morris and song shin-liquidity black holes

L iquid it y B lac k Holes
∗
Stephen Morris
Cowles Foundation,
Yale University,
P.O .Box 2 08281,
New H a v en C T 06520, U. S. A.
stephen.morris@ yale.edu
Hyun Song Shin
London School of Economics,
Houghton Street,
London WC2A 2AE
U. K.
.uk
June 20, 2003
Abstract
Traders with short horizons and privately known trading limits inter-
act in a market for a risky asset. Risk-averse, long horizon traders sup-
ply a downward sloping residual demand curve that face the short-horizon
traders. When the price falls close to the trading limits of the short horizon
traders, selling of the risky asset by any trader increases the incentives for
others to sell. Sales become mutually reinforcing among the short term
traders, and payoffs analogous to a bank run are generated. A “liquidity
black hole” is the analogue of the run outcome i n a bank run model. Short
horizon traders sell because others sell. Using global game techniques, this
paper solves for the unique trigger point at which the liquidity black hole
comes into existence. Empirical implications include the sharp V-shaped
pattern in prices around the time of the liquidity black hole.
∗
Preliminary v ersion. Commen ts welcome. We thank Guillaume Plan tin and Amil Dasgupta
for discussio ns during the prepa ration of t he paper.

1. Introduc tion
Occasionally, financial markets experience episode s of turbu lenc e of su ch an ex-
treme kind that i t appears to stop functioning. Such episodes are marked b y a
he a vily one-sided order flow, rapid price changes, and financial distress on the part
of many of the traders. The 1987 stock market crash is perhaps the most glaring
example of such a n episode, but there are other, more recent e xamples such as
the collapse of the dollar against the yen on October 7th, 1998, and instances of
distressed trading in some fixed income mark ets during the LT C M crisis in the
sum mer of 1998. Practitioners dub such episodes as “liquidit y holes” or, more
dramatically, “liquidity blac k holes” (Taleb (1997, pp. 68-9), P ersaud (2001)).
Liquidit y b lack holes are not simply instances of large price changes. Public
announc emen ts of import an t macroeconomic statistics, suc h as the U .S. employ -
ment report or GDP g rowth estimates, are sometimes marke d by large, discrete
price c hanges at the time of announcement. Howev er, such price changes are
arguably the signs of a smoothly functioning m arket that i s able to incorporate
new information quickly. The market typically finds composure quite rapidly
after suc h discrete price changes, as show n b y Fleming and Remolona (1999) for
the US Treasury securities market.
In contrast, liqu idity black ho les have the feature that they seem to gather
momentum from the endogenous re sponses of the mark et participants t hemselves.
Rathe r lik e a tropical storm, they appear to gather more energy as they develop.
P art of the explanation for the endogenous feedback mechanism lies i n the idea
that the incentives facing traders undergo changes when prices change. For
instanc e, ma rket d istress can feed on itself. When asset pric es f all, some traders
ma y get close to their trading limits and are induced to sell. But this selling
pressure sets off further dow n ward pressure on asset pric es, which induces a further
2
round of selling, and so on. P ortfolio insurance based on delta-hedging rules is
perhaps the best-kno w n example of such feedback, but similar forces w ill operate
whenever traders face constraints on their beha viour that shorten their decision

horizons. Daily trading limits and other controls on traders’ discretion arise as a
response to agency problems within a financial institution, and are there for good
reason. However, t hey have the effe ct of shortening the decision horizons of the
traders.
In what follo ws, we study traders with short decision horizons w ho hav e exoge-
nously given trading limits. Their short decision horizon arises from the threat
that a breach of the trading limit results in dismissal - a bad outcome for the
trader. However, the trading limit of each tra d er is private informa tion to that
trader. Also , although the trading limits across traders can differ, they are closely
correlated, ex ante. The traders inte ract in a market for a risky asset, where risk-
av erse, long horizon traders supply a do w n ward sloping residual demand curve.
When the p rice falls close to the trading limits of the short horizon traders, selling
of the risky asset by any trad er increases the incentives for others to sell. This is
because s ales tend to driv e down the market-clearing p rice , and the probabilit y
of breac hing one’s o wn trading limit increases. This sharpens the incentive s for
other traders to sell. In this w ay, sales become rei nforcing bet ween the short term
traders. In particular, the payoffs facing the short horizon traders are analogous
to a bank run game. A “liquidity black h ole” is the analogue o f the run outcome
in a bank run model. Short horizon traders sell because others sell.
If the trading limits were common knowledge, the payoffshavethepotential
to generate multiple equilibria. Traders sell i f they believe others sell, but if they
believ e that others w ill h old their nerve and not sell, they will refrain from selling.
Such multiplicity of equilibria is a w ell-kno wn feature of the bank run model of
Diamond and Dybvig (1983). However, when trading limits are not common
3
knowledge, as is more reasonable, the global game techniques of Morris and Shin
(1998, 2003) and G oldstein and Pauzner (2000) can be emplo y ed to solve for the
unique trigger point at which the liquidity black hole comes into existence.
1
The idea that the residual demand curv e facing activ e trade rs is not infinitely

