Hedging with physical or cash settlement under transient multiplicative price impact

Beche e , Di k; Bila e , Todo
A icle — Published Ve sion
Hedging wi h physical o cash se lemen unde ansien
mul iplica i e p ice impac
Finance and S ochas ics
P o ided in Coope a ion wi h:
Sp inge Na u e
Sugges ed Ci a ion: Beche e , Di k; Bila e , Todo (2024) : Hedging wi h physical o cash se lemen
unde ansien mul iplica i e p ice impac , Finance and S ochas ics, ISSN 1432-1122, Sp inge ,
Be lin, Heidelbe g, Vol. 28, Iss. 2, pp. 285-328,
h ps://doi.o g/10.1007/s00780-024-00531-7
This Ve sion is a ailable a :
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Finance and S ochas ics (2024) 28:285–328
h ps://doi.o g/10.1007/s00780-024-00531-7
Hedging wi h physical o cash se lemen unde ansien
mul iplica i e p ice impac
Di k Beche e 1·Todo Bila e 2
Recei ed: 12 July 2018 / Accep ed: 7 Ma ch 2023 / Published online: 15 Ma ch 2024
© The Au ho (s) 2024
Abs ac
We sol e he supe hedging p oblem o Eu opean op ions in an illiquid ex ension
o he Black–Scholes model, in which ansac ions ha e ansien p ice impac and
he cos s and s a egies o hedging a e a ec ed by physical o cash se lemen e-
qui emen s a ma u i y. Ou analysis is based on a con enien choice o educed
e ec i e coo dina es o magni udes a liquida ion o geome ic dynamic p og am-
ming. The p ice impac is ansien o e ime and mul iplica i e, ensu ing nonnega-
i i y o unde lying asse p ices while main aining an a bi age- ee model. The basic
(log-)linea example is a Black–Scholes model wi h a ela i e p ice impac p opo -
ional o he olume o sha es aded, whe e he ansience o impac on log-p ices
is modelled like in Obizhae a and Wang (J. Financ. Ma k. 16:1–32, 2013) o nomi-
nal p ices. Mo e gene ally, we allow nonlinea p ice impac and esilience unc ions.
The iscosi y solu ions desc ibing he minimal supe hedging p ice a e go e ned by
he ansien cha ac e o he p ice impac and by he physical o cash se lemen
speci ica ions. The p icing equa ions unde illiquidi y ex end no-a bi age p icing à
la Black–Scholes o comple e ma ke s in a non-pa adoxical way (c . Çe in e al. (Fi-
nance S och. 14:317–341, 2010)) e en wi hou addi ional ic ions, and can eco e
i in base cases.
Keywo ds T ansien p ice impac ·Mul iplica i e impac ·Hedging ·Op ion
se lemen ·Resilience ·Viscosi y solu ion ·Geome ic dynamical p og amming ·
E ec i e coo dina es
Ma hema ics Subjec Classifica ion 49L20 ·49L25 ·60H30 ·91G20 ·93E20
Disclaime : The opinions exp essed in his publica ion a e hose o he au ho s. They do no pu po
o e lec iews o hei ins i u ions.
