture is
that the uninformed speculator will short-sell, the uninformed trader will not buy as well.
Therefore, it is always suboptimal for the uninformed trader to buy and Step 1 is complete.
Step 2: It is optimal for the uninformed speculator to choose d = −1 under the
conjecture that the uninformed speculator chooses d = −1 when α < α1 for some α1.
If the uninformed speculator chooses to short-sell, then from his perspective q ∼ N(−1, σ2n)
and qi ∼ N(−1, σ2n + σ2
ε).
29
Conditional upon observing q, the market maker will set
P (q) = E[(1− l)(θ − δl)|q]
From Bayes’rule and given the speculators’strategies,
Pr(θ = H|q)
=g(q|θ = H)
g(q|θ = H) + g(q|θ = L)
=αφ( q−1√
σ2n+σ2ε) + (1− α)φ( q+1√
σ2n+σ2ε)
αφ( q−1√σ2n+σ2ε
) + (2− α)φ( q+1√σ2n+σ2ε
)
In addition,
l(q)
= Pr(qi ≤ q∗A|q)
= Φ(q∗A −Qσε
)
Thus
P (q)
=αφ( q−1√
σ2n+σ2ε) + (1− α)φ( q+1√
σ2n+σ2ε)
αφ( q−1√σ2n+σ2ε
) + (2− α)φ( q+1√σ2n+σ2ε
)[1− Φ(
q∗A − qσε
)][H − δΦ(q∗A − qσε
)]
+φ( q+1√
σ2n+σ2ε)
αφ( q−1√σ2n+σ2ε
) + (2− α)φ( q+1√σ2n+σ2ε
)[1− Φ(
q∗A − qσε
)][L− δΦ(q∗A − qσε
)]
30
Therefore, if the speculator chooses to short-sell, he expects to get
E[P (q)]
=
∫ +∞
−∞
αφ( q−1√σ2n+σ2ε
) + (1− α)φ( q+1√σ2n+σ2ε
)
αφ( q−1√σ2n+σ2ε
) + (2− α)φ( q+1√σ2n+σ2ε
)[H − δΦ(
q∗A − qσε
)][1− Φ(q∗A − qσε
)]
× 1
σnφ(q + 1
σn)dq
+
∫ +∞
−∞
φ( q+1√σ2n+σ2ε
)
αφ( q−1√σ2n+σ2ε
) + (2− α)φ( q+1√σ2n+σ2ε
)[L− δΦ(
q∗A − qσε
)][1− Φ(q∗A − qσε
)]
× 1
σnφ(q + 1
σn)dq
where the last equality comes from law of iterated expectations.
He expects the security to pay off
E[V ]
=
∫ +∞
−∞[H + L
2− δΦ(
q∗A − qσε
)][1− Φ(q∗A − qσε
)]1
σnφ(q + 1
σn)dq
Therefore, the speculator’s payoff from short-selling is
E[P (q)]− E[V ]
=
∫ +∞
−∞[αφ( q−1√
σ2n+σ2ε) + (1− α)φ( q+1√
σ2n+σ2ε)
αφ( q−1√σ2n+σ2ε
) + (2− α)φ( q+1√σ2n+σ2ε
)− 1
2][1− Φ(
q∗A − qσε
)](H − L)
× 1
σnφ(q + 1
σn)dq
=
∫ q∗A
−∞[αφ( q−1√
σ2n+σ2ε) + (1− α)φ( q+1√
σ2n+σ2ε)
αφ( q−1√σ2n+σ2ε
) + (2− α)φ( q+1√σ2n+σ2ε
)− 1
2][1− Φ(
q∗A − qσε
)](H − L)
× 1
σnφ(q + 1
σn)dq
+
∫ +∞
q∗A
[αφ( q−1√
σ2n+σ2ε) + (1− α)φ( q+1√
σ2n+σ2ε)
αφ( q−1√σ2n+σ2ε
) + (2− α)φ( q+1√σ2n+σ2ε
)− 1
2][1− Φ(
q∗A − qσε
)](H − L)
× 1
σnφ(q + 1
σn)dq
31
When σε → 0,
E[P (q)]− E[V ]
→∫ +∞
q∗A
[αφ( q−1√
σ2n+σ2ε) + (1− α)φ( q+1√
σ2n+σ2ε)
αφ( q−1√σ2n+σ2ε
) + (2− α)φ( q+1√σ2n+σ2ε
)− 1
2](H − L)
1
σnφ(Q+ 1
σn)dQ
When δ < δ, and α ≤ 1−δ2−L
H− δ2
, q∗A = −∞ and
E[P (q)]− E[V ]
→∫ +∞
−∞[αφ( q−1√
σ2n+σ2ε) + (1− α)φ( q+1√
σ2n+σ2ε)
αφ( q−1√σ2n+σ2ε
) + (2− α)φ( q+1√σ2n+σ2ε
)− 1
2](H − L)
1
σnφ(q + 1
σn)dq
=
∫ +∞
−∞[αe
2q
σ2n + (1− α)
αe2q
σ2n + (2− α)
− 1
2](H − L)
1
σnφ(q + 1
σn)dq
= (H − L)
∫ +∞
−∞
α(e2q
σ2n − 1))
2[αe2q
σ2n + (2− α)]
1
σnφ(q + 1
σn)dq
≡ (H − L)