elastic was suggested by Grossman and M iller (1988), who posited a role for risk-
averse market maker s who accommodate order flo ws and are compensated w ith
higher expected return. Campbell, Grossman, and Wang (1993) find evidence
consistent with this hyp othesis by showing that returns accompanied by high
volume tend t o be rev ersed more strongly. P astor a nd Stambaugh (2002) provide
further evidence for this hyp othesis by finding a role for a liquidity factor in an
emp irical asset pricing model, based on the idea that price reversal s often follow
liquidity shortages. Bernardo and Welch (2001) and B runnermeier and Pedersen
(2002) hav e used this device in m odelling limited liquidity facing activ e traders
2
.
More generally, the l imited capacity of the m arket to absorb sales of assets has
figured prominently in the literature on banking and financial crises (see Allen and
Gale ( 2001), Gorton and Huang (2003) and Sc hnabel and Shin (2002)), where the
price repercussions of asset sales hav e importan t adverse welfare consequences.
Similarly, the ineffecient liquidation o f long assets in Diamond and Rajan (2000)
hasananalogouseffect. The shortage of aggregate liquidity that such liquidations
bring a bout can g enerate c ontagious failures in the banking system .
1
Global game techniques have been in use in economics for s ome time, b ut they are less well
established in the finance literature. Some exceptions include Abreu and Brunnerm eier (2003),
Plantin (2003) and Bruche (2002).
2
Lustig (2001) emphasizes solvency constraints in giving rise to a liquidity-risk factor in
addition to aggregate consumption risk. Acharya and Pedersen (2002) develop a model in
whic h each asset’s return is net of a stochastic l iquidity cost, and expected returns are related
to return covariances with the aggregate liquidity cost (as well as to three other covariances).
Gromb and Vayanos (2002) build on the intuitions of Shleifer and Vishny (1997) and show that
margin constraints have a similar effect in limiting the ability of arbitrageurs to exploit price
differences. Holmström and Tirole (2001) prop ose a role for a related notion of liquidity arising

from the limited pledgeability of assets held by firms due to agency problems.
4
Some mar k et microstructure studies sho w evidence consisten t with an endoge-
nous trading response that m agnifies the initial price c hange. Cohen and Shin
(2001) show that the US Tr easury securities market exhibit evidence of positive
feedback trading during periods of rapid price c hanges and heavy order flow. In-
deed, even for m acroeconomic announcements, Evans and Ly ons (2003) find that
the foreign exchange mark et relies on the order flow of the traders in order to
interpret the signific ance of the macro announcement. H asbrouck (2000) finds
that a flow of new market orders for a stock are accompanied by the withdrawal of
limit orders on the opposite side. Danielsson and Pa yne’s (2001) study of f oreign
exchange trading on the Reuters 2000 trading system shows ho w the demand or
supply curve disappears from the market when the price is m oving against it, o nly
to reappear when the mark et has regained composure. The interpretation that
emerges from these studies is that smalle r v ersions of such liquidity gaps are per-
v a siv e in activ e markets - that the market undergoes m any “mini liquidit y gaps”
several times per day.
The next section presents the model. We then procee d to solve for the equi -
librium in the trading game using global game tec hniques. We conclude with a
discussion of the empirical implications and the endogenous nature of market risk.
2. Model
An asset is traded a t tw o consecutiv e d ates, and then is liquidated. We i ndex the
two trading dates by 1 and 2. The liquidation value of the asset at date 2 when
viewed from date 0 is given by
v + z (2.1)
where v and z are tw o independen t random variables. z is normally distributed
with mean zero and variance σ
2
, and is realized after trading at date 2. v is
5

realized after trading at date 1. We do not need to im pose any assumptions on
the d istribution of v. Th e important feature for our exercise is that, at date 1
(after the realization of v), the liquidation value of the a sset is norma l w i th m ea n
v and variance σ
2
.
There are two gr oups of traders in the market, and the realizatio n of v at
date 1 is common kno wledge among all of them. There is, first, a con tinuum
of risk neutral traders of measure 1. E ach trader holds 1 unit of the asset.
We may think of them as proprietary traders (e.g. at an in vestmen t bank or
hedge fund). T hey are subject to an incentive con tract in which their pa y off is
proporti ona l to the final liquidation value of t he asset. However, these traders are
also subject to a loss limit at date 1, as will be described in more detail below. If a
trader’s loss between dates 0 and 1 exceeds this lim it, then the trader is dismissed.
Dism issal is a bad outcome for the trader, and the trader’s decision re flects the
tradeoff between keeping his trading position open (and reaping the rewards if
the liquidation value o f the asset is h igh), against the risk of dismissal a t date 1
if his loss limit is breached at date 1. We do not model explicitly the agency
problems that motiva te the loss limi t. The loss li mit is taken to be exog enous for
our purpose.
Alongside this group of r isk-neutral traders i s a risk-a verse market-making sec-
tor o f the economy. The mark et-making sector provides the residual demand curve
facing the risk-neutral traders as a whole, in the manner envisaged by Grossman
and Miller (1988) and Campbell, Grossman and Wang (1993).
We represent the market-making sector by mean s of a repres e ntative trad er
with constan t a bsolute risk ave rsion γ who posts limit buy orders for the asset
at date 1 that coincides with his competitive demand c urv e. At date 1 (after v
is rea lized), the liquidation value of the asset is norm ally distributed with mean
v and variance σ
2