D. Beche e
[email p o ec ed]
T. Bila e
bila e [email p o ec ed]
1Humbold Uni e si ä zu Be lin, Ins i u ü Ma hema ik, Un e den Linden 6, D-10099 Be lin,
Ge many
2Fac Se Resea ch Sys ems Inc., 2 S ebama S ., 1407 So ia, Bulga ia
286 D. Beche e , T. Bila e
JEL Classifica ion C61 ·G12 ·G13
1In oduc ion
By using me hods o s ochas ic a ge p oblems (see Sone and Touzi [29]) and ge-
ome ic dynamic p og amming in sui ably chosen educed e ec i e coo dina es o
magni udes a liquida ion, we sol e he supe hedging p oblem o Eu opean de i a-
i es in a ma ke model wi h mul iplica i e ansien p ice impac . I he ma ke o
he unde lying is illiquid o i la ge olumes a e o be aded, he e is p ice impac
and eedback e ec s om hedging can a ec he minimal supe hedging p ices (see
F ey [19], Schönbuche and Wilmo [28], F ey and Pol e [20], Bank and Baum [3])
and he co esponding hedging s a egies which almos su ely supe eplica e he op-
ion. Since ades a ma u i y can al e he p ice o he unde lying and he eby he
de i a i e payou , se lemen speci ica ions o he op ion (in cash o in physical uni s)
become ele an and his shows in p icing and hedging equa ions. As ou esul s ad-
d ess hedging in e ms o liquida ion alues, i.e., “ eal” ins ead o “pape ” alues
(see Ja ow [24]), we eco e such e ec s, whe eas F ey [19], F ey and Pol e [20],
Boucha d e al. [13] s udy hedging in e ms o book (“pape ”) alues. The se le-
men cons ain s imposed o hedging (in Sec . 2) in combina ion wi h s abili y by
a sui ably chosen no ion o alue, which depends con inuously on ading s a egies,
mo eo e help o a oid some known pa adoxical e ec s in p ice impac modelling
(see Bank and Baum [3], Çe in e al. [16], Beche e e al. [9, Rema k 3.3], and
c . he commen s abou di e en no ions o weal h a e (2.9) and (2.10)) and o e ly
excessi e oppo uni ies o manipula ing de i a i e payo s (as in Schönbuche and
Wilmo [28, Sec . 4.1]).
The bes -known model o ansien p ice impac is p obably he one due o
Obizhae a and Wang [26]. I s a es ha he dynamic holdings o a la ge ade
ha e addi i e linea impac (wi h pa ame e λ>0) on he p e ailing p ice so he
unde lying asse ia
(log-)p ice: ds =d¯s +λdY ,wi h
impac le el: dY =−βY d +d =: −h(Y )d +d ,(1.1)
whe e ¯sis a gi en una ec ed ( undamen al) p ice e olu ion o he unde lying, while
Yis a ma ke impac le el p ocess, whose mean- e e ing dynamics is d i en by 
and is linea in he asse holdings o he la ge ade and ansien o e ime, e-
co e ing a some esilience a e gi en by he pa ame e β>0 in he linea esilience
unc ion h.
Assuming p ice impac o be addi i e helps o ma hema ical ac abili y (in pa ic-
ula i ¯sis a ma ingale) and can se e o app oxima e mul iplica i e impac on a sho
ho izon. This is common in he li e a u e on op imal ade execu ion, as explained in
Busse i and Lillo [15, Sec . 6], who u he desc ibe [15, Sec . 5] how ansien im-
pac is calib a ed addi i ely o log-p ices, hence mul iplica i ely in p ices. See also
he compa ison in Beche e e al. [7, Example 5.5] o a gumen s in a ou o impac
o be mul iplica i e i combined wi h mul iplica i e p ice dynamics o Black–Scholes
ype. A la ge s and o li e a u e in es iga es (linea –)quad a ic con ol p oblems in
Hedging wi h ansien p ice impac 287
his ealm (see e.g. Bank e al. [4], Acke mann e al. [1]) which a e di e en om
supe hedging and i s espec i e p icing p oblem. An undesi able p ope y o he ad-
di i e impac (1.1) in his con ex is ha i can lead o nega i e p ices s o he un-
de lying asse . I is plausible ha ading a quan i y o s ocks, ha is, a ac ion o
company owne ship, should ha e a ela i e (hence mul iplica i e) e ec on he p ice.
Indeed, al eady Be simas and Lo [10, Sec . 3] ha e a gued ha ela i e (pe cen -
age) p ice impac which is p opo ional o he aded numbe o s ocks (i.e., addi i e
impac wi h espec o log-p ice in i s o de app oxima ion) is mo e plausible han
absolu e p ice impac , and hey ci e empi ical e idence.
A simple way o ob ain a mul iplica i e impac a ian is by a log-linea in e -
p e a ion o he addi i e Obizhae a–Wang model (1.1), simply by aking s=logS,
¯s=log ¯
S o be log-p ices ins ead o nominal p ices S,¯
S(a ec ed, espec i e un-
damen al). Then he impac on S=¯
Sexp(λY ) is mul iplica i e and log-linea , wi h
he esilience and he (log-)p ice impac unc ions om (1.1) being linea . This is
he basic log-linea example (see Example 2.1) which is co e ed by and mo i a es
ou ansien mul iplica i e impac model, wi h una ec ed p ice p ocess ¯
S o he
unde lying asse o Black–Scholes–Me on ype. Ou analysis mo eo e allows non-
linea and nonpa ame ic esilience and p ice impac unc ions hand in (2.1), (2.2).