2
∫ +∞
−∞f(α, q, σn)dq
where
f(α, q, σn)
=α(e
2q
σ2n − 1)
[2 + α(e2q
σ2n − 1)]
1
σnφ(q + 1
σn)
Thus
sgn[E[P (q)]− E[V ]]
= sgn[
∫ +∞
−∞f(α, q, σn)dq]
32
Note that we can write∫ +∞−∞ f(α, q, σn)dq as
∫ +∞
−∞f(α, q, σn)dq
=
∫ 0
−∞f(α, q, σn)dq +
∫ +∞
0f(α, q, σn)dq
=
∫ +∞
0f(α,−q, σn)dq +
∫ +∞
0f(α, q, σn)dq
=
∫ +∞
0[f(α, q, σn) + f(α,−q, σn)]dq
where we used change of variables to arrive at the second inequality. Note that f(α, q, σn) >
0 > f(α,−q, σn) as f(α, q, σ2n) > 0 if and only if q > 0. Note that
f(α, q, σn)
−f(α,−q, σn)=
2 + α(e− 2q
σ2n − 1)
2 + α(e2q
σ2n − 1)
< 1
as
e− 2q
σ2n < e2q
σ2n
Therefore f(α, q, σn) + f(α,−q, σn) < 0 and therefore∫ +∞−∞ f(α, q, σn)dq < 0, resulting
in E[P (q)] − E[V ] < 0. Short-selling is therefore not optimal for the uninformed speculator
when δ < δ, and α ≤ 1−δ2−L
H− δ2
.
When δ < δ, and α > 1−δ2−L
H− δ2
, q∗A < 0 and
E[P (q)]− E[V ]
→ H − L2
∫ +∞
q∗A
f(α, q, σn)dq
Through changing variables, for any N > 0, we have
∫ 0
q∗A
f(α, q, σn)dq
=1
N
∫ 0
Nq∗A
f(α,q
N, σn)dq
=1
N
∫ −Nq∗A0
f(α,− q
N, σn)dq
33
Therefore
∫ +∞
q∗A
f(α, q, σn)dq
=1
N
∫ −Nq∗A0
f(α,− q
N, σn)dq +
∫ +∞
0f(α, q, σn)dq
=
∫ −Nq∗A0
[1
Nf(α,− q
N, σn) + f(α, q, σn)]dq +
∫ +∞
−Nq∗Af(α, q, σn)dq
Recall that when σ2ε → 0,
q∗A →σ2n
2[ln(α− (1−
δ2 − LH − δ
2
))− lnα]
Thus, when δ < δ, q∗A is increasing in α as
∂q∗A∂α
=σ2n
2
1−δ2−L
H− δ2
α[α− (1−δ2−L
H− δ2
)]> 0
Therefore, the derivative of∫ +∞q∗A
f(α, q, σn)dq with respect to α is
∂
∂α
∫ +∞
q∗A
f(α, q, σn)dq
= [1
Nf(α, q∗A, σn) + f(α,−Nq∗A, σn)]N
∂(−q∗A)
∂α
+
∫ −Nq∗A0
∂
∂α[
1
Nf(α,− q
N, σn) + f(α, q, σn)]dq
−f(α,−Nq∗A, σn)N∂(−q∗A)
∂α+
∫ +∞
−Nq∗A
∂
∂αf(α, q, σn)dq
= f(α, q∗A, σn)∂(−q∗A)
∂α
+
∫ −Nq∗A0
∂
∂α[
1
Nf(α,− q
N, σn) + f(α, q, σn)]dq
+
∫ +∞
−Nq∗A
∂
∂αf(α, q, σn)dq
Now take the limit of ∂∂α
∫ +∞q∗A
f(α, q, σn)dq when N → +∞. The first term is positive as
34
f(α, q∗A, σn) < 0 and ∂(−q∗A)∂α = −∂q∗A
∂α < 0. The second term is positive as
limN→+∞
∂
∂α[
1
Nf(α,− q
N, σn) + f(α, q, σn)]
=∂
∂αf(α, q, σn)
→
√2πe− (q+1)2
2σ2n
(e2q
σ2n − 1
)σn
(α
(e2q
σ2n − 1
)+ 2
)2
> 0 when q > 0
The third term converges to zero as −Nq∗A → +∞. Therefore ∂∂α
∫ +∞q∗A
f(α, q, σn)dq > 0
when N → +∞, implying that ∂∂α
∫ +∞q∗A
f(α, q, σn)dq > 0 as we can choose a number N0
suffi ciently large so that we can change variables and express∫ +∞q∗A
f(α, q, σn)dq as
∫ +∞
q∗A
f(α, q, σn)dq
=
∫ −N0q∗A0
[1
N0f(α,− q
N0, σn) + f(α, q, σn)]dq +
∫ +∞
−N0q∗Af(α, q, σn)dq
Thus∫ +∞q∗A
f(α, q, σn)dq is increasing in α. When α → 1 −δ2−L
H− δ2
, q∗A → −∞ and∫ +∞q∗A
f(α, q, σn)dq < 0. When α → 1, q∗A →σ2n2 ln
δ2−L
H− δ2
. Note that when δ < δ,δ2−L
H− δ2
< 1.
