Item # 4 SEMINAR IN LAW, ECONOMICS, AND ORGANIZATION RESEARCH
Professors Louis Kaplow, Lucian Bebchuk, Oliver Hart, and Kathryn Spier Monday, October
18 Pound 108, 12:30 p.m.
“ Collateral Damage? Derivatives Trading and Empty Voting”
Holger Spamann
Collateral Damage? Derivatives Trading and
Empty Voting
Holger Spamann
Harvard University
Octob er 12, 2010
[Preliminary and incomplete]
AbstractThispaperanalyzesam
o d e l o f p a r a l l e l t r a d i n g o f vo t i n g s e c u r i t i e s a n d
d e r i va t i ve s w i t h p e r f e c t l y n e g a t i ve l y c o r r e l a t e d c a s h
- ow s , s u ch a s b o n d s a n d c r e d i t d e f a u l t s wa p s . I n t h e p r e
s e n c e o f n o i s e f r o m l i q u i d i ty t r a d e r s , a h e d g e f u n d c a n s
o m e t i m e s p r o t a b l y a c q u i r e a vo t i n g b l o ck o f s e c u r i t i e s
ove r - h e d g e d by a n e ve n l a r g e r b l o ck o f d e r i va t i ve s , s u ch t
h a t i t s i n c e nt i ve s a r e t o vo t e f o r d e c i s i o n s t h a t red u ce t h e
va l u e o f t h e s e c u r i ty ( " e m p ty vo t i n g " ) . At o t h e r t i m e s , t h e
h e d g e f u n d vo t e s t o i n c r e a s e t h e va l u e o f t h e s e c u r i ty. T h
e r e s u l t i n g u n c e r t a i nty a n d t h e h e d g e f u n d s p r i va t e i n f o
r m a t i o n w i t h r e s p e c t t o i t s ow n vo t i n g p l a n s e n a b l e t h e h
edgefundtoreaptradinggains.Ascollateraldamage,
s o c i a l va l u e i s d e s t r oye d e a ch t i m e t h e h e d g e f u n d vo t e s t
o m i n i m i z e t h e va l u e o f t h e s e c u r i ty. D e r i va t i ve m a r ke t p a
r t i c i p a nt s c a n p r o t e c t a g a i n s t t h i s s t r a t e g y by i n c l u d i n
g p o s i t i o n l i m i t s a n d p ayo u t i n f o r m a t i o n p o o l i n g i n t h e i r
d e r i va t i ve c o nt r a c t s .
In rec ent years, comm entators have voiced grave con cerns ab ou t "e mpty
votin g" en abled by d erivative s (esp e cially Hu an d Black 2006, 2007, 2008a,
2008b ). The f ear is that ce rtain owne rs of voting se curities such as b on ds or
sh are s will us e d erivatives to he dge more than all th e cash - ow attached to
the sec urities . As a cons equen ce , like an over-insu re d h om e-own er might p
ref er to s ee his hou se bu rn d own, thes e owne r-voters might f avor an
outcome that reduces the valu e of their sec urity. Thu s de rivative s might s et
ince ntives for value- des troyin g votin g b e havior. This conc ern h as b een
partic ularly strong with resp ect to cred it d efau lt s waps (CDS ) an d b on ds.
Ref erring to e mpty votin g, George S oros (2010) h as calle d CDS a "lice nse to
kill," an d fe llow f und manager David Einh orn has calle d CDS "in herently u nsaf
e" (Se nde r 2009).
Te r e n c e M . C o n s i d i n e Fe l l o w , H a r va r d L a w S ch o o l ; h s p a m a n n @ l a w . h a r
va r d . e d u . Fo r h e l p f u l d i s c u s s i o n s , I t h a n k R u ch i r A g a r w a l , P h i l i p p e A g h i
o n , L u c i a n B e b ch u k , E ¢ B e n m e l e ch , S t e f a n o G i g l i o , O l i v e r H a r t , S c o t t H i
r s t , A n d r e i S h l e i f e r , t h e H a r va r d L a w S ch o o l c o r p o r a t e f e l l o w s l u n ch g r
o u p , a n d s e m i n a r p a r t i c i p a n t s a t H a r va r d L a w S ch o o l a n d H a r va r d U n i v e r
s i ty s D e p a r t m e n t o f E c o n o m i c s . Fo r g e n e r o u s n a n c i a l s u p p o r t , I t h a n k t
h e P r o g r a m o n C o r p o r a t e G o v e r n a n c e a t H a r va r d L a w S ch o o l , t h e J o h n M .
O l i n C e n t e r f o r L a w , E c o n o m i c s , a n d B u s i n e s s a t H a r va r d L a w S ch o o l , a n
d H a r va r d U n i v e r s i ty.
1
Not all agree , h owever, th at d erivatives, and CDS in particu lar, p os e s uch
ris ks . The Intern ational Swap s and De rivative s Asso ciation (IS DA) claims
that the fears are not ju sti ed with res p ect to CDS (Men gle 2009, p p. 10-11). A
wou ld -b e em pty voter, so th e argum ent go e s, wou ld have to p ay a high
CDS price re ecting the in creased de fault probability of th e b on d. Give n this h
igh price, the strate gy wou ld n o lon ger b e p ro tab le .
This pap er provid es a m o d el to as ses s such claim s, an d shows th at fe
ars of a de stru ctive e/ec t of a parallel derivatives market on corp orate voting are
ind eed j usti ed . To do so, it e mb ed s into a s tan dard trading mo del (Kyle
1984; Kyle and Vila 1991) parallel trading of (a) voting s ecu ritie s an d (b )
derivatives with exactly opp os ite cas h- ows and n o voting rights. (For con
cretene ss, one may think of the s ecu rity as a b on d and the derivative as a
credit def ault s wap.) In the m o d el, a large trad er a "h edge f un d" acquires
a blo ck of voting se curities . In parallel, the fu nd also ac qu ires a p osition in de
rivatives that is smaller or large r than its sec urities p osition, as the case m ay b
e. Whe n th e he dge fu nd s de rivative p osition is larger than its s ecu ritie s p
osition (i.e., th e he dge fun d is "net short"), it maximizes its payo/ by votin g to
reduce the valu e of th e se curity. Ration al market particip ants ("market makers")
anticipate th e he dge f und s gene ral strategy. Bu t in any given c ase, the y d o
n ot know the size of the he dge fu nd s de rivative p osition b e caus e th e h ed
ge f und s trades are con foun ded with th os e of liquid ity trad ers with varyin g
exogenou s d emand for de rivative s. Liquidity trad ers thus p rovide c am ou age
for th e hed ge fu nd, and prices of se curities and d erivative s re ect an ave rage
of th e h igh and low p ayo/ of the sec urity. Th e h edge fu nd gain s by b uyin g se
curities relatively cheaply (and p oss ib ly sellin g d erivatives exp en sive ly) wh en
it votes to maximize the se curities value, and it gains by buying derivatives
relatively cheap ly when it votes to min im iz e th e se curities value . Th at is, the
hed ge fu nd gains by trad ing on the un certainty th at its own voting b eh avior
gene rates. Th e cos ts are b orn e by the liquidity traders. As collateral d amage ,
so cial value is des troyed each time the he dge f un d votes to min imiz e the
value of a s ecu rity.
1The mo del s hows that the problem of value- des troyin g vote s is more seve re
in more "liqu id " d erivatives marke ts , that is, markets in wh ich trad es by
liquidity trad ers are su b jec t to greater u ctuations an d trading cos ts are lower.
