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Optimal portfolio liquidation with execution cost and risk
Idris Kharroubi, Huyen Pham
To cite this version:
Idris Kharroubi, Huyen Pham. Optimal portfolio liquidation with execution cost and risk. SIAM Journal on Financial Mathematics, Society for Industrial and Applied Mathematics 2010, 1, pp.897- 931. �10.1137/09076372X�. �hal-00394997�
Optimal portfolio liquidation with exeution ost and risk
IdrisKHARROUBI
LaboratoiredeProbabiliteset
ModelesAleatoires
CNRS,UMR7599
UniversiteParis7,
andCREST,
e-mail: kharroubiensae.fr
Huy^en PHAM
LaboratoiredeProbabiliteset
ModelesAleatoires
CNRS,UMR7599
UniversiteParis7,
CREST,and
Institut UniversitairedeFrane
e-mail: phammath.jussieu.fr
June 14,2009
Abstrat
Westudy theoptimalportfolio liquidationproblemoveranite horizonin alimit
orderbook withbid-askspreadandtemporarymarketprieimpatpenalizing speedy
exeutiontrades. Weuseaontinuous-timemodelingframework,butin ontrastwith
previous related papers (see e.g. [24℄ and [25℄), we do not assume ontinuous-time
tradingstrategies. We onsider instead real trading that ourin disrete-time, and
thisisformulatedasanimpulseontrolproblemunderasolvenyonstraint,inluding
the lag variable traking the time interval between trades. A rst important result
of our paper is to showthat nearlyoptimal exeution strategies in this ontext lead
atuallytoanitenumberoftradingtimes,andthisholdstruewithoutassumingadho
anyxedtransationfee. Next,wederivethedynamiprogrammingquasi-variational
inequalitysatisedbythevaluefuntioninthesenseofonstrainedvisositysolutions.
We also introdue a family of value funtions onverging to our value funtion, and
whihisharaterizedastheuniqueonstrainedvisositysolutionsofanapproximation
ofourdynamiprogrammingequation. Thisonvergeneresultisusefulfornumerial
purpose,postponedinafurther study.
Keywords: Optimal portfolio liquidation, exeution trade, liquidity eets, order book,
impulseontrol, visositysolutions.
MSC Classiation (2000) : 93E20, 91B28, 60H30, 49L25.
WewouldliketothankBrunoBouhardforusefulomments.WealsothankpartiipantsattheIstanbul
workshoponMathematialFinaneinmay2009,forrelevantremarks.
Understanding trade exeution strategies is a key issue for nanial market pratitioners,
and has attrated a growing attention from theaademi researhers. An important pro-
blem faed by stok traders is how to liquidate large blok orders of shares. This is a
hallenge due to the following dilemma. By trading quikly, the investor is subjet to
higher osts due to market impat reeting the depth of the limit order book. Thus,
to minimizeprie impat, it is generally beneial to break up a large order into smaller
bloks. However, more gradual trading over time results in higher risks sine the asset
value an vary more during the investment horizon in an unertain environment. There
has been reentlya onsiderable interest in the literatureon suh liquidityeets, taking
into aount permanent and/or temporary prie impat, and problems of this type were
studiedbyBertsimasand Lo[7 ℄,AlmgrenandCriss[1 ℄,Bank andBaum[5 ℄,Cetin,Jarrow
and Protter [8 ℄, Obizhaeva and Wang [18 ℄, He and Mamayski [13 ℄, Shied an Shoneborn
[25℄,LyVath,Mnifand Pham[17 ℄,Rogers andSingh [24 ℄,andCetin,Sonerand Touzi [9℄,
to mentionsome of them.
There are essentially two popular formulation types for the optimal trading problem
in the literature: disrete-time versus ontinuous-time. In the disrete-time formulation,
we maydistinguishpapersonsidering thattradingtake plae atxed deterministitimes
(see [7℄),at exogenous randomdisretetimes given forexamplebythejumpsofa Poisson
proess (see [22 ℄, [6 ℄), or at disrete times deided optimally by the investor through an
impulseontrol formulation (see [13 ℄ and [17 ℄). In this lastase, one usuallyassumes the
existeneof a xedtransationost paidat eah tradinginorder to ensurethatstrategies
donotaumulateintimeandourreallyat disretepointsintime(seee.g. [15 ℄ or[19 ℄).
Theontinuous-timetrading formulation isnotrealistiinpratie,butisommonly used
(asin[8 ℄,[25 ℄or[24 ℄),duetothetratabilityandpowerfultheoryofthestohastialulus
typiallyillustratedbyIt^o'sformula. Inaperfetlyliquidmarketwithouttransationost
and market impat,ontinuous-time trading isoften justied by arguingthat it is a limit
approximationof disrete-timetradingwhenthetimestepgoestozero. However, onemay
questionthevalidityofsuhassertion inthepresene ofliquidityeets.