In addition,δ2−L
H− δ2
increases in δ, resulting in q∗A increasing in δ. Since when fixing α,∫ +∞q∗A
f(α, q, σn)dq increases in q∗A when q∗A < 0,∫ +∞q∗A
f(α, q, σn)dq increases in δ. When
δ → δ,δ2−L
H− δ2
→ 1 and q∗A → 0 and∫ +∞q∗A
f(α, q, σn)dq > 0. When δ → 0,δ2−L
H− δ2
→ r, imply-
ing that q∗A is increasing in r. When r → 0, then q∗A → −∞ and∫ +∞q∗A
f(α,Q, σn)dQ < 0.
When r → 1 then q∗A → 0 and∫ +∞q∗A
f(α, q, σn)dq > 0. Therefore there exists a unique
α1 ∈ (1−δ2−L
H− δ2
, 1) which is also a function of r and δ such that short-selling is optimal form
the uninformed speculator if α > α1 and either
1) r > cr1 or
2) r ≤ cr1 but δ > cδ.
35
where cr1 is the unique solution to
∫ +∞
σ2n2
ln cr1
f(1, Q, σn)dQ = 0 (9)
, cδ is the unique solution to
∫ +∞
σ2n2
lncδ2 −LH− cδ2
f(1, Q, σn)dQ = 0
, α1 is the unique solution to
∫ +∞
σ2n2
[ln(α1−(1−δ2−LH− δ2
))−lnα1]f(α1, Q, σn)dQ = 0 (10)
Step 2 is thus proved.
Step 3: It is optimal for the uninformed speculator to choose d = −1 under the
conjecture that the uninformed speculator chooses d = 0 when α < α2 for some
α2.
When the conjecture is that uninformed speculator chooses d = 0, the investors will
withdraw if and only if q ≤ q∗NT .
If the speculator chooses to short-sell, he expects to get
E[P (q)]
=
∫ +∞
−∞
αφ( q−1√σ2n+σ2ε
) + (1− α)φ( q√σ2n+σ2ε
)
αφ( q−1√σ2n+σ2ε
) + αφ( q+1√σ2n+σ2ε
) + 2(1− α)φ( q√σ2n+σ2ε
)[1− Φ(
q∗NT − qσε
)][H − δΦ(q∗NT − qσε
)]
× 1
σnφ(q + 1
σn)dq
+
∫ +∞
−∞
αφ( q+1√σ2n+σ2ε
) + (1− α)φ( q√σ2n+σ2ε
)
αφ( q−1√σ2n+σ2ε
) + αφ( q+1√σ2n+σ2ε
) + 2(1− α)φ( q√σ2n+σ2ε
)[1− Φ(
q∗NT − qσε
)][L− δΦ(q∗NT − qσε
)]
× 1
σnφ(q + 1
σn)dq
36
whereas he expects the security to pay off
E[V ]
=
∫ +∞
−∞[1− Φ(
q∗NT − qσε
)][H + L
2− δΦ(
q∗NT − qσε
)]1
σnφ(q + 1
σn)dq
Therefore, the speculator’s payoff from short-selling is
E[P (q)]− E[V ]
=
∫ +∞
−∞[
αφ( q−1√σ2n+σ2ε
) + (1− α)φ( q√σ2n+σ2ε
)
αφ( q−1√σ2n+σ2ε
) + αφ( q+1√σ2n+σ2ε
) + 2(1− α)φ( q√σ2n+σ2ε
)− 1
2](H − L)[1− Φ(
q∗NT − qσε
)]
× 1
σnφ(q + 1
σn)dq
=
∫ q∗NT
−∞[
αφ( q−1√σ2n+σ2ε
) + (1− α)φ( q√σ2n+σ2ε
)
αφ( q−1√σ2n+σ2ε
) + αφ( q+1√σ2n+σ2ε
) + 2(1− α)φ( q√σ2n+σ2ε
)− 1
2](H − L)[1− Φ(
q∗NT − qσε
)]
× 1
σnφ(q + 1
σn)dq
+
∫ +∞
q∗NT
[αφ( q−1√
σ2n+σ2ε) + (1− α)φ( q√
σ2n+σ2ε)
αφ( q−1√σ2n+σ2ε
) + αφ( q+1√σ2n+σ2ε
) + 2(1− α)φ( q√σ2n+σ2ε
)− 1
2](H − L)[1− Φ(
q∗NT − qσε
)]
× 1
σnφ(q + 1
σn)dq
When σε → 0,
E[P (q)]− E[V ]
→ (H − L)
∫ +∞
q∗NT
[αφ( q−1√
σ2n+σ2ε) + (1− α)φ( q√
σ2n+σ2ε)
αφ( q−1√σ2n+σ2ε
) + αφ( q+1√σ2n+σ2ε
) + 2(1− α)φ( q√σ2n+σ2ε
)− 1
2][1− Φ(
q∗NT − qσε
)]
× 1
σnφ(q + 1
σn)dq
=1
2(H − L)
∫ +∞
q∗NT
αφ( q−1σn
)− αφ( q+1σn
)
αφ( q−1σn
) + αφ( q+1σn
) + 2(1− α)φ( qσn
)
1
σnφ(q + 1
σn)dq
=1
2(H − L)
∫ +∞
q∗NT
αφ( q+1σn
)
αφ( q−1σn
) + αφ( q+1σn
) + 2(1− α)φ( qσn
)
1
σn[φ(
q − 1
σn)− φ(
q + 1
σn)]dq
37
Thus
sgn{E[P (q)]− E[V ]}
= sgn(
∫ +∞
q∗NT
αφ( q+1σn
)
αφ( q−1σn
) + αφ( q+1σn
) + 2(1− α)φ( qσn
)
1
σn[φ(
q − 1
σn)− φ(
q + 1
σn)]dq)
= sgn(
∫ +∞
q∗NT
h(α, q, σn)dq)
where
h(α, q, σn) ≡αφ( q+1
σn)
αφ( q−1σn
) + αφ( q+1σn
) + 2(1− α)φ( qσn
)
1
σn[φ(
q − 1
σn)− φ(
q + 1
σn)]
=α(e
2q
σ2n − 1)
2 + α(e2q
σ2n − 1) + 2(1− α)(e2q+1
2σ2n − 1)
1
σnφ(q + 1
σn).