Th is is a cau se for con cern give n d erivatives markets explosive growth over
last d ecad e or so. At the s ame tim e, the mo de l also sh ows that the m a jor
costs of th e em ptyvotin g p rob le m are intern alize d by the participants of the
derivatives market, notably liquidity traders. The se p artic ip ants th ere fore have
th e (almos t) corre ct ince ntives to ad opt a contractual s olution to the problem,
which the pap er outlin es .The pap er als o b rie y revie ws legal d o ctrines th at
may stand in way of value -de stroying emp ty votin g, bu t it argu es that the cu
rrent law d o e s not protect against all su ch voting.
Abs ent le gal or contractual c on straints, th e marke t s ability or inability to inf
er the hed ge f und s b e havior f rom ob served trade s is th e key d eterminant
of
1T
h e i n c e n t i v e s f o r s e l f - r e g u l a t i o n a r e s l i g h t l y t o o w e a k b e c a u s e d e r i va
t i v e m a r k e t p a r t i c i p a n t s d o n o t i n t e r n a l i z e t h e n e g a t i v e e /e c t s o n o t h e r s
e c u r i ty h o l d e r s .
2
pro tability of the he dge f un d s s trategy. On the one hand , if th e market cou ld
p e rf ectly obs erve th e he dge fu nd s p os itions , it c ou ld p erfe ctly predict
the voting outcome (sin ce only a hed ge fu nd with more de rivative s th an sec
urities wou ld rationally vote to red uc e th e valu e of th e sec urities ). In that case
, however, th e he dge f und would have to p ay f or the de rivative e xactly wh at
the d erivative contract is going to pay ou t, an d so th e hed ge f un d could not m
ake any pro ts. On the othe r h and, if th e market c ou ld not learn ab out th e h
edge f un d s p osition un de r any circ ums tan ces , then th e hed ge fu nd c ou ld
take ve ry large p ositions without a/ec tin g th e p rice an d make ve ry large p
ro ts by goin g short derivative s if the market pric e presu mes (f alse ly, given
th e he dge fu nd s interve ntion) that the value -re du cing de cision will b e ad
opted, an d by going long de rivatives in the op p osite case . In re ality, and in th e
mo del, large trade s will a/ec t th e price of b oth sec urities an d derivatives. The
trades reveal information , in th is cas e ab out th e h edge fu nd s p osition. Th is
inf ormation , howe ver, is partially hid den by the n ois e from liqu id ity trade s. Th
e p rob le m f or marke t makers, b oth in th e real world an d in the mo del, is wh
at inf erenc es to d raw f rom th e imp erfe ctly ob served in formation.
The emp hasis on trading and m arket makers in feren ce p rob le m sets th e
pres ent mo de l ap art from Bolton an d Oe hmke (2010). Bolton and Oeh mke
also mo del the interaction of d erivatives and c ontrol rights. Con cretely, they
analyz e the e /ect of CDS availability on ren egotiation in an inc om plete
contracting mo del of deb t with s trategic d ef au lt. In th eir mo d el,
creditors ability to hed ge the ir exp osure to the d ebtor w ith CDS c ontrac ts in
creases creditors b argain ing p ower in rene gotiation . This red uc es th e inc id
en ce of strate gic de fault and the ref ore has the b en e c ial e /ect of inc re as in
g the de bt cap acity of the rm; at the same time, overin surance may lead to an
ine¢ c ie ntly high frequen cy of bankruptcy. Crucially, Bolton and Oe hmke (2010)
ass ume that the CDS se llers obse rve the exact p os ition of the b uyer-cred itor,
who th ere fore n ever gain s f rom dealing in CDS as s uch. By contrast, th e p res
ent pap e r b uilds on the assu mption th at derivative (CDS) sellers know neithe r
how m any se curities (b ond s) the he dge f un d h old s, nor h ow many d
erivatives (C DS contracts) th e he dge fu nd en ds up h old in g. Th e latter ass
ump tion app e ars ju sti ed b e cause de rivative s (CDS) are available from multip
le s elle rs who do n ot know how many contracts have b een sold by the others,
and to whom.
2The ide a that a he dge fu nd can us e its emp ty voting p owe r to create unc
ertainty ab out the se curity payo/ an d p ro t on a n et long or a ne t sh ort p
osition is als o pres ent in Brav and M atthe ws (f orth coming). In their mo de l, h
oweve r, the on ly trade d asse ts are sh are s, an d the only way to create a s hort
p osition is by sh ortin g th e sh ares . To retain voting p ower wh ile b e in g s hort
the s hare s, i.e ., to bu ild an empt y votin g p os ition, the hed ge fu nd acquires n
ake d vote s f rom other s hare holders th rough th e s hare lend in g marke t. Brav
and Matthews ass ume th at th e h edge fu nd can d o so f or f re e u p to a ce
rtain amount, and not at all b e yon d th at amount.This ass umption do es not
work ou tside the sh are
2C
f . C h r i s t o /e r s e n e t a l . ( 2 0 0 7 ) , w h o d o c u m e n t t h a t t h e a v e r a g e v o t e d o e
sindeedsellforapriceofzerointhesharelendingmarket.
3
lend ing marke t, howe ver, b ec ause the h edge fu nd will have to pay f or votes b
un dled with a c as h- ow into a s ecu rity. M oreove r, trad in g in de rivative s p
res ents add itional pro t opp ortun ities for the hed ge f und .
The tre atm ent of d erivatives an d empty voting in th is p ap er is entirely the
ore tic al. Comm entators ge nerally agree that n o p re sently available d ata
either proves or disp roves that th e issu es c on side red here are a seriou s
problem, and that su ch data would b e d i¢ cult to come by (Hu and Black 2008a,
2008b ; M en gle 2009, p . 9; Brav an d M atthe ws, forth coming, msc r. p. 34). Hu
and Black (2006, 2007, 2008a, 2008b ) p re sent at least anec dotal eviden ce , h
oweve r, of th e problem o c currin g in ind ividu al in stan ce s. Given the
explosive growth of de rivative s marke ts over the las t dec ade or so, it is als o p
ossib le that the prob le m was n ot serious in th e past bu t may b ecome so in th
e fu tu re . If this were so, the normative recomme ndations of th is p ap er cou ld b
e s een as forestallin g a p otential problem from turning into a real one .
The res t of this pap e r is stru ctured as f ollows. Sec tion I se ts u p the mo
del. Se ction I I e xp lains the e qu ilibriu m c on cep t. S ection I I I solve s f or the
equilib riu m. S ection IV prese nts c omparative statics. Sec tion V d is cus ses
contractual remed ie s th at parties to d erivative c ontrac ts might adopt to solve
th e p rob lem ide nti e d in this pap er; it als o addres ses exis tin g le gal limits. S
ection VI con clude s.
1 Mo del Setup
This sec tion intro duc es th e se tu p of th e mo de l: the two typ e s of traded
asse ts (sec urities and de rivative s), th e three typ es of market p articipants (he
dge f un d, liquid ity traders, and marke t makers), an d trading includ ing inf
ormation. It con clude s with s ome remarks on th is se tu p.
The basic s etup an d the equilib rium c once pt disc uss ed in th e ne xt se
ction is similar to Kyle and Vila (1991), who m o d el the tradin g of a raide r who
has the p ower to de clare a value- in cre asing takeover. In c ontras t to Kyle and
Vila (1991), howe ver, th e present mo de l cons iders th e p arallel trad in g of two
corre lated asse ts , in wh ich th e acquisition of a s u¢ cient amount of one of the
m is require d f or th e p ower to de clare th e "takeover" in the rst p lac e.