The model is a mul iplica i e a ian o he (nonlinea ) addi i e impac model om
P edoiu e al. [27], whe e p ice impac can be in e p e ed in e ms o a limi o de
book shape ha is s a ic wi h espec o ela i e p ice pe u ba ions wi h Ybeing hei
olume e ec p ocess (see Beche e e al. [7, Sec . 2.1]).
The con ibu ions o he p esen pape a e h ee old:
(1) We sol e he supe hedging p oblem unde a ansien p ice impac which is
mul iplica i e, ins ead o addi i e.
(2) Ou esul s accoun o se lemen speci ica ions imposed a ma u i y which
equi e analysis in liquida ion alues ins ead o book (pape ) alues, so ha physical
uni s o he unde lying isky asse and cash ma e a ma u i y (and as well a he ini ial
ime), i.e., e minal (ini ial) p ice impac canno be ea ed as null. Following he
e minology by Boucha d e al. [13], his means ha we sol e he hedging p oblem
o non-co e ed ins ead o co e ed op ions (howe e , see Rema k 7.1 o ex ensions
o co e ed op ions unde ansien mul iplica i e p ice impac ).
(3) In his ealm, he model we s udy is basically comple e, wi h he ansien p ice
impac being he only dig ession om he ic ionless Black–Scholes model assump-
ions ( o ¯
S), and i yields non i ial ex ensions o he classical no-a bi age p icing
and hedging, while a oiding pa adoxical e ec s om illiquidi y modelling as men-
ioned in Çe in e al. [16], wi hou adding u he ic ions (like ansac ion cos s,
o cons ain s on ading s a egies o be “small”). In pa icula , he la ge ade nei-
he has he abili y o “manipula e” (see Ja ow [24], Bank and Baum [3]) he ma -
ke o achie e un easonable p o i s (see Rema k 2.4 and subsequen ema ks), no
can he sides ep liquidi y cos s en i ely and ade in e ec like a small ade by ex-
ploi ing modelling a e ac s ha occu due o a lack o sensible con inui y p ope ies
(c . Beche e e al. [9, Sec . 3]).
We o mula e he supe hedging p oblem as a s ochas ic a ge p oblem and p o e
a dynamic p og amming p inciple (DPP) along educed coo dina es o he e ec i e
p ice and impac p ocesses, which ep esen he p ice and impac le els ha would
288 D. Beche e , T. Bila e
p e ail i he la ge ade we e o unwind he (long o sho ) posi ion in he unde lying
isky asse immedia ely. Along he educed coo dina es, he DPP p o ides a way o
compa e a s opping imes he ins an aneous liquida ion weal h and he (minimal) su-
pe hedging p ice. This pe mi s cha ac e ising he supe hedging p ice as he iscosi y
solu ion o a nonlinea p icing PDE, which is a semilinea ex ension o he Black–
Scholes equa ion, wi h he non-linea i y in ol ing he (nonpa ame ic) p ice impac
and esilience unc ions as well. I he PDE has a su icien ly egula solu ion, i yields
an op imal s a egy which e en eplica es he op ion payo in he equi ed se lemen
uni s. This s a egy inco po a es he ansien na u e o impac in ha i depends on
he e ec i e le el o impac . Ou analysis is also mo i a ed by analy ical ac abili y.
I shows how e ec s om ansience o p ice impac a ise in a basically comple e
model wi hou o he addi ional ic ions om ansac ion cos s o cons ain s, wi h
scope o esul s beyond hose o he p esen pape , as ou lined in Sec . 7.