f(α, q, σn)
=α(e
2q
σ2n − 1)
[2 + α(e2q
σ2n − 1)]
1
σnφ(q + 1
σn)
Hence h(α, q, σn) > 0 if and only if q > 0. We also know that q∗NT < 0 and that dq∗NTdα > 0.
We can similarly derive the derivative of∫ +∞q∗NT
g(α, q, σn)dq with respect to α as
∂∫ +∞q∗NT
g(α, q, σn)dq
∂α
= g(α, q∗NT , σn)∂(−q∗NT )
∂α
+
∫ −q∗NT0
∂ 1N (g(α,− q
N , σn) + g(α, q, σn))
∂αdq
+
∫ −q∗NT0
g(α, q, σn))
∂αdq
Taking the limit N → +∞, we can similarly prove that when choosing N suffi ciently large,∂∫+∞q∗NT
g(α,q,σn)dq
∂α dq > 0.
When α→ 1, we have q∗NT →σ2n2 ln
δ2−L
H− δ2
. When α→ 0, q∗NT → −∞. Hence, there exists
38
α2 ∈ (0, 1) such that the uninformed speculator will deviate to short-selling if and only if
α > α2 and either 1) r > cr1 2) r < cr1 but δ > cδ where cr1 and cδ are the same as above
and α2 ∈ (1−δ2−L
H− δ2
, 1) is the unique solution of
∫ +∞
q∗NT (α2)g(α2, q, σn)dq = 0 (11)
Step 3 is thus proved.
Finally, choose
α = max(α1, α2) ∈ (1−δ2 − LH − δ
2
, 1) (12)
and the proposition is proved.
7.3 Proof of Proposition 1:
Proof. We have already shown in the proof of Lemma 1 that d∗(s = H) = 1 and negatively
informed speculator will not buy, resulting in d∗(s = L) = 0. Now consider the uninformed
speculator. Suppose that the conjecture is that the uninformed speculator will buy, then
from his perspective q ∼ N(1, σ2n) and qi ∼ N(1, σ2
n + σ2ε).
From Bayes’rule and given the speculators’strategies,
Pr(θ = H|q)
=g(q|θ = H)
g(q|θ = H) + g(q|θ = L)
=φ( q−1√
σ2n+σ2ε)
αφ( q√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)
In addition,
l(q)
= Pr(qi ≤ q∗B|q)
= Φ(q∗B − qσε
)
39
where we denote the threshold when uninformed speculator chooses to buy by q∗B and q∗B > 0.
Thus
P (q) = Pr(θ = H|q)[1− l(q)][H − δl(q)] + Pr(θ = L|q)[1− l(q)][L− δl(q)]
=φ( q−1√
σ2n+σ2ε)
αφ( q√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)[1− Φ(
q∗B − qσε
)][H − δΦ(q∗B − qσε
)]
+αφ( q√
σ2n+σ2ε) + (1− α)φ( q−1√
σ2n+σ2ε)
αφ( q√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)[1− Φ(
q∗B − qσε
)][L− δΦ(q∗B − qσε
)]
Therefore, if the speculator chooses to buy, he expects to pay
E[P (q)]
=
∫ +∞
−∞
φ( q−1√σ2n+σ2ε
)
αφ( q√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)[1− Φ(
q∗B − qσε
)][H − δΦ(q∗B − qσε
)]1
σnφ(q − 1
σn)dq
+
∫ +∞
−∞
αφ( q√σ2n+σ2ε
) + (1− α)φ( q−1√σ2n+σ2ε
)
αφ( q√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)[1− Φ(
q∗B − qσε
)][L− δΦ(q∗B − qσε
)]1
σnφ(q − 1
σn)dq
whereas he expects the security to pay off
E[V ]
=
∫ +∞
−∞[1− Φ(
q∗B − qσε
)][H + L
2− δΦ(
q∗A − qσε
)]1
σnφ(q − 1
σn)dq
40
Therefore, the speculator’s payoff from short-selling is
E[V ]− E[P (q)]
=
∫ +∞
−∞[1
2−
φ( q−1√σ2n+σ2ε
)
αφ( q+1√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)][1− Φ(
q∗B − qσε
)](H − L)
× 1