. From the linearity of demand with Gaussian uncertainty
6
and exponential utility, the market-making sector’s limit orders define the linear
residual d emand curve :
d =
v − p
γσ
2
where p is the p ri ce of th e a sset a t da te 1. Thus,iftheaggregatenetsupplyof
the asset fro m the risk-neutral traders is s,priceatdate1satisfies
p = v − cs (2.2)
where c is the constant γσ
2
. Since the market-ma kin g sect or is risk- averse, it
must be compensated for taking ov er the risky asset a t date 1, s o that the price
of the asset falls short of its expected payoff by the amount cs.
2.1. Loss limits
In the absence of any artificial impediments, the efficient allocation is for the risk-
neutral traders to hold all o f the risky asset. However, the risk-neutral traders
are su bject to a loss limi t that con strains their actions. T he lo ss limit is a trigger
price or “stop price” q
i
for trad er i suc h that if
p<q
i
then trader i is dismissed at d ate 1. Dismissal is a bad outcome for the trader,
and results in a payoff of 0. T he loss limits of the traders should be construed
as being determined in part b y the ov erall r isk position and portfolio composition
of their em plo yers. Loss limits therefore differ across traders, and information
regarding suc h limits are closely guarded. Am ong other things, the loss limits

fail to be common knowledge among the traders. This w ill be the cr ucial feature
of o ur model that drive s the main results. We will a lso assume th a t, conditional
on being dismissed, the trader prefers to maximize the value of his trading book.
The idea he re is that the trader is traded more leniently if the loss is smaller.
7
We will model the loss limits as random variables that are closely correlated
across the traders. Trader i’s loss limit q
i
is given by
q
i
= θ + η
i
(2.3)
where θ is a uniformly distributed random variable with support
£
θ
,
¯
θ
¤
,represent-
ing the common component of all loss limits. The idiosyncratic component of
i’s loss limit is given by the random variable η
i
, which is uniformly distributed
with support [−ε, ε],andwhereη
i
and η
j

for i 6= j are independent, and η
i
is
independent of θ. Crucially, trader i knows only of his own loss limit q
i
.He
m ust infe r t h e l oss limits of th e o ther traders, based on h i s k n owledg e o f the joint
distribution of {q
j
}, and his own loss limit q
i
.
2.2. Execution of sell orders
The trading at date 1 takes place by matching the sales of the risk-neutral traders
with the limit buy ord er s posted by the market -making sector. However, the
sequence in which the sell orders are executed is not under the control of the
sellers. We will assume that if the aggregate sa le o f the asset by the risk-neutral
traders is s, then a seller’s place in the queue for execution is uniformly distributed
in the interval [0,s]. Th us the expected price at which trader i’s sell order is
executed is given by
v −
1
2
cs (2.4)
and depends on the aggregate sale s. This feat ure of our model captures two
ingredients. The first is the idea that the price receive d by a seller depen ds on
the amo unt sold by other tra ders. When there is a flood of sell orders (large s),
then the sale pr ice that can be expected is low. The second ingr edient i s the
departure from the assumption that the transaction price is known w ith certaint y
when a trader decides to sell. Ev en though traders may h a v e a good indication

8
of the p ri ce that they can expect by selling (say, through indicativ e prices), the
actual ex ecution price cannot be guaranteed, and will depend on the overa ll selling
pressure in the mark et. T his second feature - the unc ertainty of transactions price
- is an important feature of a mark et under stress, and is emphasized by m an y
practitioners (see for instance, Kaufman (2000, pp.79-80), Taleb (1997, 68-9)).
The pa y off to a seller now depends on whether the execution price is high
enough as not to breach the loss limit. Let us denote by ˆs
i
thelargestvalueof
aggregate sales s that guarantees t hat trader i can execute his sell order without
breaching the loss limit. That is, ˆs
i
is definedintermsoftheequation:
q
i
= v − cˆs
i
(2.5)
where the expression o n the right hand side is the lowest possible price received by
a seller when the aggregate sale is ˆs
i
. Thus, whenever s ≤ ˆs
i
,traderi’s expected
payoff to selling is given by (2.4). However, when s>ˆs
i
, there is a positive
probability that the loss limit is breached, which leads to the bad payoff of 0.
Whe n s>ˆs