1.1 Traded assets
The re are two traded3ass ets with p erfec tly ne gatively correlated payo/s : sec
urities, which will th rou ghout b e d enoted by th e letter X , an d d erivatives,
which will throughou t b e den ote d by the le tte r Y . If th e sec urity pays v , th e
de rivative p ays 1 v . Cons equently, th e derivative can b e interpreted as an in
surance claim on the s ecu rity. In particu lar, if th e se curity we re a b on d, the d
erivative
3"
Tr a d i n g " d o e s n o t n e e d t o b e u n d e r s t o o d l i t e r a l l y i n t h i s m o d e l . I n p a r t i
c u l a r , i t i s p o s s i b l e t h a t t h e d e r i va t i v e i s a c o n t r a c t t h a t i s s o l d o v e r t h e c
ounter.Whatmattersisthattherebeanactivemarketforthecontractin
w h i ch va r i o u s p a r t i e s c a n a c t a s s e l l e r s o r b u y e r s , w h i ch i s t r u e f o r m a n y d
e r i va t i v e m a r k e t s .
4
cou ld b e a cre dit def ault s wap; if th e se curity we re a sh are , th e derivative
cou ld b e a total equity return s wap.
The s ecu rity payo/ v de p en ds on a binary ch oic e b etwee n two actions,
which is d etermined by a vote of the sec urity h old ers. For e xamp le , if the se
curity is a b ond , the choice cou ld b e whe th er or not to agree to a p rop os ed
restructu ring; if th e sec urity is a s hare, it c ould b e whethe r or not to agree to a
m erge r. Normalize th e p ayo/ whe n the "right" dec is ion is taken to v = 1 , and
when the "wron g" dec is ion is taken to v = 0 .
Each s ecu rity provid es on e vote; d erivative s d o not p rovide any vote s.
Natu rally, a ration al, in formed, and un hedged se curity holder wou ld always
vote for th e "right" de cision in orde r to rec eive v = 1 . As will b e d iscu sse d in
the ne xt s ubs ection and the conclu ding remarks on the mo d el setup , this is
ind eed what all s ecu rity holders are assu med to d o. The one e xc eption is the
hed ge f und if an d b e cause the hed ge f und own s more de rivative s than sec
urities , i.e., if the he dge fu nd is over-he dged an d thus h as an "e mpty" voting p
os ition. Th e he dge fu nd s ability to b lo ck the "right" d ecision de p end s on wh
ethe r th e hed ge f und s se curity-h old in g is ab ove some voting thresh old .
Th e votin g thresh old is ass umed to b e a ran dom variable u niformly d is
tributed on the unit interval. Th e ran domn ess c aptures un exp ec te d variation
in voter p articipation , u nce rtainty arisin g f rom legal c on cerns , di/e re nt
formal thresh old s for di/erent typ es of de cisions, etc. The votin g thresh old is as
sum ed to b e inde p en dent of other exogenou s variab le s in th e mo de l, and it
will b e in dep e nd ent of any trad ing activity sinc e it will b e only re veale d af ter
all trad in g o ccu rs .
The numb er of sec urities is normalize d to one (of which in n ites im al divis
ion s are trad ed). S hort-s ellin g is allowed f or b oth de rivative s and se curitie s.
1.2 Market Participants
The re are thre e typ es of risk-ne utral marke t participants: liquidity traders, on e
he dge fun d, and comp etitive market makers.
1.2.1 Li qui di ty traders The liqu idity trad ers d o n ot act strategically. The y
exogenou sly trad e qu antities ~x and ~y of s ecu ritie s and derivatives, resp ectively, where p os itive numb
e rs ind ic ate that the liquid ity trade rs are sellin g, and ne gative numb e rs ind
icate that the y are buying. The se trade s are n ot s ens itive to p rice , an d the
sou rce of thes e trad es is n ot mo d elled. To motivate th ese trad es and th eir
pric e in se nsitivity, one may th in k of large ins titutional inve stors and the ir
regulatory con straints . For example, ce rtain p en sion fu nds might b e forced to
sell b onds f ollowing a cred it d owngrad e of the b orrower. Similarly, n ancial in
stitu tions might b e forced to purch ase cred it def ault swaps on certain b on ds
the y hold. Or one may th ink of mutual fun ds h avin g to liqu idate part of the ir p
ortfolio to mee t re demp tion re qu ests .
Liqu id ity trade rs su pply of d erivatives, ~y , is s to chastic (ke eping in mind
that the "sup ply" can b e n egative). W ith probability , the su pp ly is h igh ( ~y
= y
5
state h ), wh ile with probability (1
), s up ply is low ( ~y = y state l ). De n e
the di/e ren ce b etween the se sup ply re alizations as
y y > 0. For simplic
ity, liquid ity trad ers su pply of sec urities, ~x, is ass ume d to b e cons tant.
1.2.2 Hedge f und The he dge f un d d o e s act s trategically. In itially, th e he dge
f un d d o e s n ot h old
4any s ecu ritie s or de rivative s. It pu rchase s quantities x and y of sec urities an
d de rivative s, resp e ctive ly, taking into account th e e/e ct of its trad es on the
price (as explain ed b elow), its own votin g p ower, and its own votin g in centive
s. As explained in the p re vious sub sec tion , holding x 0 se curities give s the h
ed ge f und the voting p ower to imp lement the "wrong" d ecision with prob ability
x (assu ming th at it is not p os sible to own more th an all of the ou tstandin g s
ecu rities). Of c ou rs e, th e he dge f un d will on ly have an inc entive to us e this
p ower if y > x. The he dge fun d in curs a nan cing cos tc 2 x2+ y2 .1.2.3 Mar ket
maker s The m arket make rs ab sorb any e xc ess s upp ly or de mand ( ^x; ^y ) =
( ~x x; ~y y ).Sin ce th ey are risk-ne utral an d in p e rf ect comp e tition with
one another, th ey pu rchas e or sell th ese quantities at p rice s that e qu al exp
ec te d value, as explained in more d etail b e low. Market makers h ave rational
exp ec tations, i.e. th ey obs erve the net s up ply of se curitie s ( ^x; ^y ) ; and up
d ate the ir b e lief s ab ou t ~y .
1.3 Trading and Information
Trad in g pro c eed s as follows. 1. Liquidity trade rs exoge nous ly s upp ly ( ~x;
~y ), whe re ~y is s to chastic as d esc rib ed in th e p revious sub sec tion. 2. The hedge fund observes ( ~x; ~y )
b ef ore s ub mitting its own market orde r
(x; y ). 3. The marke t makers obse rve only net marke t s upp ly ( ^x; ^y ) = (
~x x; ~y y ).Base d on th is ob servation , the y up date th eir b e lief s ab out
th e exp ected value of the s ecu rities an d de rivative s, as explained in more
de tail b elow. At the se value s (p rices ), th ey ll all n et orders ( ^x; ^y ).
The as sum ption th at only th e hed ge fu nd obs erves liquidity traders ord
ers is c ru cial, bu t it n eed not b e interpreted literally. A m ore lib eral and
realistic interpretation is th at b oth th e hed ge f un d an d the market makers obs
erve n et marke t su pply, but s in ce only the hed ge f un d knows its ow n orders,
on ly th e he dge f und is ab le to back out liquidity trad ers orde rs. In a fu ller, d
yn amic mo del, the he dge f und could grad ually adju st its trades to the availab
le sup ply.