σnφ(q − 1
σn)dq
=
∫ q∗B
−∞[1
2−
φ( q−1√σ2n+σ2ε
)
αφ( q√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)][1− Φ(
q∗B − qσε
)](H − L)
× 1
σnφ(q − 1
σn)dq
+
∫ +∞
q∗B
[1
2−
φ( q−1√σ2n+σ2ε
)
αφ( q√σ2n+σ2ε
) + (2− α)φ( q−1√σ2n+σ2ε
)][1− Φ(
q∗B − qσε
)](H − L)
× 1
σnφ(q − 1
σn)dq
41
When σε → 0,
E[V ]− E[P (q)]
→∫ +∞
q∗B
[1
2−
φ( q−1σn
)
αφ( qσn
) + (2− α)φ( q−1σn
)]
1
σn(H − L)φ(
q − 1
σn)dq
=H − L
2
∫ +∞
q∗B
α[φ( qσn
)− φ( q−1σn
)]
αφ( qσn
) + (2− α)φ( q−1σn
)
1
σnφ(q − 1
σn)dq
=H − L
2
∫ +∞
q∗B
αφ( q−1σn
)
αφ( qσn
) + (2− α)φ( q−1σn
)
1
σn[φ(
q
σn)− φ(
q − 1
σn)]dq
=H − L
2
∫ +∞
q∗B
α
αe− 2q−1
2σ2n + (2− α)
1
σn[φ(
q
σn)− φ(
q − 1
σn)]dq
=H − L
2{ α
αe− 2q−1
2σ2n + (2− α)
[Φ(q
σn)− Φ(
q − 1
σn)]|+∞q∗B
−∫ +∞
q∗B
d
dq[
α
αe− 2q−1
2σ2n + (2− α)
][Φ(q
σn)− Φ(
q − 1
σn)]dq}
=H − L
2{− α
αe−2q∗B−1
2σ2n + (2− α)
[Φ(q∗Bσn
)− Φ(q∗B − 1
σn)]
−∫ +∞
q∗B
d
dq[
α
αe− 2q−1
2σ2n + (2− α)
][Φ(q
σn)− Φ(
q − 1
σn)]dq}
< 0
where the second-to-last inequality is because of integration by parts. The inequality is
because both terms are negative as Φ( qσn
) > Φ( q−1σn
) for any q and ddq [ α
αe− 2q−12σ2n +(2−α)
] > 0.
Therefore buying is not optimal when the conjecture is that the uninformed speculator will
not buy. We can follow proofs in step 1 of Proposition 2 to show that when the conjecture is
that the uninformed speculator will not trade, the uninformed speculator will not buy and
the proposition is proved.
7.4 Proof of Lemma 2:
Proof. Recall that
q∗B = 2q∗A +1
2
42
Therefore q∗A < q∗B if and only if
q∗A > −1
2
Recall that when σε → 0,
q∗A =σ2n
2ln[1−
δ−δH− δ
2
α]
is increasing in δ. When δ → δ, q∗A → 0 > −12 . When δ → 0, q∗A →
σ2n2 ln(1 − δ
α) =
σ2n2 ln[1 − 1−r
α ] which is increasing in r. When r → 1 − α, q∗A → −∞ whereas when r → 1,
q∗A → 0. Therefore there exists a unique cr2 < 1 such that −12 < q∗A < 0 when r > cr2 where
cr2 is defined asσ2n
2ln[1− 1− cr2
α] = −1
2(13)
In addition, since q∗A > −12 , we also have q
∗B < q∗A + 1, resulting in q∗A < q∗B < q∗A + 1 when
r > cr where
cr = max(cr1, cr2) (14)
7.5 Proof of Lemma 3:
Proof. Note that
∆εHH
= εHIB − εHIA
= Φ(q∗B − 1
σn)− Φ(
q∗A − 1
σn) > 0
and
∆εLH
= εLIB − εLIA
= Φ(q∗A + 1
σn)− Φ(
q∗Bσn
) > 0
43
as q∗A < q∗B < q∗A + 1.
Part 1 is thus proved.
In addition,
∆εHU
= εHUB − εHUA
= Φ(q∗Bσn
)− Φ(q∗A + 1
σn) < 0
as q∗B < q∗A + 1. In addition,
∆εLU
= εLUB − εLUA
= −[Φ(q∗Bσn
)− Φ(q∗A + 1
σn)] > 0
= ∆εLH
Thus, part 2 and 3 are proved.
7.6 Proof of Proposition 3:
Proof. The proposition is proved in four steps.
Step 1: Prove that short-selling is uniquely optimal for the uninformed spec-
ulator if α > α(δ).
This follows from Proposition 2. Note that we explicitly write the dependence of α on δ.
Step 2: Prove that ∂∆V∂δ < 0 and ∂∆V
∂α < 0.