i
,traderi’s expected payo ff to selling is
ˆs
i
s
¡
v −
1
2
cˆs
i
¢
(2.6)
If trader i decides to hold on to the asset, then the pa yo ff is given by the
liquidation value of the asset at date 2 if the l oss limit is not b reac hed, and 0 if
it is breach ed. Thus, the expected pa yoff to tra der i of holding the asset, as a
function of aggregate sales s,is
u (s)=
½
v if s ≤ ˆs
i
0 if s>ˆs
i
(2.7)
Bringing together (2.4 ) and (2.6), we can write the expec ted payoff of trader i
9
from selling the asset as
w (s)=






v −
1
2
cs if s ≤ ˆs
i
ˆs
i
s
¡
v −
1
2
cs
¢
if s>ˆs
i
(2.8)
The p ayoffs are depicted in Figure 2.1. Holding the asset does better when s<ˆs
i
,
butsellingtheassetdoesbetterwhens>ˆs
i
. The trader’s optimal action depends
on the d ensity over s. We now solv e for equilibrium in this trading game.
.
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s
v
0
q
i
1
bs
i
p = v − cs
u(s)
w(s)
Figure 2.1: Payoffs
3. Equilibrium
At date 1, v is realized, a nd is common knowledge a mong all traders. Th us, at
date 1, it is comm on knowledge that the liquidation value at date 2 has mean v
10
and variance σ
2
. E ac h trader decides whether to sell or hold the asset on the basis
of the realization of v and his own loss limit. Trader i’s stra tegy is a function
(v, q
i
) 7−→ {hold, sell}

that maps realizations of v and q
i
to a trading decision. W hen v is either very
high or v ery lo w, trader i has a dominant action. When v i s v ery high relativ e
to q
i
(so that the realization of v is considerably higher than the loss limit for i),
trader i will prefer to hold. I n partic ular, since the period 1 price p cannot fall
below v − c,traderi’s dominant action is to hold when:
v ≥ q
i
+ c (3.1)
This is because the loss limit for trader i will not be breached even if all other
traders s ell. Conversely, when v is so low that
v<q
i
(3.2)
then the loss limit is breached even if all other trader s hold. Give n our assumption
that the traders prefers to maximize the v alue of his trading book conditional
on being dism issed, selling is the dominant action when v<q
i
.However,for
inte rmedi ate va lue s of v where
q
i
≤ v<q
i
+ c (3.3)
trader i’s optimal action depends on the incidence of se lling by other traders. If
trader i believes that others are se lling, he will sell als o. If, however, the others

are not selling, then he will hold. If the loss limits were common knowledge,
then such interdepende nce of actions w ould lead to multiple equilibria, and an
indeterminacy in the predicted outcome. W he n the loss lim its are not common
knowledge (as in our case), we can largely elim inate the multiplicity of equilibria
through g lobal game techniques.
11
In particular, w e will solv e for the unique eq uilibrium in t hreshold strategies
in which trader i has the threshold v
∗
(q
i
) fo r v that depends on his o w n loss limit
q
i
suc h that the equi librium strategy is give n by
(v, q
i
) 7−→
½
hold if v ≥ v
∗
(q
i
)
sell if v<v
∗
(q
i
)
(3.4)

In other w ords, v
∗
(q
i
) is the trigger level of v for trader i such that he sells if
and only if v falls below this critical level. We will show that there is precisely
one equilibrium of this kind, and proceed to solve for it by solving for the trigger
points {v
∗
(q
i
)}. Our claim can be summarized in terms of the following the orem.
Theorem 1. There is an eq uilibrium i n threshold strategies where the threshold
v
∗
(q
i
) for trader i is giv en by the unique value of v that solves
v − q
i
= c exp
½
q
i
− v
2(v + q
i
)
¾
(3.5)

There is no other threshold equilibrium.
The left hand side of (3.5) is increasing in v and passes through the origin,
whiletherighthandsideisdecreasinginv and passes through (0,c), so that there
is a unique solution to (3.5). At this s olution, w e must hav e v − q
i
> 0,sothat
the trigger point v
∗
(q
i
) is strictly above the loss limit q
i
. Traders ado pt a pre-
emp tive selling strategy in w hich the tr igger level lea ves a “margin for prud ence”.
The intuition here is that a trader anticipates the negativ e consequences of other
traders selling. Other traders’ pre-emptive selling str ategy must be met by a
pre-emptive selling strategy on my part. In equilibrium, ev ery trader adopts an
aggressive, pre-emptiv e selling strategy because others d o so . If the trade rs hav e
long decision horizons, they can ignore the short-term fluctuations in price and
hold the asset for its fundamen tal value. Howeve r, traders subject to a loss limit
have a short decision horizon. Ev en though the fundamentals are good, short term
12
price fluctuations can cost him his job. Thus, loss limits inevitably shorten the
decision horizon of t he traders. The fact there there is a pre-emptive eq uilibrium
of this kind is perhaps not so rema rkable. Ho wever, what is of interest is the f act
that there is no ot her threshold eq uilibrium. In parti cular, the “nice” s trategy
in which the traders disarm by collectively lowering their threshold points v
∗
(q
i