4O
t h e r w i s e , t h e h e d g e f u n d s p r o b a b i l i ty o f w i n n i n g t h e v o t e w o u l d b e mi n
f x; 1 g . I n p r i n c i p l e , i t i s c o n c e i va b l e t o a l l o w x > 1 i f s h o r t - s e l l i n g i n c r e a s
e s t h e a m o u n t o f " s e c u r i t i e s " a va i l a b l e i n t h e m a r k e t . To s i m p l i f y m a t t e r s
,itisnotallowedinthispaper.
6
1.4 Remarks on the mo del setup
The mo d el assu mes th at the hed ge f und is able to ac qu ire any amou nt x of
se curities that it des ires . In partic ular, this ab ility do es n ot dep end on th e
amount ~x su pp lied by liqu id ity trade rs. In re ality, it may of te n b e di¢ cu lt or
imp oss ib le to acquire large blo cks of shares or b onds . Th ere are , howe ver,
many situation s in which e xogen ous sales of se curities ~x are large , an d th e
re ader m ay restric t th e applicability of th e mo del to such s itu ations . For e
xamp le , many institution al investors sell all th eir h old in gs of a b ond if th e b
ond s credit rating drops b e low a ce rtain th re shold. Moreover, in the mo del,
an upp er b oun d on the amount x of secu rities th at th e h edge fun d c an
acquire wou ld not change the h edge f und s s trategy, and th e only ch ange
from th e re sults prese nted b elow wou ld b e that th e h ed ge f und might have
to s ettle for the u pp e r b ound rather than its pref erred , higher p osition (i.e ., on
e wou ld obs erve corner solutions).
Relatedly, the assu mption that large p urch as es h ave no p rice imp act b
eyond the probability u p date by the market m ake rs is not literally true . To go b
ack to the ac qu is ition of se curitie s, it would p re sumab ly b ec ome harder and
h arde r to n d add itional sec urities as the hed ge fu nd s p os ition grows , an d
this would b e re e cted in higher trad ing c osts f or larger p os itions. M ath
ematic ally, h oweve r, the ass ump tion of a n ancin g cost for the h edge f und h
as th e s ame a/ect as ass uming inc reas in g trading c os ts for larger blo cks, so
that nothing s ub stantive se ems to h in ge on th e as sum ption of con stant pric
es con ditional on th e u p dated probab ility.
Fin ally, it is a s trong as su mption that on ly th e on e hed ge f un d is ready to
bu y large s take s, an d to c onside r ove r- hed gin g its se curities p osition and
to vote the s ecu rities f or the "wrong" dec is ion . This e xclu des , rst, that any of
the other market particip ants in the mo del, name ly in divid ual market m ake rs
and liquid ity trade rs, who mu st hold th e re main in g s up ply of se cu ritie s,
would ever hold more derivatives th an sec urities, or if they did, th at the y n
everth eles s voted f or th e "right" de cision . On e ju sti cation f or this cou ld b e
in stitu tion al, namely that re pu tational conce rn s or she er ap athy prevent
market makers an d liquidity trad ers to vote for the "wrong" d ecision, or to overhed ge th eir sec urities p osition in th e rst p lac e.
Se cond , the ab ove as sump tions rule out s trategic comp e tition with a s
econd large player. For examp le, one can imagine a sec ond hed ge f un d trying
to s hare the s p oils, or to b uy u p en ough of the s ecu rity at a low p rice to
prevent th e rst he dge f und from e ver winn ing a vote for th e "wron g" d
ecision. One way to j ustify th is res triction is to as sume ge nu in ely private in
form ation on b eh alf of th e he dge fu nd , in p articular ab out liquidity trad er s
trades . From a prac tic al p oint of view, ad ding anoth er strategic p laye r would
c omplic ate the mo d el bu t not eliminate the u nd erlying ec on omic problem.
For e xamp le , even if the sec urity we re trad ing at de ep dis count b e caus e of
he dge fu nd s p re sen ce, an oth er large player c ould not ne ces sarily
pro tably interven e by b uyin g up the entire su pply of s ecu rities if an d b e caus
e that sec ond large player in cu rs s imilar nan cing cos t as the rst hed ge fu
nd. More over, even if the s econ d large player cou ld pro tably do th is, then in
exp ectation th e price of th e sec urity would re- ad ju st
7
to 1, so that the strate gy wou ld e nd up b eing n ot pro tab le afte r all.
2 Equilibrium Concept
The ab ove as sum ption s on in divid ual b ehavior allow su mmarizing the s
trategic interaction in two s im ple equilib riu m cond itions (Kyle 1984; Kyle and
Vila 1991).
First, th e ass umption that liqu id ity trad ers trade s are exogenou s mean s
that liquid ity trad ers d ecisions ne ed n ot b e consid ere d at all.
Se cond , th e assu mption that market-makers act as comp etitive price takers
and ab sorb or su pply any deman ded qu antities ( ^x; ^y ) mean s that th eir b
eh avior can b e sum marized by the pric e f unc tion. More s p eci c ally, p rices
are entirely pin ned down by marke t- make rs rational exp ec tations of th e s
ecu rity and de rivative p ayo/s. M oreove r, the re is e /ec tively on ly one pric e
fu nction b e cause the payo/s of de rivative s an d sec urities are p e rf ectly ne
gatively correlated. Let Px( ^x; ^y ) b e th e pric e of se curitie s and P 2y( ^x; ^y )
b e th e pric e of de rivative s. Fu rthe rmore, let ( ^x; ^y ) : R! [0; 1] de scrib e
th e c orrectly in fe rred probability that th e s ecu rity will pay z ero con ditional
on obs erved net sup ply of se curities and derivatives , ( ^x; ^y ). Th en
market-makers equilibrium b e havior is fu lly cap tu re d by the following cond
ition:
i ( ^x; ^y ) = 1
i
i ( ^x; ^y ) =
( ^x; ^y )
and Py
=1P
( ^x; ^y )
i
i;
yi
( ; ) , x( ^x; ^y ) : Fin ally, obs erve that at th e voting stage , th e h ed ge f
ned
down by its h old in gs of sec urities an d derivatives . If th e hed ge fu n
ritie s th an derivatives (x > y ), it will vote f or th e "right" d ecision; othe
the "wrong" dec is ion (with mixin g p os sible only if x = y ). Conse qu e
only has two true ch oic e variab le s, name ly its trad es x and y . Le t
un d s pro t in s tate i 2 f h; l g , and (x) the he dge f un d s (p oss ib ly m
os e s tates . Then the hed ge f un d s e qu ilibriu m strate gy is capture
on dition :
; yi) maximize
( ; yi
Pro t maximization : for eve ry i 2 f h; l g , (x s
x j ^x;
^y .
8
) . In cho
osing its pro t-maximizing p osition s (x; y ), th e he dge f und will of
cou rse take into acc ou nt the e/e ct of its trad es on pric es, i.e ., on th e prob
ability inf ere nce of the m arket makers, ( ^x; ^y ). In th at s ens e, th e e¢ c ie nt
marke t con dition imp lie s an inverse d emand cu rve agains t which the h ed ge
fu nd s pro t maximization op e rates.
Cons truc tin g an equilib rium p rinc ip ally require s th e de nition of s om e in
fe re nce fu nction ( ; ) for which b oth c on dition s can hold simu ltane ously.
In principle , ( ; ) equals th e probability that the he dge f und is able to
implement the "wrong" votin g de cision, provid ed that it wou ld want to d o so
in the rst place . That is , in p rinc ip le , ( ^x; ^y ) = E max f 0; min fx;
E¢ cient markets : for some
Px
1gg 1f y >xgNote , however, th at ( ^x; ^y ) is ob jec tively d e ned on ly at p
oints ( ^x; ^y ) that are actually obse rved in equilibriu m. At other p oints, ( ^x;
^y ) is derive d from s ubje ctive o/-e qu ilibriu m b e lief s of marke t- makers .