44
When short-selling is allowed, α > α(δ) and σε → 0,
Vallowed
=1
2(E[V |θ = H] + E[V |θ = L])
=1
2{α[1− Φ(
q∗A − 1
σn)] + (1− α)[1− Φ(
q∗A + 1
σn)]
−αr[1− Φ(q∗A + 1
σn)]− (1− α)r[1− Φ(
q∗A + 1
σn)]}
When short-selling is not allowed and σε → 0,
Vban
=1
2(E[V |θ = H] + E[V |θ = L])
=1
2{α[1− Φ(
q∗B − 1
σn)] + (1− α)[1− Φ(
q∗Bσn
)]
−αr[1− Φ(q∗Bσn
)]− (1− α)r[1− Φ(q∗Bσn
)]}
If δ = 0 and α→ 1,
q∗B = 2q∗A +1
2= (σ2
n + σ2ε) ln
α− [1− (δ2−rL
rH− δ2)]
α+
1
2
If δ = 0
q∗B = 2q∗A +1
2= (σ2
n + σ2ε) ln
α− [1− (−rLrH )]
α+
1
2
45
Therefore the payoff difference when α > α(δ) is
∆V
= Vban − Vallowed
=1
2{α[Φ(
q∗A − 1
σn)− Φ(
q∗B − 1
σn)] + (1− α)[Φ(
q∗A + 1
σn)− Φ(
q∗Bσn
)]
−r[Φ(q∗A + 1
σn)− Φ(
q∗Bσn
)]}
=1
2{α[Φ(
q∗A − 1
σn)− Φ(
q∗B − 1
σn)] + (1− α)[Φ(
q∗A + 1
σn)− Φ(
q∗Bσn
)]
−δ2 + r
1− δ2
[Φ(q∗A + 1
σn)− Φ(
q∗Bσn
)]
+(δ2 + r
1− δ2
− r)[Φ(q∗A + 1
σn)− Φ(
q∗Bσn
)]}
Differentiate with respect to δ results in
∂∆V
∂δ
=1
2{αφ(
q∗A − 1
σn) + (1− α)φ(
q∗A + 1
σn)−
δ2 + r
1− δ2
φ(q∗A + 1
σn)}∂q
∗A
∂δ
−1
2{αφ(
q∗B − 1
σn) + (1− α)φ(
q∗Bσn
)−δ2 + r
1− δ2
φ(q∗Bσn
)}∂q∗B
∂δ
+1
2{−[Φ(
q∗A + 1
σn)− Φ(
q∗Bσn
)] + [Φ(q∗A + 1
σn)− Φ(
q∗Bσn
)]}∂(
δ2
+r
1− δ2
)
∂δ
+(δ2 + r
1− δ2
− r) 1
σn[φ(
q∗A + 1
σn)− 2φ(
q∗Bσn
)]∂q∗A∂δ
= (δ2 + r
1− δ2
− r) 1
σn[φ(
q∗A + 1
σn)− 2φ(
q∗Bσn
)]∂q∗A∂δ
where the third inequality is because the first two terms are zero because of the definition
of q∗A and q∗B.
Note that ∂∆V∂δ < 0 when −1
2 < q∗A < 0, i.e. when q∗A < q∗B < q∗A + 1. To see this, note
that it is straightforward to show that ∂q∗A∂δ > 0. In addition, when −1
2 < q∗A < 0, we have
0 < q∗B < 12 < q∗A + 1. Therefore φ(
q∗A+1σn
) < φ(q∗Bσn
), resulting in φ(q∗A+1σn
)− 2φ(q∗Bσn
) > 0.
46
Similarly, differentiate with respect to α results in
∂VB∂α
= Φ(q∗A − 1
σn)− Φ(
q∗B − 1
σn)− [Φ(
q∗A + 1
σn)− Φ(
q∗Bσn
)]
+(δ2 + r
1− δ2
− r) 1
σn[φ(
q∗A + 1
σn)∂q∗A∂α− φ(
q∗Bσn
)∂q∗B∂α
]
= Φ(q∗A − 1
σn)− Φ(
q∗B − 1
σn)− [Φ(
q∗A + 1
σn)− Φ(
q∗Bσn
)]
+(δ2 + r
1− δ2
− r) 1
σn[φ(
q∗A + 1
σn)− 2φ(
q∗Bσn
)]∂q∗A∂α
< 0
when −12 < q∗A < 0, i.e. when q∗A < q∗B < q∗A + 1. The reason is that Φ(
q∗A−1σn
) < Φ(q∗B−1σn
)
as q∗A < q∗B, Φ(q∗A+1σn
) − Φ(q∗Bσn
) > 0 as q∗A + 1 < q∗B and φ(q∗A+1σn
) − 2φ(q∗Bσn
) > 0 for the same
reason as stated above.
Step 3: Prove that ∆V (δ = 0, α) < 0, ∆V (δ = δ, α = α(δ) + ε) > 0 and ∆V (δ =
δ, α = 1) < 0 where ∆V is defined as the equilibrium payoffof banning short-selling
relative to not banning short-selling.
This is established now. We know that when δ = 0, uninformed short-selling is optimal
when α > α(0). We first show that ∆V (δ = 0, α) < 0 when α > α(0).