In this paper, we propose a ontinuous-time framework taking into aount the main
liquidity features and risk/ost tradeo of portfolio exeution: there is a bid-ask spread
in the limit order book, and temporary market prie impat penalizing rapid exeution
trades. However, in ontrast with previous related papers ([25℄ or [24 ℄), we do not as-
sumeontinuous-timetrading strategies. We onsider insteadreal trading thattake plae
in disrete-time, and without assuming ad ho any xed transation ost, in aordane
with the pratitioner literature. Moreover, a key issue in line of the banking regulation
andsolvenyonstraintsistodeneinaneonomially meaningfulwaytheportfolio value
of a position in stok at any time, and this is addressed in our modelling. These issues
areformulatedonvenientlythroughanimpulseontrolprobleminludingthelag variable
trakingthetimeintervalbetweentrades. Thus,weombinetheadvantagesofthestohas-
ti alulustehniques,and therealistimodelingof portfolio liquidation. Inthisontext,
we studytheoptimalportfolio liquidationproblemovera nitehorizon: theinvestorseeks
minal liquidation wealth, and undera natural eonomi solveny onstraint involving the
liquidationvalueof a portfolio.
A rst important result of our paper is to show that that nearly optimal exeution
strategies inthis modeling lead atually to a nite number of trading times. While most
models dealing with trading strategies via an impulse ontrol formulation assumed xed
transation ost in order to justify a posteriori the disrete-nature of trading times, we
prove here that disrete-time trading appear naturally as a onsequene of liquidity fea-
tures represented by temporary prie impat and bid-ask spread. Next, we derive the
dynamiprogrammingquasi-variationalinequality(QVI)satisedbythevaluefuntionin
thesenseof onstrainedvisosity solutionsinorder to handlestate onstraints. There are
some tehnial diÆulties related to the nonlinearity of the impulse transation funtion
induedbythemarketprieimpat,andthenon smoothnessof thesolvenyboundary. In
partiular,sinewedonotassumea xedtransationfee,whihpreludestheexisteneof
a stritsupersolutionto theQVI, wean notprovediretly aomparison priniple(hene
a uniqueness result) for the QVI. We then onsider two types of approximations by in-
troduing familiesof value funtions onverging to our original value funtion, and whih
are haraterized as uniqueonstrained visositysolutions to their dynamiprogramming
equations. Thisonvergene result isusefulfornumerialpurpose, postponedina further
study.
Theplanofthepaperisorganizedasfollows. Setion2presentsthedetailsofthemodel
andformulatestheliquidationproblem. InSetion3,weshowsomeinterestingeonomial
and mathematial properties of the model, in partiular the niteness of the number of
tradingstrategies underilliquidityosts. Setion4isdevotedto thedynamiprogramming
and visosityproperties ofthe value funtionto ourimpulseontrolproblem. We propose
in Setion 5 an approximation of the original problem by onsidering small xed tran-
sation fee. Finally, Setion 6 desribesanother approximation of the model with utility
penalization by smallost. As a onsequene, we obtain that ourinitial value funtionis
haraterized as the minimal onstrained visosity solution to its dynami programming
QVI.
2 The model and liquidation problem
We onsider a nanial market where an investor has to liquidate an initial position of
y > 0 shares of risky asset (or stok) by time T. He faes with the following risk/ost
tradeo: if he trades rapidly, this results in higher osts for quikly exeuted orders and
market prie impat; he an then split the order into several smaller bloks, but is then
exposedto theriskofpriedepreiation duringthetradinghorizon. Theseliquidityeets
reeived reently aonsiderableinterest startingwiththepapers byBertsimasand Lo [7℄,
and Almgrenand Criss [1 ℄ in a disrete-time framework, and further investigated among
others inObizhaeva and Wang [18℄, Shied an Shoneborn [25 ℄, or Rogers and Singh [24 ℄
in a ontinuous-time model. These papers assume ontinuous trading with instantaneous
trading rate induing prie impat. In a ontinuous time market framework, we propose
timethroughanimpulseontrolformulation,andwithatemporaryprieimpatdepending
on thetimeintervalbetweentrades, and inludinga bid-askspread.
Wepresentthedetailsofthemodel. Let(;F;P)beaprobabilityspaeequippedwith
altrationF =(F
t )
0tT
satisfyingtheusualonditions,andsupportingaonedimensional
Brownian motionW ona nitehorizon[0;T℄,T <1. We denotebyP =(P
t
) themarket
prieproess of therisky asset, byX
t
the amount of money(or ash holdings), byY
t the
numberof shares in thestokheld by the investorat time t, and by
t
the time interval
betweentimet andthe lasttrade before t. We setR
+
=(0;1)and R
=( 1;0).