While he e is a la ge li e a u e on op imal execu ion and po olio op imisa ion
p oblems unde ansien p ice impac , mos ly o p ice impac being addi i e bu also
o mul iplica i e impac (see Obizhae a and Wang [26], Al onsi e al. [2], Busse i
and Lillo [15] o Guo and Ze os [21], Beche e e al. [8,9] and e e ences he ein),
he li e a u e on supe hedging (o pe ec hedging, i.e., eplica ion) unde p ice im-
pac , as s a ed abo e, mos ly ea s pe manen and pu ely ins an aneous p ice impac
( ansac ion cos s, possibly nonlinea ) o a combina ion o he wo (see F ey [19],
Schönbuche and Wilmo [28], Bank and Baum [3], Çe in e al. [16], F ey and
Pol e [20]), wi h he impac o en aken in mul iplica i e o m. Fo he implica ions
o op ion se lemen speci ica ions on hedging, only ew pape s allow a p ice impac
also a ma u i y. Clea ly, i equi es some ele an non-ze o p ice impac a ma u i y
o ob ain di e ences be ween se lemen speci ica ions o op ions in physical o in
cash uni s, as in Boucha d e al. [12]. Howe e , mos a icles (see [19,16,20]) ea
ano he hedging p oblem which is no posed in e ms o hedgable uni s o asse s, bu
ins ead in e ms o book ( ha is “pape ”) alue, wi h he p ice impac a ma u i y (and
possibly ini ia ion) in he analysis e ec i ely aken o be ze o. Tha ela es o a di e -
en hedging p oblem o “co e ed” op ions (see Rema k 7.1). A majo di e ence o
he wo k by Boucha d e al. [12], which o e ed a esh iew o he hedging p oblem
and inspi ed ou s, is ha he analysis in [12] is o pe manen and addi i e impac . In
con as , ou hedging esul s show non i ial e ec s om ansience o p ice impac ,
gi en ha he impac is mul iplica i e. While he basic example o [12] is he Bache-
lie model wi h addi i e impac , ou basic example is a Black–Scholes- ype model
wi h ansien mul iplica i e p ice impac (see Example 2.1 and Rema k 2.6) which
is he log-linea a ian o he model by Obizhae a and Wang [26]. Mo e de ailed
compa isons a e p o ided h oughou he pape .
The pape is o ganised as ollows. Sec ions 2and 3in oduce he model o an-
sien mul iplica i e p ice impac and o mula e he hedging p oblem. E ec i e co-
o dina es o dynamic p og amming in “liquida ion magni udes” a e explained in
Sec . 4. Sec ion 5iden i ies hedging p ices by iscosi y solu ions o semilinea PDEs
(possibly degene a e, wi h del a cons ain s), wi h echnical p oo s de e ed o he Ap-
pendix. The esul s a e illus a ed by nume ical examples in Sec . 6. Finally, Sec . 7
ex ends he esul s o combined ansien and pe manen impac , poin s ou u he
possible ex ensions o c oss-impac wi h mul iple asse s, and commen s on ela ed
esul s o he di e en hedging p oblem o co e ed op ions.

Hedging wi h ansien p ice impac 289
2 A mul iplica i e ansien p ice impac model
This sec ion desc ibes he model o his pape . An ex ension wi h addi ional pe ma-
nen impac is desc ibed in Sec . 7.Le (, F,P)be a comple e p obabili y space
wi h coun ably gene a ed F, a il a ion F=(F ) ≥0sa is ying he usual condi-
ions and an F-B ownian mo ion W. We ake semima ingales o ha e càdlàg pa hs,
R++ =(0,∞)and in ∅=+∞.
The una ec ed p ice p ocess ¯
So he unde lying isky asse e ol es, i he la ge
ade (she) is inac i e, acco ding o he s ochas ic di e en ial equa ion
d¯
S =¯
S (μ d +σdW ), ¯
S0∈R++,
wi h a cons an σ>0 and a bounded p og essi e p ocess μ. The càdlàg adap ed
p ocess deno es he e olu ion o he holdings (in uni s o sha es) in he isky asse ,
say a s ock, which is he unde lying o he de i a i e con ingen claim in he hedging
p oblem. The ma ke impac p ocess Y=Yis de ined pa hwise in he Sko ohod
space o càdlàg pa hs by
dY
=−h(Y
)d +d ,Y
0−=y∈R,(2.1)
o a esilience unc ion h:R→Rwhich is a Lipschi z-con inuous unc ion wi h
sgn(x)h(x) ≥0, as in Beche e e al. [7,9]. When he la ge ade ades dynamically
acco ding o a s a egy , he isky asse p ice obse ed on he ma ke , which is he
ma ginal p ice a which an addi ional in ini esimal quan i y could be aded, is
S := S
:= (Y
)¯
S , ≥0,(2.2)
whe e he p ice impac unc ion :R→R++ is inc easing and in C1wi h
(0)=1. In pa icula , λ:= / is a nonnega i e and locally in eg able
C0- unc ion sa is ying
(x)=exp x
0λ(u) du,x∈R.(2.3)
Example 2.1 The basic example is a ansien p opo ional p ice impac wi h una -
ec ed p ices ¯
Sgi en by geome ic B ownian mo ion, as in he Black–Scholes model
wi h μ∈Rcons an , o esilience h(y) =βy and log-p ice impac log (y) =λy
linea unc ions wi h cons an s β,λ ∈R++. Then he mul iplica i e p ice impac is
p opo ional o he numbe o sha es  = − − aded a ime , ha is, linea
in log-p ices wi h
logS +δ −log S −=λ
wi h exponen ial decay logS +δ=log(S )exp(−βδ) o e ime when he e a e no
u he ades wi hin he ime pe iod ( , +δ]. Fo such a linea choice o hand log ,
he log-asse p ices log Sunde mul iplica i e impac e ol e like nominal asse p ices
in he seminal model by Obizhae a and Wang [26] o addi i e ansien p ice impac ,
as desc ibed in (1.1).