Recall that when α > α(0),
∆V
=1
2{α[Φ(
q∗SA − 1
σn)− Φ(
q∗SB − 1
σn)]− rα[Φ(
q∗SA + 1
σn)− Φ(
q∗SBσn
)]
+(1− α)(1− r)[Φ(q∗SA + 1
σn)− Φ(
q∗SBσn
)]}
47
Note that
d∆V
dα
=1
2{[Φ(
q∗SA − 1
σn)− Φ(
q∗SB − 1
σn)]− [Φ(
q∗SA + 1
σn)− Φ(
q∗SBσn
)]
+[αφ(q∗SA − 1
σn) + (1− α)φ(
q∗SA + 1
σn)− rφ(
q∗SA + 1
σn)]∂q∗SA∂α
−[αφ(q∗SB − 1
σn) + (1− α)φ(
q∗SBσn
)− rφ(q∗SBσn
)]∂q∗SB∂α}
When δ = 0, we have
αφ(q∗SA − 1
σn) + (1− α)φ(
q∗SA + 1
σn) = rφ(
q∗SA + 1
σn)
and
αφ(q∗SB − 1
σn) + (1− α)φ(
q∗SBσn
) = rφ(q∗SBσn
)
Therefore the last two terms vanish and
d∆V
dα
=1
2{[Φ(
q∗SA − 1
σn)− Φ(
q∗SB − 1
σn)]− [Φ(
q∗SA + 1
σn)− Φ(
q∗SBσn
)]}
Note that q∗SA + 1 > q∗SB. When −12 ≤ q∗SA < 0, then q∗SA ≤ q∗SB and
d∆Vdα < 0. When
−32 ≤ q
∗SA < −1
2 , then q∗SA > q∗SB > q∗SA − 1. The mean value theorem then results in
d∆V
dα
=1
2[φ(t1)
q∗SA − q∗SBσn
− φ(t2)q∗SA + 1− q∗SB
σn]
for some t1 ∈ (q∗SB−1σn
,q∗SA−1σn
) and some t2 ∈ (q∗SBσn,q∗SA+1σn
). Since q∗SB − 1 < q∗SA − 1 <
q∗SB < 0 and |q∗SA−1| > 32 > |q
∗SA+ 1|, φ(t1) < φ(t2), resulting in d∆V
dα < 0. When α→ 1− r,
q∗SA → −∞ and q∗SB → −∞, resulting in ∆V = 0. Since α(0) > 1− r, ∆V (δ = 0, α) < 0 for
all α > α(0).
We now show that ∆V (δ = 0, α) < 0 for all α < α(0).
48
When α < α(0), not trading is optimal for the uninformed. Recall that when not trading
is optimal for the uninformed and when short-selling is not banned, each investor runs if and
only if q ≤ q∗NT , where q∗NT is the unique solution of
αφ(q∗NT+1σn
)
φ(q∗NTσn
)+ 1− α
αφ(q∗NT−1
σn)
φ(q∗NTσn
)+ 1− α
=1− δ
2δ2 + r
We also know that when δ < δ, q∗NT < 0.
When short-selling is banned, the threshold is q∗B where q∗B is defined as
αφ(
q∗B−1√σ2n+σ2ε
)
φ(q∗B√σ2n+σ2ε
)+ 1− α =
δ2 + r
1− δ2
Thus, when α < α(δ), the payoff difference of not banning short-selling to be
Vallowed
=1
2(E[V |θ = H] + E[V |θ = L])
=1
2{α[1− Φ(
q∗NT − 1
σn)] + (1− α)[1− Φ(
q∗NTσn
)]
−αr[1− Φ(q∗NT + 1
σn)]− (1− α)r[1− Φ(
q∗NTσn
)]}
and banning short-selling to be
Vban
=1
2(E[V |θ = H] + E[V |θ = L])
=1
2{α[1− Φ(
q∗B − 1
σn)] + (1− α)[1− Φ(
q∗Bσn
)]
−αr[1− Φ(q∗Bσn
)]− (1− α)r[1− Φ(q∗Bσn
)]}
49
The payoff difference between banning versus not-banning when α < α(δ) is therefore
∆V (α, δ)
=1
2{α[Φ(
q∗NT − 1
σn)− Φ(
q∗B − 1
σn)] + (1− α)[Φ(
q∗NTσn
)− Φ(q∗Bσn
)]
−αr[Φ(q∗NT + 1
σn)− Φ(
q∗Bσn
)]− (1− α)r[Φ(q∗NTσn
)− Φ(q∗Bσn
)]}
=1
2{α[Φ(
q∗NT − 1
σn)− Φ(
q∗B − 1
σn)] + (1− α)(1− r)[Φ(
q∗NTσn
)− Φ(q∗Bσn
)]
−αr[Φ(q∗NT + 1
σn)− Φ(
q∗Bσn
)]}
When δ < δ, q∗NT + 1 > q∗B. To see this, note that when σε → 0,
αφ(
q∗B−1σn
)
φ(q∗Bσn
)+ 1− α =
αφ(q∗NT−1σn
)
φ(q∗NTσn
)+ 1− α
αφ(q∗NT
+1
σn)
φ(q∗NTσn
)+ 1− α
=δ2 + r
1− δ2
When δ < δ,δ2
+r
1− δ2
< 1. This results inφ(q∗B−1σn
)
φ(q∗Bσn
)< 1 and
φ(q∗NT−1σn
)
φ(q∗NTσn
)<
φ(q∗NT+1σn
)
φ(q∗NTσn
). Therefore
q∗B < 12 and q
∗NT < 0. Thus, when −1
2 < q∗NT < 0, q∗NT +1 > q∗B. In addition, αφ(q∗NT+1σn
)
φ(q∗NTσn
)+1−
α < 1, resulting inφ(q∗B−1σn
)
φ(q∗Bσn
)>
φ(q∗NT−1σn
)
φ(q∗NTσn
)and thus q∗B > q∗NT . In addition, it is straightforward
to show that q∗B increases with respect to q∗NT . In addition, q
∗B → −1
2 when q∗NT → −1
2 and
q∗B → 12 when q
∗NT → 0. Thus −1
2 < q∗B < 12 when −
12 < q∗NT < 0, which is satisfied when
r > cr. When q∗NT ≤ −12 ,
φ(q∗NT+1σn
)
φ(q∗NTσn
)≥ 1, resulting in α
φ(q∗NT+1σn
)
φ(q∗NTσn
)+ 1 − α ≥ 1. Therefore
αφ(q∗NT=1σn
)
φ(q∗NTσn
)+ 1− α ≥ αφ(
q∗B−1σn
)
φ(q∗Bσn
)+ 1− α, resulting in q∗NT ≥ q∗B and thus q∗NT + 1 > q∗B.