Trading strategies. We assume that the investor an only trade disretely on [0;T℄.
Thisismodelledthroughanimpulseontrolstrategy=(
n
;
n )
n0 :
0
:::
n
:::T
are nondereasing stopping times representing the trading times of the investor and
n ,
n0,are F
n
measurable random variables valued in R and giving the number of stok
purhased if
n
0 or selled if
n
<0 at these times. We denote by A theset of trading
strategies. The sequene (
n
;
n
) may be a priori nite or innite. Notie also that we
do not assume a priorithat the sequene of trading times (
n
) is stritly inreasing. We
introduethelagvariable trakingthetime intervalbetweentrades:
t
= inf
t
n :
n
tg; t2[0;T℄;
whih evolvesaording to
t
= t
n
;
n
t<
n+1
;
n+1
=0; n0: (2.1)
Thedynamis of thenumberof sharesinvested instokisgiven by:
Y
t
= Y
n
;
n
t<
n+1
; Y
n+1
= Y
n+1 +
n+1
; n0: (2.2)
Cost of illiquidity. The market prie of the risky asset proess follows a geometri
Brownian motion:
dP
t
= P
t
(bdt+dW
t
); (2.3)
with onstants b and > 0. We do notonsider a permanent prie impat on the prie,
i.e. the lastingeet of large trader, butfous here on theeet of illiquidity,that is the
prie at whih an investorwill trade the asset. Suppose now that the investor deides at
timet to make an orderin stokshares ofsize e. If theurrent market prieisp, and the
timelag fromthelastorder is ,thenthepriehe atuallyget fortheorder eis:
Q(e;p;) = pf(e;); (2.4)
where f is atemporary prieimpat funtionfrom R [0;T℄into R
+
[f1g. We assume
thattheBorelianfuntionf satisesthefollowingliquidityandtransationostproperties:
(H1f) f(0;) =1,and f(:;) isnondereasing forall 2[0;T℄,
(H2f) (i) f(e;0) =0 fore<0,and (ii)f(e;0) =1 fore>0,
(H3f)
b
:= sup
(e;)2R
[0;T℄
f(e;) <1 and
a := inf
(e;)2R
+ [0;T℄
f(e;) >1.
= p, and a purhase (resp. a sale) of stok shares indues a ost (resp. gain) greater
(resp. smaller)thanthemarketprie,whihinreases(resp. dereases)withthesizeofthe
order. In other words, we have Q(e;p;) (resp. ) pfor e (resp. ) 0,and Q(:;p;)
is nondereasing. Condition (H2f) expresses the higher osts for immediay in trading:
indeed, the immediate market resilieny is limited, and the faster the investor wants to
liquidate(resp. purhase)theasset,thedeeperintothelimitorderbookhewillhaveto go,
andlower(resp. higher)willbetheprieforthesharesoftheassetsold(resp. bought),with
a zero (resp. innite)limitingprie forimmediate bloksale (resp. purhase). Condition
(H2f) also prevents theinvestor to pass orders at onseutive immediate times, whih is
thease inpratie. Instead ofimposinga xedarbitrary lagbetweenorders, we shallsee
that ondition (H2) implies that trading times are stritly inreasing. Condition (H3f)
aptures atransation ost eet: at time t, P
t
is themarket ormid-prie,
b P
t
is thebid
prie,
a P
t
is the ask prie, and (
a
b )P
t
is the bid-ask spead. We also assume some
regularity onditionson thetemporaryprieimpatfuntion:
(Hf) (i) f isontinuouson R
(0;T℄,
(ii)f isC 1
on R
[0;T℄and x 7!
f
is boundedon R
[0;T℄.
Ausualform(seee.g. [16 ℄,[23 ℄,[2 ℄)oftemporaryprieimpatandtransationostfuntion
f,suggestedbyempirialstudiesis
f(e;) = e j
e
j
sgn(e)
a 1
e>0 +1
e=0 +
b 1
e<0
; (2.5)
with the onvention f(0;0) = 1. Here 0 <
b
< 1 <
a ,
a
b
is the bid-ask spread
parameter, > 0 is the temporary prie impat fator, and > 0 is the prie impat
exponent. In our illiquidity modelling, we fous on the ost of trading fast (that is the
temporary prie impat), and ignore as in Cetin, Jarrow and Protter [8℄ and Rogers and
Singh [24 ℄ thepermanent prieimpat of a largetrade. This lasteet ould be inluded
inour model, byassuming a jump of theprie proess at the trading date, dependingon
theorder size,see e.g. He and Mamayski [13 ℄ and LyVath,Mnif andPham [17℄.