)
down to th ei r loss limi ts q
i
cannot figure in a ny equilibrium beha vi our.
0.0 1.0 2.0 3.0 4.0 5.0
c
0.0
1.0
2.0
3.0
4.0
5.0
v
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v − c = q
i
v = q
i
v
∗
hold dominant
sell dominant
Figure 3.1: v
∗
as a function of c. q
i

=1
Figure 3.1 plots v
∗
as a function of the parameter c as given b y (3.5), while
fixing q
i
=1. Recall that c = γσ
2
,whereγ is the coefficient o f absolute risk
aversion. We can see that the critical value v
∗
can be substantially higher than
the loss limit (given by 1). When v is very high, so that v − c>q
i
, holding the
asset is the dominan t action. T his dominance region is the area abov e the upw ard
sloping dashed line in figure 3.1. Conv ersely when v<q
i
, the dominant action is
to sell, and this area is indicated as the region belo w the horizon tal dashed line.
13
The large “w edge” betw een these tw o dominance regions is the region in which
the outcome depends on the resolution of the strategic trading gam e bet ween the
traders. The equilibrium trigger point v
∗
bisects this wedge, and determines
whether trader i holds or sells. The solid line plots the equilibrium trigger point
give n by the solution to (3.5).
Te chnically, the global gam e analysed here does not conform to the canonical
case discussed in Morris and Shin (2003) in which the pay o ffs satisfy strategic

complementarity, and uniqueness can be proved by th e it er ated d eletion of do mi-
nated strate gies. I n our ga me , the p ayoff difference between holding and selling is
not a monotonic function of s. We can see this best from figure 2.1. The pa y off
difference rises initially, but then drops discontinuously, and then rises thereafter,
much like the bank run game of Goldstein and Pauzner ( 2000). Our argumen t
for the uniqueness o f the threshold equilibrium rests on the interaction between
strategic uncertain t y (uncertainty concerning the actions of other traders) and
fundamen tal uncertainty (uncertain ty concerning the fundamentals). Irrespec -
tiv e of th e s everi ty of fundamen tal unce rtainty, the strate gic u n cer tainty persists
in equilibrium, and the pre -emptiv e a ction of the traders reflects the op tim al re-
sponse to strate gic unc ertain ty. Our solution method below will bring this feature
out explicitly.
3.1. Strategic uncertain ty
The payoff difference between holding the asset and selling the asset when ag-
gregate sales are s is given b y u (s) − w (s). T he expec ted payoff advantage of
holding the asset over selling it is given by
Z
1
0
f (s|v, q
i
)[u (s) − w (s)] ds
where f (s|v,q
i
) is the density over the equilibrium value of s (the proport ion of
traders who sell) conditional on v and trader i’s own loss limit q
i
.Traderi will
14
hold if the integral is positive, and sell if it is neg ative. Thus, a dire ct way to

solv e for our equilibrium is to solv e for the density f (s|v,q
i
).
It is convenient to v iew the trader’s thre shold strategy as the ch oice of a
thresho ld f or q
i
as a function of v.Thus,letusfix v and s uppose that all traders
follow the threshold strategy around q
∗
, so that tr ader i sells if q
i
>q
∗
and holds
if q
i
≤ q
∗
. Suppose that trader i’s loss limit q
i
happens to be e xactly q
∗
.Wewill
derive trader i ’s subjective density ov er the aggregate sales s. S ince the traders
have unit measure, a g gregate sales s is giv en by the proportion of traders w ho sell.
From trader i’s point of view, s is a random variable with support on the unit
interval [0, 1]. The cumulative distribution function over s viewed from trader i’s
viewpoint can be obtained from the answ er to the follo wing question.
“My l oss limit is q
∗

. W hat is the probability that s is less than z?” (Q)
The answ er to t his question will yield F (z|q
∗
) - the probabilit y that the pro-
portion o f trade rs who sell is at mo st z, conditional on q
i
= q
∗
. Since all traders
are hypothesized to be using the threshold strategy around q
∗
, the proport ion
of traders who sell is giv en by the propor tion of traders whose loss limits hav e
realizations to the right of q
∗
. When the common element of the loss limi ts is θ,
the individual loss limits are distributed uniformly o v e r the interval [θ − ε, θ + ε].
The traders who sell are those whose loss limits are abo ve q
∗
. Hence ,
s =
θ + ε − q
∗
2ε
When do we hav e s<z? This happens when θ is low enough, so that the area
under the densit y to the right of q
∗
is squeezed to a size below z. There is a v alue
of θ at which s is precisely equal to z. Thisiswhenθ = θ
∗

,where
θ
∗
= q
∗
− ε +2εz
15
We have s<zif and only if θ < θ
∗
. Thus, we can answer question (Q) by finding
the posterior probability t hat θ < θ
∗
.
Fo r this, we m u st turn to trader i’s posterior density over θ conditional on
his loss limit being q
∗
. This posterior density is uniform o v er the interval
[q
∗
− ε,q
∗
+ ε]. T his is because the ex an te distribution o ver θ is uniform and
the idiosyncratic element of the loss limit is uniformly distributed around θ.The
probability that θ < θ
∗
is then the area under the posterior density o ver θ to the
left of θ
∗
.Thisis,
θ