3 Equilibrium
This p ap er f o c use s on a f ull p o oling equilibrium, th at is, an equ ilibrium in
which the obs erve d n et market s upp ly ( ^x ; ^y ) is id entical in b oth s tates of
the world (h and l ), i.e., the h edge f und f ully hide s b ehin d th e n ois e. W hen
liqu id ity trad ers are large sellers of d erivative s (or equ ivalently, small b uyers),
the hed ge f und acqu ires a large amou nt of derivatives , vote s for the "wron g" d
ecision, an d, if s ucc ess fu l, gain s on its derivative p os ition. W hen liquid ity
trad ers are sm all se llers of de rivative s (or e qu ivalently, large b uyers), the h
ed ge fu nd acquire s fe w de rivative s or e ven s ells th em, vote s f or the "right"
voting d ecision, and gains on its holdings of the sec urity.
5To nd the equilibrium, the p ap er rst conj ec tu re s that it exis ts , and then pro
cee ds to ve rify all the e qu ilibriu m cond itions. S om e of the more tech nical con
ditions are re le gate d to the app e nd ix, inc lu ding a d is cus sion of th e
exogenous parameter s pace for wh ich the equilib rium exists. It c an b e s hown
that the equilib riu m con sidered he re is u niqu e for su ¢ ciently low value s of c,
i.e ., f or su ¢ ciently low nanc in g costs for the hed ge f und .
3.1 Direct implications of the p o oling conjecture
Certain relationsh ips b etwe en exoge nous and en dogenou s variab le s f ollow
d irectly f rom the conj ecture that obs ervab le net market sup ply ( ^x ; ^y ) is
identic al in b oth states, and yet th e hed ge fu nd is able to a/ect the vote
di/erently in the two states .
First, the hed ge f und mu st always b uy the same am ou nt of se curities , x . Se
cond , th e h edge f un d mu st bu y d i/ere ntial qu antities of de rivative s y<
yhlthat p e rf ectly o/set th e d i/e re nce b etween y and y . That is, it mu st b e th
at ^y = y yh= y yl, implyin g yh yl= y y = :
5I
nthealternative,separatingequilibrium,thehedgefund sholding
s a r e f u l l y r e v e a l e d , a n d h e n c e t h e h e d g e f u n d e a r n s z e r o p r o t . To " e n f
o r c e " t h i s e q u i l i b r i u m , t h e r e m u s t b e a f u n c t i o n ( ; ) s u ch t h a t t h e h e
dgefundcannotdeviatetoatradewithpositivepro tsineitherstate.L
e t x = ~x ^x a n d y = y ^y . T h e n t o h o l d t h e h e d g e f u n d t o z e r o p r o t s , i t w o u l
d h a v e t o b e t h e c a s e t h a t f o r a l l ( ^x; ^y ) ,
l
0 :C o m b i n i n g t h e s e tw o i n e q u a l i t i e s , o n e c a
nobtainthecondition
h
=(y+
x)[x
( ^x; ^y )] c2 x 2 + (y + )2i 0
h= ( x y ) ( ^x; ^y ) c2 x2+ y2
2 x y 2
( ^x; ^y
+ y2 +
+x(x y
):
) c x2
y + 22
B u t t h e n f o r a n y ( ^x; ^y ) s u ch t h a t > x
y>
2
+ y2 + y + 2 i + x ( x y
)
2
( ^x; ^y ) ch x2
a n d x; y > 0 , i t w o u l d f o l l o w t h a t f o r c s m a l l e n o u g h ,
<0;
2 x y
2
9
w h i ch i s i n c o n s i s t e n t w i t h t h e d e n i t i o n o f a p r o b a b i l i ty f u n c t i o n .
T
hird,
to
crea
te
di/er
ent
sec
urity
an d
de
rivati
ve p
ayo/
s in
the
two
state
s
and
he
nce
pro t
op p
ortu
n
ities
f or
the
h ed
ge f
un
d, it
mu
st b
e th
at
the
prob
abilit
y of
th e
"wro
n g"
d
ecisi
on b
eing
ado
pted
di/er
sb
etwe
en
the
two
state
s.
On e
con
ditio
nf
or th
is is
yl<
x<
yh, s
o
that
th e
h ed
ge f
un d
will
vote
di/er
ently
in
the
two
state
s.
Th e
othe
r
con
ditio
n is
th at
x> 0
so
that
the
hed
ge
fu
nd
can
actu
ally
in u
e
nce
the
votin
g ou
tcom
e.
F
in
ally,
s
ince
po
olin
g
me
an s
that
the
m
arke
t
mak
e rs
c an
not
te ll
th e
two
state
s ap
art,
it
mus
tbe
th
at
( ^x ;
^y )
= x
6.To
ge
ther
with
th e
e¢
cie
nt
mar
ke t
cond
ition,
this
imp
lies
Py(
^x ;
^y )
=1
Px(
^x ;
^y )
= (
^x ;
^y )
= x
.
3.
2
Fi
rs
tor
d
er
c
o
n
di
ti
o
n
s
7The
p
ro t
maxi
miza
tion
cond
ition
re
qu
ires
that
th e
hed
ge f
un d
cann
ot
do b
etter
in
eith
er
state
by
ad
jus
tin g
its
trad
es.
Ass
u
ming
th at
th e
pro t
fu nc
tion
is
di/e
renti
ab
le,
this
re
qu
ires
that
th
e rs
torde
r
cond
ition
s for
an
opti
mu
m
are
satis
e d.
Th
at th
ep
ro t
fu
nctio
n is
in
dee
dd
i/ere
ntiab
le
can
be
p
rove
d
alon
g
the
lin
es of
Kyle
an d
Vila
(199
1).
Gi
ve
n
th
e
re
su
lts
of
th
e
p
re
vi
ou
s
su
b
-s
e
cti
on
,
ar
ou
nd
th
e
eq
uil
ib
riu
m
for
st
at
es
h
an
dl
,
th
e
p
ro
t
fu
nc
tio
n
ca
n
b
e
wr
itt
e
n
(w
h
er
e
F
(x)
is
th
e
pr
ob
ab
ilit
y
th
at
th
e
h
ed
ge
f
un
d
ca
n
ge
t
th
e
"w
ro
n
g"
d
ec
isi
on
im
pl
e
m
en
te
d,
gi
ve
n
its
se
c
uri
ty
ho
ldi
ng
s
x):
h
D
en
oti
ng
by
su
b
sc
rip
ts
i
th
e
de
riv
ati
ve
of
af
un
c
tio
n
wi
th
re
sp
e
ct
to
its
ith
ar
gu
m
en
t,
th
e
rs
tor
de
rc
on
dit
io
n
s
ar
e:
l
2 ( ^x; ^y ) y
= (x y ) ( ^x; ^y ) c(x; y ) =
x [1 F (x)] + y F (x) xP= (y
P
x) [x
( ^x; ^y )] c2 (x; y ) (=^x; ^y ) c + y
x xPx( ^x; ^y ) y Py2 xx2
2
y
+2 y2
x
( ^x; ^y ) c2
+ y2 2x
2x
+ y2
) = (yh
x)
2
x) [1 +
1
( ^x; ^y )] [x
@ x (x
h@
7[
) = (yh
y (yh
) = (x
l
l
) = (x
( ^x ; )] cx
^y
( ^x
yl)
1
( ^x; ^y) +
( ^x; ^y ) cx = 0:
yl)
2
( ^x; ^y )
( ^x; ^y
l
l
...]