Cashholdings. Weassumeazerorisk-freereturn,sothatthebankaountisonstant
betweentwo tradingtimes:
X
t
= X
n
;
n
t<
n+1
; n0: (2.6)
When adisretetrading Y
t
=
n+1
oursattime t=
n+1
,thisresultsina variationof
theashamount givenbyX
t := X
t X
t
= Y
t :Q(Y
t
;P
t
;
t
) dueto theilliquidity
eets. Inother words,we have
X
n+1
= X
n+1
n+1 Q(
n+1
;P
n+1
;
n+1 )
= X
n+1
n+1 P
n+1 f(
n+1
;
n+1
n
); n0: (2.7)
Notie that similarly as in the above ited papers dealing with ontinuous-time trading,
we do not assume xed transation fees to be paid at eah trading. They are pratially
insigniantwithrespetto theprieimpatandbid-askspread. We anthennotexlude
n+1
n+1 n
some n. However,notie thatunderondition(H2f), animmediatesaledoesnotinrease
the ash holdings, i.e. X
n+1
= X
n+1
= X
n
, while an immediate purhase leads to a
bankrupty,i.e. X
n+1
= 1.
Liquidation value and solveny onstraint. A key issue in portfolio liquidation is
to dene in an eonomially meaningful way what is the portfolio value of a positionon
ash and stoks. In our framework, we impose a no-short sale onstraint on the trading
strategies,i.e.
Y
t
0; 0tT;
whih isinlinewiththebankregulationfollowingthe nanialrisis,and we onsiderthe
liquidation funtionL(x;y;p;) representing the netwealth value thatan investor witha
ash amount x, would obtained byliquidating his stok position y 0 by a single blok
trade,whenthemarketprieispandgiventhetimelag fromthelasttrade. Itisdened
on RR
+ R
+
[0;T℄by
L(x;y;p;) = x+ypf( y;);
and we imposetheliquidationonstrainton tradingstrategies:
L(X
t
;Y
t
;P
t
;
t
) 0; 0tT:
We have L(x;0;p;) = x, and under ondition(H2f)(ii), we notiethat L(x;y;p;0) = x
fory 0. Wenaturally introduetheliquidationsolveny region:
S = n
(z;)=(x;y;p;)2RR
+ R
+
[0;T℄: y>0 and L(z;)>0 o
:
We denote its boundaryand its losure by
S =
y S[
L
S and
S = S[S;
where
y S =
n
(z;)=(x;y;p;)2RR
+ R
+
[0;T℄: y=0 and x=L(z;)0 o
;
L S =
n
(z;)=(x;y;p;)2RR
+ R
+
[0;T℄: L(z;)=0 o
:
We also denote byD
0
theornerlineinS:
D
0
= f0gf0gR
+
[0;T℄ =
y S\
L S:
Admissible trading strategies. Given (t;z;) 2 [0;T℄
S, we say that the impulse
ontrolstrategy =(
n
;
n )
n0
is admissible, denoted by 2 A(t;z;), if
0
=t ,
n
t, n 1,and theproessf(Z
s
;
s )=(X
s;
Y
s
;P
s
;
s
);tsTg solutionto (2.1 )-(2.2)-
(2.3)-(2.6)-(2.7), withan initialstate (Z
t
;
t
) =(z;) (andtheonventionthat (Z
t
;
t )
= (z;) if
1
> t), satises (Z
s
;
s
) 2 [0;T℄
S for all s 2 [t;T℄. As usual, to alleviate
notations,weomitted the dependeneof (Z ;)in (t;z;;), when there isno ambiguity.
th eta = 0 .1
th eta = 0 .5 y : s to c k s h a re s
y : s to c k s h a re s
y : s to c k s h a re s
y : s to c k s h a re s x:
cash
x:
cash x:
cash x:
cash D D
D D
Figure1: DomainS inthenonhathedzoneforxedp=1and evolving from1:5to 0:1.
Here
b
=0:9and f(e;)=
b exp(
e
) fore<0. Notie thatwhen goesto 0, thedomain
onverges to theopen orthant R
+ R
+ .
theta=1
x:cash amount
y: stock amount p: mid−price
Figure 2: Lower bound of the domain S for xed = 1. Here
b
= 0:9 and f(e;) =
b exp(
e
) fore<0. Notiethat whenp isxed, we obtaintheFigure1.
p=1
x:cash amount
y: stock amount theta: time−lag order
Figure 3: Lowerboundof thedomain S for xedp =1 with f(e;)=
b exp(
e
) fore<0
and
b
=0:9. Notiethat when is xed,we obtaintheFigure1.