∗
− (q
∗
− ε)
2ε
(3.6)
=
q
∗
− ε +2εz − (q
∗
− ε)
2ε
= z
In other words, the probability that s<zconditional on loss limit q
∗
is exactly
z. The cumulative distribution function F (z|q
∗
) is the identity function:
F (z|q
∗
)=z (3.7)
The density over s is then obtained by differentiation.
f (s|q
∗
)=1 for all s (3.8)
The density over s is uniform. The notew orth y feature of this result that the
constant ε does not enter into the expression for the density over s. No ma tter
how small or large is the dispersion of loss lim its, s has the unif orm density ove r the

unit interval [0, 1].Inthelimitasε → 0 , ev ery trader’s loss limit con ve rges to θ.
Th us, fundamental uncertaint y disappears. Everyone’s loss limit con verges to the
common elem ent θ, and every one knows this fact. And y et, even as fundamen tal
uncertaint y disappears, the strategic uncertainty is unc hanged.
16
3.2. Solving for Equilibrium Threshold
Having found the conditional densit y over s at the threshold poin t q
∗
,wecannow
return to the payoffs of the game. We noted earlier that the expected pay off
advantage to holding the asset is given by
Z
1
0
f (s|v, q
i
)[u (s) − w (s)] ds
At the threshold point q
∗
, we ha ve j ust sh ow n th at the densit y f (s|v,q
i
) is uniform.
In addition, the trader is indifferent betw een holding and selling. Thus, at the
threshold point, we have
Z
1
0
[u (s) − w (s)] ds =0
From this equation, we ca n solve for th e thre shold point. Figure 3.2 il lustrates
the argumen t. The in tegra l of the payoff differenc e with res pect t o a uniform

density over s must be equal to zero. This means that the area labelled A in
figure 3.2 must be equal to the area labelled B.
Substituting in the expressions for (2.7) and (2.8), and noting that ˆs =(v − q
∗
) /c,
we have
1
2
c
Z
v−q
∗
c
0
sds =
(v−q
∗
)(v+q
∗
)
2cs
Z
1
v−q
∗
c
1
s
ds
which simp lifies to

v − q
∗
=2(v + q
∗
)log
c
v − q
∗
Re-arranging this equation g iv es (3.5) of the orem 1. There is a unique solution
to th is equation as alread y noted, where v>q
∗
.
So far, we have show n that i f all traders follo w the threshold strategy around
q
∗
, t hen a trader is indifferent between holding and selling given the threshold loss
limit q
∗
. We m ust show that if q
i
>q
∗
,traderi prefers to sell, and if q
i
<q
∗
,
trader i prefers to hold. This step of the argumen t i s presen ted in the appe ndix.
17
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s
v
∗
(q
i
)
01
q
i
bs
i
u(s)
w(s)
A
B
Figure 3.2: Expected Payoffs in Equilibrium

3.3. Normal distribution ov er types
Although our argument has been giv e n in the context of the uniform distribu-
tion ov er loss limits, this particular distributi onal assumption turns out not to
be important. What m atters fo r the arg u ment is the shape of the strategic un-
certaint y facing the traders, in partciular, the density over the proportion wh o
se ll. Pro vided that the idiosyncratic te rms are s mall, the assumption of unifor-
mity is without loss of generality (see Morris and Shin (2003, section 2)). We
illustrate the point by showing how we would obtain the same solution (3.5) for
the thr eshold poin t when the density of loss limits i s G aussian.
Th us, suppose that θ is distributed normally with mean
¯
θ and p recision α,and
q
i
is the sum θ + ε
i
where ε
i
is normal with mean ze ro and precision β,where{ε
i
}
are independen t across i, and inde pendent o f θ. We can th e n pose question (Q)
18
in this new context. The answer is give n b y the area under the normal de nsity
to the rig ht of the threshold poin t q
∗
.
1 − Φ
³
p

β (q
∗
− θ)
´
= Φ
³
p
β (θ − q
∗
)
´
(3.9)
where Φ (·) is the cumulative distribution function for the standard normal. Let
θ
∗
be the common cost elemen t at whic h this proportion is exactly z.Thus,
Φ
³
p
β (θ
∗
− q
∗
)
´
= z (3.10)
Whe n θ ≤ θ
∗
, the proportion of traders that sell is z or less. So, the question
of whether s ≤ z boils down to the question of whether θ ≤ θ

∗
. Conditional on
q
i
= q
∗
, the densit y over θ is normal with me an
α
¯
θ + βq
∗
α + β
and precision α + β. Th us, the probability that θ ≤ θ
∗
is the area under the
conditional density o f θ to the le ft of θ
∗
,namely
Φ
µ
p
α + β
µ
θ
∗
−
α
¯
θ + βq
∗