@
h
= 0; @ ; ^y) + [x
( ^x; ^y )] cy= 0; @ @ x (xh= 0; @ @ y (y) cy
R e c a l l t h a t w e h a v e a s s u m e d t h a t i t i s n o t p o s s i b l e t o6o w n m o r e t h a n
alloftheoutstandingsecurities,i.e.,x 1.
10
3.3 Equilibrium values
Comb in in g th e rst-order cond itions an d the d irec t implication s of the p o oling con jec ture yie ld s th e f
ollowing equilibriu m value for x :
x
= max = x ( + c) [1
c] 2c;
1 ;
2
=x
h
; ^y
y
2c ;
( + c)y; y+ (1
c) ; ( ^x ) = c x The u pp er b ound on x come s f rom
the assu mption that it is not p os sible to own more than all the outstand in g se
curities , wh ose su pply is normalized to one . Assu ming that we are n ot at th
e c orn er solution x= 1 , th e solutions for the oth er en dogenou s variab les c
an b e written
= ( + c) [1
3c]
yh = ( + 3c) [1
c]2c ;
( ^xand the impac t of trade s on the p robab ility inf ere nce f un ction ( ; ) at the
; ^y ) = c + x
equilib riu m p oint is
(
^x
l
l
1
; ^y
(1
)=
( c) [1
c] 2c ;
( + c) [1
c] 2 c:
)=c x =c
2
3.4 Other conditions
To ve rify that th e he dge f und s trade s are in dee d pro t maximizing, it remains to verif y th e s econd -order
cond itions , an d that the hed ge fu nd s equilib rium p ro ts are p ositive in b oth s tates (since th e he dge f und
always h as the op tion n ot to trad e). As alread y mention ed , it is also ne ces sary to verif y the as sump tion
mad e ab ove th at x 2 (0; 1) and that no se curity h old in gs outside this inte rval yield s highe r pro ts. Fin ally,
since the de rivation ab ove as sum ed di/erentiability of ( ; ), it must b e checked that the hed ge f und can not
earn h igh er pro ts at p oints where ( ; ) is n ot d i/erentiable, i.e., whe re equals e ith er on e or ze ro. Verif
ying the se con dition s is re le gated to th e app e ndix, whe re it is sh own that all cond itions hold su b je ct to
certain restric tions on th e p arameter sp ace.
11
4 Comparative Statics
The primary variable of inte res t is the prob ability that value- des troyin g de
cisions will b e adop te d, that is
x = max
( + c) [1
c] ;
2c
1 :
At the internal s olu tion, this probability is in creasing in , the u ctuation in
liqu idity trade rs d erivative trades , and de cre asing in c, the h edge fu nd s n
ancin g cost. The p robab ility is in creasing in b ec ause greater u ctuation in
liquidity trad es p rovides more camou age to th e hed ge fun d. Th is e nables the
he dge fu nd to take greater n et p osition s, with c oncomitantly greater p ro ts con
ditional on obtain ing the de sired votin g re sult. This in turn make s it more
attractive f or th e h ed ge f und to ac qu ire a more p owerf ul voting p os ition. As
the noise go es to ze ro, s o do es th e he dge f und s votin g p os ition an d he
nce the probab ility of value- des troyin g de cisions.
The prob ability is dec reas in g in c b ec ause greater nan cing costs make it
more costly f or the hed ge fu nd to acquire large votin g b lo cks an d d erivative
p osition s. Th is is p erhaps mos t clear in the expres sion s for the e qu ilibriu
m holdings of de rivatives , wh ich are centered arou nd x bu t sh if ted
downwards by an amou nt c. As sh own in th e app e ndix, the h ed ge fu nd d
o e s not even make p os itive pro ts in b oth state s of the world if the trad ing
cost parameter c is greater than
8equilib riu
p 1=8 ; in th at case , fu ll p o oling is no lon ger a s
m.
ustainable
1
( ^x
( ;
^x ^y
8
; ^y
1
1
12
The slop e of th e in fe re nce an d h enc e of the pric e fu nction with resp e ct
to th e obs erve d net sup ply of sec urities , , re ects the amb igu ou s nature of
inc re asing the h edge f und s s ecu rity h old in gs. On th e one han d, greater s
ecu rity h old in gs make it less attrac tive for th e he dge fu nd to vote f or th e
"wron g" de cision. On the other h and, greater se curity holdings als o make it
easier f or th e h edge f und to get th e "wrong" d ecis ion adopte d, if it wants to.
Re ec tin g this du al ch aracter of inc re as in g th e h edge f un d s se curity p os
ition, the inf erre d probability of the "wron g" d ec is ion b eing adop te d can eith
er rise or f all if the hed ge fu nd in creases its sec urity holdings, d ep en ding on
the relative s iz e of and c. If > c, ) > 0, i.e., whe n th e hed ge fun d bu ys
more sec urities, the inf erred probab ility of a "wron g" vote go e s down . By c
ontras t, if < c, ) < 0 , i.e ., th e infe rred p robab ility of a "wrong" vote go es up
wh en the h edge f und bu ys more se curitie s.
5 Contractual Remedies
[this s ection is p articularly inc om plete]
E q u i l i b r i u m b e h a v i o r f o r c > p1 = 8 r e m a i n s t o b e v e r i e d . I t i s p r o b a b l y n
otanequilibriumforthehedgefundnevertodoanytradesbecauseata
ninitialtradeofzero,thehedgefund smarginalcostoftradingiszero
. S o i t i s l i k e l y t h a t t h e r e w i l l a l w a y s b e s o m e p o s i t i v e p r o b a b i l i ty o f t h e
hedgefundacquiringsomesecuritiestoattempttoimplementtheneg
ativevotingoutcome.
The discu ss ion thus f ar h as not cons id ered legal or contractual limitation s
on the "empty voting" s trate gy pu rsu ed by th e hed ge f un d. This sec tion
brie y outlin es two of th e main legal con straints b efore pu ttin g f orward a
contractual solution, wh ich th e p articipants of th e d erivative market sh ou ld nd
in the ir se lf -inte res t to adopt.
5.1 Legal limits
On th e legal side , the he dge fu nd might n d its b ehavior incrimin ate d un der
di/e re nt do ctrin es in the d ebt and e qu ity case . In the equity case, holding a su
¢ ciently large blo ck of voting s hare s would qualify the hed ge f un d as a
controllin g sh are hold er. Controlling shareholde rs , h owever, arguab ly h ave
a du ciary du ty n ot to exercise th eir votin g p ower agains t th e b en e t of th e
corp oration in f urth eran ce of s om e private b e ne ts , here th e p ayo/ on th e d
erivative s.
In th e d ebt c as e, creditor votes in bankruptc y c an b e disallowed if they are
not c as t in "go o d f aith ," an d cas tin g th e votes "e mpty" with the intention of
lim iting recovery by cred itors would arguab ly qualif y as not b eing in "go o d f
aith."
In each case , h oweve r, app lication of th ese d o c trines requires that the e
mptine ss of the h ed ge f un d s voting p osition b e kn own by fe llow sh areh old
ers, cred itors, or the ju dge, as th e case may b e . At p res ent, this is n ot ne ces
sarily guarante ed by e xisting disc los ure obligation s (Hu and Black ...).
Moreover, limits on "e mpty votin g" do n ot ap p ear to apply to creditors outside
of bankru ptcy.
5.2 Contractual p osition limits and information p o oling
Participants in the derivative market have the p ossibility and the in centive ,h oweve
r, to adopt a contractual solution to the "empty voting" problem.