Thus ∆V (δ, α) < 0 if q∗NT ≤ q∗B, i.e. when q∗NT ≤ q∗B < q∗NT + 1, implying that when
uninformed speculator does not trade, banning short-selling is always bad, implying that
∆V (δ = 0, α) < 0. In addition, when δ → δ, q∗A → 0, α → 0 as both α1 and α2 → 0 and
50
q∗B → 12 . Then
∆V → 1
2{α[Φ(− 1
σn)− Φ(− 1
2σn)] + (1− α)[Φ(
1
σn)− Φ(
1
2σn)]
−r[Φ(1
σn)− Φ(
1
2σn)]}
=1
2[(1− 2α− r)[Φ(
1
σn)− Φ(
1
2σn)],
which is positive if and only if α < 12(1 − r). Therefore, ∆V (δ = δ, α = α(δ) + ε) =
∆V (δ = δ, α = α(δ) + ε) > 0.
Finally,
∆V (δ = δ, α = 1)
=1
2rH [(−1− r)[Φ(
1
σn)− Φ(
1
2σn)] < 0
Thus, step 3 is proved.
Step 4: Define an indifference curve of
∆V (δ, α∗(δ)) = 0
If α > α(δ), then
∆V (δ, α∗(δ)) = ∆V (δ, α∗(δ)) = 0 (15)
Note that dα∗(δ)dδ < 0 as differentiating equation (15) with respect to δ results in
dα∗(δ)
dδ= −
∂∆V (δ,α∗(δ))∂δ
∂∆V (δ,α∗(δ))∂α
< 0
Now define
α∗∗(δ) =
α∗(δ) if α > α(δ)
α(δ) if α < α(δ)
Then ∆V > 0 if and only if α ∈ (α(δ), α∗(δ)), which may be empty but clearly is not
empty when δ is suffi ciently close to δ, as ∆V (δ = δ, α = α(δ) + ε) = ∆V (δ = δ, α =
51
α(δ) + ε) > 0 shown in the third step and by continuity.
7.7 Proof of Lemma 4:
Proof. To prove that α(δ) is decreasing in δ, it is suffi cient to show that both α1 and α2 as
defined in equation (10) and (11) is decreasing in δ. We only show that α1 is decreasing in δ
as the proof for α2 to be decreasing in δ is essentially the same. Recall that α1 is defined as
∫ +∞
σ2n2
[ln(α1−(1−δ2−LH− δ2
))−lnα1]f(α1, Q, σn)dQ = 0
, or, equivalently ∫ +∞
q∗A(α1(δ),δ)f(α1, Q, σn)dQ = 0
Take derivative with respect to δ results in
∫ +∞
q∗A(α1,δ)
∂
∂α1f(α1, Q, σn)dQ
∂α1(δ)
∂δ+f(α1, q
∗A(α1, δ), σn)dQ[
∂q∗A(α1, δ)
∂α1
∂α1(δ)
∂δ+∂q∗A(α1, δ)
∂δ] = 0
Rearranging terms result in
∂α1(δ)
∂δ= −
f(α1, q∗A(α1, δ), σn)
∂q∗A(α1,δ)∂δ∫ +∞
q∗A(α1,δ)∂∂α1
f(α1, Q, σn)dQ+ f(α1, q∗A(α1, δ), σn)∂q∗A(α1,δ)
∂α1
The numerator is positive as
∂q∗A(α1, δ)
∂δ
=σ2n
2
1
α1 − (1−δ2−L
H− δ2
)
∂
∂δ(δ2 − LH − δ
2
) > 0
The denominator is also positive as it has been shown in the proof of Proposition 2 that
52
∫ +∞q∗A(α1,δ)
∂∂α1
f(α1, Q, σn)dQ > 0 and ∂q∗A(α1,δ)∂α1
> 0 as
∂q∗A(α1, δ)
∂α1
=σ2n
2
α1
α1 − (1−δ2−L
H− δ2
)
1−δ2−L
H− δ2
(α1)2> 0
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