α + β
¶¶
(3.11)
This expr ession gives F (z|q
∗
). Substituting out θ
∗
by using (3.10) and re-
arranging, we can re-write (3.11) to give:
F (z|q
∗
)=Φ
³
α
√
α+β
¡
q
∗
−
¯
θ
¢
+
q
α+β
β
Φ
−1
(z)

´
(3.12)
Diffe rentiation of this expression with respect to z will giv e us the subjective
density over s. Let us denote the density funct ion o ver s by
f (s|q
∗
)
19
In the limit where β →∞, t he idiosyncratic components of loss limits around θ
shrink to zero, so that everyone’s loss limit is given by θ. The limiting case of
(3.12) when β becomes large is given b y:
F (z|q
∗
) → Φ
¡
Φ
−1
(z)
¢
= z
so that F (z|q
∗
) is the iden tity function. Since the cumulative distribution func-
tion over s is the identity function, the density over s is uniform. T hat i s
f (s|q
∗
)=1, for all s
Th us, the solution for the threshold poin t v
∗
is unchanged from (3.5). Also, in

the limiting case, when v>v
∗
, the density over the proportion who sell is the
degenerate spike at 0,whilewhenv<v
∗
, the density is the degenerat e spike at
1. Hence, the switching strategy around v
∗
constitutes a symmetric eq uilibrium
strategy.
4. Discuss ion
Liquidity b lack holes are associated with a sharp V-shaped price path for prices.
The price at date 1 is given by v − cs, while the expected value of the asset at
date 1 is v. Th u s, the expecte d retu rn from date 1 to d ate 2 is given by
v
v−cs
.In
the limiting case, where the loss limits are perfectly correlated across traders, s
takes th e valu e 1 below v
∗
, and tak es the value 0 abo v e v
∗
.Thus,whenthereis
a liquidit y blac k hole at date 1, the expected return is
v
v − c
which is strictly l arger than the actuarially fair rate of 1 for r isk-netural traders.
The larger is c, the greater is the likely bounce in price. The parameter c is
given by c = γσ
2

,whereγ is the coeffic ien t of absolute risk av ersion and σ
2
is
20
the variance of the fundamentals. Since c gives the slope of the residual de mand
curve facing the active traders, we can interpret c as representing the d egree of
illiquidit y of the mark et. The larger i s c, the sm aller is the capacity t o absorb
the selling pressure from the a ctive traders. Th us, when a liquidit y black h ole
comes into existence, a large c is as sociated with a sharper decline in prices, and
a comm ensurate bounce bac k in prices i n the final period.
Another implication of our model is that the trading v olume at the time of
the liquidit y black hole and its aftermath will be considerable. When the m ark et
strike s the liquidity black hole, the w hole o f the asset holding in the risky asset
changes hands from the risk-neutral short horizon traders to the risk-averse market
making sector. Although w e ha ve not modelled the d ynamic s, we could envisage
that immediately afterw a rds, once the loss limits ha ve been adjusted do wn given
the new price, there w ill be an imm ediate reversal of the trades in which the
risky asset ends up back in the hands of the risk neutral traders once more. The
large tra ding v o lume that is generated by these reversals will be associated wi th
the sharp V-shaped price dynamics already noted. The association bet ween
the V-shaped pattern in prices and the large trading v olume is consisten t with
the evidence found in Campbell, Grossm an and Wang (1993) and Pastor and
Stambaugh (2002).
Traders who are aware of their env ironment take accou nt of limited liquidity in
the marke t. The equilibrium strategies of the traders there fore al so ta ke acc ount
of the degree of illiquidity of the market. The solution for the threshold point
(3.5) shows that when c increases, the gap bet we en v
∗
and q
i

increases also, as
shown in figure 3.1. In other words, the when c is large, a trade r’s trigger poin t v
∗
is m uch higher than his true loss limit q
i
. The trader bails out at a muc h higher
price than his loss limit because he is apprehensive about the effec t of other trader s
bailing out. Just as in the run outcome in a b ank run game, the traders in the
21
illiquid market bail out more aggressiv ely when they fear the bailing out of other
traders. Since in o ur model the efficient outcome is for the risk-neutral trade rs
to h old the risky a sset, the increase i n c results in a gre ater w elfare loss, ex an te .
This last point raises some thorny questions for regulatory policy. While the
trigger-happy beha viour of the individual traders is optimal from the point of view
of that trader alone, the resulting equilibrium is socially inefficient. In particular
when the loss limit of one trader is raised, this has repercussions beyond that
individual. For other traders in the market, the raising of the loss lim it by one
trader imposes an unwelcome negative externality in the form of a more volatile
interim price. The natural response of the other traders would be to raise their
own trading limits to match. T he analogy here is with an arms race.
More generally, when the endogenous nature of price fluctuations is taken into
accoun t, the regulatory response to market risk may take on quite a different
flavour from the orthodox appr oach using value at risk using historical prices.
Danielsson, Shin and Zigrand (2002) and Danielsson and Shin (2002) explore
these issues further.
Appendix.
In this appendix, we complete the argumen t for theorem 1 by sho wing that
if q
i
>q