In the mo d el, the hed ge f und s gains trans late dire ctly into loss es for
liquid ity trad ers in th e d erivative market. Wh en liquidity trade rs bu y many d
erivative s, the he dge fu nd bu ys fe w, votes the "right" way, and the reby d
ecrease s the payo/ of the derivatives . W he n liqu id ity trad ers s ell m any de
rivatives , the he dge fu nd b uys many, votes the "wrong" way, and the re by in cre
ases the p ayo/ of the derivatives . Liqu idity trade rs sh ould th erefore p re fe
r and comp e titive marke t m ake rs would b e forc ed to provid e de rivative
contracts that exclude the p ossib ility of "emp ty voting."
At prese nt, c ertain de rivative contracts, in p artic ular CDS, alread y c ontain
claus es that purp ort to forc e the c ontrac t own er to exerc ise any voting rights
in the corres p ond in g se curity with a vie w to maximizin g the valu e of that se
cu rity. As Hen ry Hu an d Be rn ie B lack n ote , however, the se c laus es are
hardly e nf orce able b e caus e th e required in formation is not gen erally availab
le (Hu an d Black 2008a, p. 733; 2008b , p p. 682-3). It would app e ar di¢ cu lt
and cu mb ersome to collect votin g inf ormation on a frequ ent basis.
A more eas ily impleme ntable p rotec tive mechanism would b e to limit th e
amount of h edging th at u sers of derivatives are allowed to do, i.e., to in sert p
osition limits into derivatives contracts. This would re nde r th e "empty voting"
13
strate gy relative ly u nattractive to h edge fu nd s who nee d to b e able to
overhe dge a s iz eable p os ition of voting s ecu ritie s to make the strategy
pro table. To b e s ure , p osition limits could b e circumvente d by b uying d
erivatives from mu ltiple se llers. S elle rs cou ld rese rve the contractu al right,
however, to p o ol inf ormation at th e pay-out stage to en sure that p os ition limits
were res p ec te d. The in formation c ou ld b e p o oled in th e hand s of a third
party s ub je ct to s trict con de ntiality p rovisions. The third party would colle ct p
ay- ou t inf ormation f rom all d erivative c ounte rp arties and only re lease any in
form ation if it f ound that ind ividual recip ie nts had e xc eed ed th eir p osition
limits.
6 Conclusion
Commentators have b e en worryin g ab ou t the votin g inc entives of a sec urity
holde r wh o is "ove r-h ed ge d," i.e., wh o holds more o/-s ettin g d erivatives th
an votin g se cu ritie s. Th is p ap er p rovides th e rs t rigorous de monstration
that bu ild in g su ch an "empty voting p os ition" c an b e a p ro tab le strate gy for
a he dge fu nd if sec urities and de rivative s trade in p arallel noisy m arkets. Th e
strate gy will b e all th e more pro table, an d valu e-d estroyin g votes all th e
more like ly, as trading c os ts de crease an d market liquidity in creases , as has
ge nerally b e en the trend in d erivatives marke ts over the last de cade. This p ap
er the re fore add s reason to take the "e mpty votin g" prob lem s erious ly.
At the s ame tim e, the mo d el sh ows that the h edge fu nd s gains come at
the exp ens e of liqu id ity trad ers in the derivative markets, with los ses to s ecu
rity holders merely arisin g as collateral d am age . Henc e liquidity trad ers in th e
de rivative markets have th e right in centive s to pu sh for a contractual solution of
th e "emp ty voting problem" dis cus sed in this pap er. In p articu lar, they may
insis t on us in g contracts with p osition limits.
14
A
p
p
e
n
d
i
x
A
d
d
it
i
o
n
a
l
E
q
u
ili
b
r
i
u
m
C
o
n
d
it
i
o
n
s
T
h
i
s
a
p
p
e
n
d
i
x
v
e
ri
co
n
si
de
re
d
in
th
e
m
ai
n
te
e
s
r
e
m
a
i
n
i
n
g
c
o
n
d
it
i
o
n
s
f
o
r
t
h
e
f
u
ll
p
o
o
li
n
g
e
q
u
ili
b
ri
u
m
xt.
7x
Th
e
d
eri
va
tio
n
of
th
e
he
d
ge
fu
nd
s
pr
ot
m
ax
im
izi
ng
tra
de
s
as
su
m
ed
th
at
th
e
he
dg
e
fu
nd
s
pr
ob
ab
ilit
y
of
wi
nn
in
g
th
e
vo
te
ar
2 (0; 1)
ou
nd
th
e
eq
uil
ib
riu
m
p
oi
nt
eq
ua
ls
x,
an
d
th
at
th
e
de
riv
ati
ve
of
th
e
pr
ob
ab
ilit
yf
un
cti
on
is
1.
Th
at
is
,
th
e
de
riv
ati
on
of
op
ti
m
al
tra
de
s
on
ly
co
ns
id
er
ed
x
2
(0;
1).
It
n
ee
d
s
to
b
e
ve
ri
ed
,
rst
,
th
at
th
e
op
ti
m
u
m
tra
de
ac
tu
all
yf
all
s
int
o
thi
s
int
e
rv
al,
an
d,
s
ec
on
d,
th
at
no
x
ou
tsi
de
thi
s
int
er
va
l
yi
el
ds
hi
gh
er
p
ro
ts
.
Fo
r
x = ( + c) [1
1)
h (x; y ) jx 0
c]2 c 2 (0;
wou ld not b e pro table.
8 Positive pro ts
15
0
:
S
o
2x
+ y2
to h old tru e requires th e followin g re strictions on th e exoge nous
parameters: ( + c) [1
c] > 0() 1
> c: ( + c) [1
c] < 2c()
2c( + c) [1
c] :
<
h
Next, che ck that n o x0outside (0; 1) wou ld yie ld h igh er pro ts for the he dge
= (x y ) ( ^x; ^y ) c2 f=und , given ( ^x; ^y ). First, do es n ot de p en d on F (x), s o the global
0 c2 2x+ y2
maximization of llhas the same cond itions as the re stricte d optimization ab
ove, with the s am e outcome. S econ d, h(x0) < h(x ) for any x0 1 sinc e we f
ou nd that x 2 (0; 1) maximize s heve n u nd er the assu mption th at F (x) = x
throughou t, while in reality F (x) = 1 for all x0 1 , an d is increasing in F (x).
Third , f or x 0, F (x) = 0 , s o marke t makers know that v = 1 f or s ure, i.e
., ( ~x x; ) = 0 , and he nce
t
h
e
h
e
d
g
e
f
u
n
d
w
o
u
l
d
b
e
e
a
r
n
i
n
g
n
o
n
p
o
s
it
i
v
e
p
r
o
t
s
,
a
n
d
t
h
i
s
d
e
v
i
a
ti
o
Th
e
he
dg
ef
un
d
al
w
ay
s
h
as
th
e
op
tio
n
no
t
to
tra
de
an
d
e
ar
n
ze
ro
pr
ot
s.
Th
us
f
ull
p
o
oli
ng
ca
n
on
ly
b
e
an
eq
uil
ibr
iu
m
if
th
e
n
h
ed
ge
f
un
d
m
ak
es
p
os
iti
ve
p
ro
ts
in
b
ot
h
s
tat
e
s.
U
s
in
g
th
e
eq
uil
ibr
iu
m
va
lu
es
de
riv
e
d
in
th
e
m
ai
n
te
xt,
w
e
m
u
st
ha
ve
h
(x ; yh) = (yh
] c2
2
x
) [x
x
x
+ y2 h
2
:
2
2 ( + c)
c) 2 ( + c) [1
c] 2c4 c 0() 2 ( + c)
[1
c] 2c ( + c)
We als o mus t
) = (x
c2 ) x yl
x + y2 l
2
have l(x ; yl
2 [1 ( + c)]: The p ositive pro ts cond ition c an thu s b e
2
stated
2 max 2n ( + c)2; 13 c2; [1 ( + c)] o
2
+
2( + c) [1 ( + c)]
c() c 2 "0;
13
r1 8# ^ 2 2
13
=
2
2
2(1
2
The up p er b ou nd for
1
c + 23
2( +
+1
c)
3 c
=
1 p 1 6c2
implies
c + 13 > c;
3
p 1 6c 2 ( + c) [1
c] 2 c4c
[1
c] 2 c
0() 2 ( + c)
2
2 (0;
1).
i.e ., it contain s the con dition 1
9 Second-order conditions
To
ens
ure
that
we
are
at
an
opti
mu
m,
we
nee
d to
ch
eck
the
sec
on d
d
eriva
tive
s.
To f
acilit
ate
th is
, we
res
trict
th e
sear
ch to
lin
ear f
unc
tions
(;
),
> c requ ired f or x
2
[1
c]
p1 6c
with
slop
es 1
and
2;
^y a
s
give
n in
the
previ
ous
sub
se
ction
. As
p
ointe
d ou
t in
the
main
te
xt,
(; )
is ob
j
ectiv
ely
de n
ed
only
at
the
p
oint
(
^x),
so th
ere
are
p
ote
ntiall
y
man
y
di/er
ent f
unc
tions
(;
)
that
c an
s
ustai
n th
e
equil
ib
rium
.
The
re
fore,
to th
e
exte
nt
that
the s
econ
d
-ord
er
cond
ition
sf
or lin
ear
(; )
con
sider
ed h
ere
yie
ld
restr
ic
tions
on
the
para
m
eter
s, it
is p
oss
ible
th at
th
ese
restr
ic
tions
wou
ld
not
ap
ply if
a
broa
der
c
lass
of
fun
ction
s
were
c on
side
red .
U
n
d
e
r
t
h
e
l
i
n
e
a
r
i
t
y
a
s
s
u
m
p
t
i
o
n
,
t
h
e
s
16
e
c
o
n
d
d
e
r
i
v
a
t
i
v
e
s
a
r
e
:
1
@2 h
@ x2
1=
2 2
=22
2h
2
@ y2
2h
2l
@ x2
2l
@ y2
2l
c
@
c
@
@
y
@
x
=
1
+
1
@
=
2
c
@
=
2
2
1
c
@
@
y
@
x
=
1
2
The con ditions
for an op timum
are thus9
2 c < 0 (2 c) (2 2 c) ( 11 22)> 0 Using the
resu lts f rom the rs t- orde r con dition s, the rs
t of the se con dition s
c = 2 c +2=
x
require s
2c
+c
2 1
< 0 ) c (1 2 c)
< [1
]) 2 12 1 2
9T
hecondition 2 2
2
1
4c + 1
p8
c
c<0isimpliedby
2 1c<0:Thecondition
2 2c<0isimpliedbythe rstconditionand
c ) (2 2 c ) ( 1
2)2
( 2 1
)2
1
17
(1
[1
)
]c
c
> 0 : F i n a l l y, t h e c o n d i t i o n
( 2 2
c)(22 c) ( 1
2
>0isimpliedbytheprecedingcondition.
Fo
x
x
(1
)
2= 2 (
+ c) [1
c] + c [2 (1
) 3 c] (1
)= 3 22 5c 6 c + 4
0) c 2 [0; r1 6) ^ 2 23 c 21 3p 1 6c2;which is gu caranteed by
2 th
p os itive p ro t c ondition . As c an b e checked algeb raically,
th
2 r)
pro t con dition als o implie s th e
up p er b ound f or from th e( rst cond ition, bu2t n ot the lower b oun
x .
2=7
1
T
h
e Corner solutions [NEED TO CHECK]
10
The
rst-orde r ap proach assu med that ( ; ) is d i/erentiable in b o
s
)
nts.
e This, however, can only b e true on a limited interval of ( ^x; ^y ) s
e ( ; ) 2 [0; 1]....
+
c
o
3
11
n Summary of parameter restrictions
In
s
um,
we
have
the
f
ollowing
parameter re strictions f rom the re qu
d
nt that x 2 (0; 1) and that pro tsc b e p os itive in b oth states:
c
o
n
d
i
t
i
o
n
r
e
q
u
i
r
e
s
c 1
< 2 c( + c) [1
c] :
3 p 21 6c; 13 c + 1 p 1 6c2
3
c
2
"
0
;
2
2
3
r
1
8
#
ra
lin
e
ar
f
un
c
tio
n
an
d
c
>
2=
7,
th
e
se
c
on
dor
d
er
c
on
dit
io
n
s
al
so
re
qu
ire
> 12
Ref
er
ence
s 1.
Bolt
on,
Patri
ck,
and
M
artin
Oeh
mke
.
201
0.
Cred
it
Def
1 p2 8c2 4c + 1:
18
ault
Swa
ps
and
th
e
Emp
ty C
red
itor
Prob
lem.
Wor
kin g
pap
er,
Colu
mb
ia
Un
ive
rsity.
2.
Brav
,
Alon
,
and
Rich
mon
d
Matt
hew
s.
Fort
h
comi
ng.
E
mpty
Voti
ng
and
th
e
E
¢
ci
en
cy
of
C
or
p
or
at
e
G
ov
er
na
nc
e.
Jo
ur
na
l
of
Fi
na
nc
ial
Ec
on
o
mi
cs
.
3. Chris to/e rs en, Sus an, Ch ristoph er Gecz y, David Mus to, an d Adam
Reed . 2007. Vote Trading an d Inf ormation Aggre gation. Journal of Finance
62:2897-2939.
4. Hu , Henry, and Bernard Black. 2006. The New Vote Bu ying: Emp ty Voting
an d Hid den (Morph able) Owne rsh ip. Southern C alifornia Law Review
79:811-908.
5. Hu , Hen ry, an d Bernard Black. 2007. Hedge Fund s, Insid ers , an d the
Decou plin g of Ec onomic and Voting Ownersh ip : E mpty Voting and Hid den
(Morphab le ) Own ership. Journal of C orporat e Finance 13:343-367.
6. Hu , Hen ry, and Be rn ard Black. 2008a. Equity an d De bt Dec oupling and
Emp ty Votin g I I: Imp ortanc e and Extens ion s. Universit y of Pennsyl vania
Law Review 156:625-739.
7. Hu , Hen ry, an d Be rn ard B lack. 2008b. De bt, Equity an d Hybrid De
coupling: Governanc e an d S ys temic Risk Imp lic ations. European Journal of
Financial Managemen t 14:663-709.
8. Kyle , Alb ert. 1984. A Th eory of Futures M arket Man ip ulation s. In The In
dustrial Organization of Fut ures M arkets, ed . Ronald And erson, 141173. Le
xington Mass .: LexingtonBo oks .
9. Kyle , Alb ert, an d Jean -Luc Vila. 1991. Noise Trading and Take ove rs .
RAND Journal of Economics 22:54-71.
10. M en gle, David . 2009. The E mpty Cred itor Hyp othes is . IS DA Res earch
Note s 3/2009.
11. S en der, Henny. Gree nlight Capital foun de r calls for CDS ban. Finan cial
Times, Novemb er 6, 2009.
12. S oros, George. 2010. Ame rica mus t f ac e up to the d angers of derivatives
. Fin ancial Times, April 22, 2010.
19