∗
,traderi prefers to sell, and if q
i
<q
∗
,traderi prefers to hold. For
this step of the argument, we again appeal to the conditional densit y o v er s.Let
us consider a variant of q uestion (Q) for a trader whose loss limit ex ceeds the
thresho ld poin t q
∗
. Thus, consider the following question.
“My loss limit is q
i
and all others use the thre shold strategy around
q
∗
. What is the probability that s is less than z?”
22
The answer to this question will yield F (z|q
i
,q
∗
) - the probability that the
proportion of traders who sell is at most z, conditional on q
i
when all others
use the threshold strategy around q
∗
. When all traders a re using the threshold
strategy around q

∗
, the proportion of traders w ho sell is giv en by the proportion of
traders whose loss limits ha ve realizations to th e rig ht of q
∗
. Whe n the com mon
elem ent of loss l imits i s θ, the individual loss limits are distributed uniformly ov er
the interval [θ − ε, θ + ε]. The traders who sell are those w hose loss limits are
above q
∗
. Hence,
s =
θ + ε − q
∗
2ε
When do we hav e s<z? This happens when θ is low enough, so that the area
under the densit y to the right of q
∗
is squeezed to a size below z. There is a v alue
of θ at which s is precisely equal to z. Thisiswhenθ = θ
∗
,where
θ
∗
= q
∗
− ε +2εz
We have s<zif and only if θ < θ
∗
. Th us, we can answer the question posed
above by finding the posterior probability that θ < θ

∗
.
For t his, w e must turn to trader i’s posterior density over θ conditional on his
loss limit being q
i
. T his posterior densit y is uniform o ver the in terval [q
i
− ε,q
i
+ ε],
since the e x ante distribution over θ is uniform and the idiosyncratic element of
the loss limit is uniformly distributed around θ. The conditional probabilit y that
θ < θ
∗
is then the area under the posterior density ov e r θ to the lef t of θ
∗
.This
is,
θ
∗
− (q
i
− ε)
2ε
(4.1)
=
q
∗
− ε +2εz − (q
i

− ε)
2ε
= z +
q
∗
− q
i
2ε
23
This gives the cumulative distribution function F (z|q
i
,q
∗
), which falls under three
cases.
F (z|q
i
,q
∗
)=



0 if z +
q
∗
−q
i
2ε
< 0

1 if z +
q
∗
−q
i
2ε
> 1
z +
q
∗
−q
i
2ε
otherwise
Hence, the corresponding density over s will,ingeneral,haveanatomateither
s =0or s =1. Thus, let us consider q
i
where q
i
>q
∗
. We sho w that trader i
does strictly better by selling, than by holding. The conditional density ov er the
half-open inverval s ∈ [0, 1) is given by
f (s|q
i
,q
∗
)=
½

0 if s<
q
i
−q
∗
2ε
1 if s ≥
q
i
−q
∗
2ε
and there is an atom at s =1with w eight
q
i
−q
∗
2ε
.
Meanwhile, from (2.7) and (2.8), the expected pay off advantage of holding
relative to selling is given by
u − w =
½
1
2
cs if s ≤ ˆs
i
−
ˆs
i

s
¡
v −
1
2
cs
¢
if s>ˆs
i
This payoff function sa tisfies the single-crossing propert y in that, u − w is non-
negative when s ≤ ˆs
i
, and is ne gative when s>ˆs
i
. T he density f (s|q
i
,q
∗
) can
be obtained from the uniform densit y b y transferring weigh t from the interval
£
0,
q
i
−q
∗
2ε
¤
to the ato m on point s =1.Since
R

1
0
(u (s) − w (s)) ds =0,wemust
have
Z
1
0
(u (s) − w (s)) f (s|q
i
,q
∗
) ds < 0
Thus, the trader with loss limit q
i
>q
∗
strictly prefers to sell. There is an e xactly
analogous argument to show that the trader with loss limit q
i
<q
∗
str ictly prefe rs
to hold. This c omplete s the argum en t for theore m 1.
24
References
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Econometrica, 71, 173-204
[2] Achary a, Viral V. and Lasse Heje P edersen (2002) “Asset pricing with liq-
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[3] Allen, Fra nklin, and Douglas Gale (2002): “Financial Fragility,” Working

P aper 0 1-37, Wharton Financial I nstitutions C en ter, Univ ersity of Pennsyl-
vania, http://fic.wharton.upenn.edu/fic/papers/01/0137.pdf.
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paper, Yale University.
[5] B ruc he , Max (2002) “A Structural Model of Corporate Bond Prices with
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[8] Cohen, Benjamin and Hyun Song Shin (2001) “P ositive Feedbac k Trading
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lished paper, http://ww w.nuff.ox.ac.uk/users/shin/working.htm
[9] Danielsson, Jon and Ric hard Payne (2001) “Measuring and explaining liq -
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25