290 D. Beche e , T. Bila e
Ou se ing also allows he esilience a e β(hence h) o be ze o, which makes he
p ice impac pe manen (c . Sec . 7) and he log-p ice impac log(S /S0−)linea in
Y −Y0−= −0−.
Nex , we speci y he la ge ade ’s p oceeds (nega i e expenses) L, which a e he
a ia ions o he cash accoun o und he dynamic holdings in he isky asse . Fo
simplici y, we assume ze o in e es and a iskless asse wi h cons an p ice 1 as cash,
i.e., p ices a e discoun ed in uni s o his nume ai e asse . Fo con inuous s a egies
o ini e a ia ion,
L() =−·
0Sd(2.4)
a e he p oceeds, and he e is a unique con inuous ex ension o he unc ional
→ L() in (2.4) o gene al (bounded) semima ingale s a egies , gi en by
L() := ·
0F(Y
)d¯
S −·
0¯
S ( h)(Y
)d −¯
SF(Y)−¯
S0F(Y
0−),(2.5)
as shown in Beche e e al. [9, Theo em 3.8], wi h he unc ion
F(x):= x
0 (u)du, x ∈R.(2.6)
Mo e p ecisely, e e y (càdlàg) semima ingale can be app oxima ed (in p obabili y)
in he Sko okhod space D([0,T])o càdlàg pa hs wi h he Sko okhod M1- opology
(c . [9, Sec . 3.1]) by a sequence o con inuous p ocesses o ini e a ia ion, and o
semima ingales nP
−→in (D([0,T]), M1)con e ging o a semima ingale ,we
hen ha e L(n)P
−→L() in (D([0,T]), M1). To de ine Lby (2.5) is hus na u al as
he con inuous ex ension o L om (2.4) o all semima ingales.
Rema k2.2 In ela ion o he abo e con inuous ex ension, we o e wo gene al com-
men s wi h ega ds o 1) li e a u e and 2) subsequen esul s on hedging, which may
be skipped a i s eading.
1) Fo o he po en ial applica ions, i seems help ul o no e ha mo e gene ally,
he e is a unique con inuous ex ension e en beyond semima ingale s a egies; see
[9, Sec . 3], and also Ho s and Ki man [22] and Acke mann e al. [1] o simila
con inui y a gumen s in di e en applica ions. Fo ou hedging p oblem in Sec . 4,
howe e , semima ingale s a egies will su ice; see e.g. in (4.3).
2) In Sec . 4, he supe hedging p oblem o De ini ion 3.2 and he supe hedging
p ice (4.4) a e going o be de ined wi h espec o a pa icula se o admissible s a e-
gies (see (4.3)). The o m o his se (which is as in Boucha d e al. [12]) plays a
echnical ole in p oo s o he geome ic dynamical p og amming p inciple (c . The-
o em 4.1). I would be na u al o ask o which ex en he pa icula choice o his se
a ec s he supe hedging p ice. We can o e wo (pa ial) answe s o his ques ions,
one o which is again ela ed o sui able con inui y p ope ies. A i s , we see ha
in base cases, he supe hedging p ices wbasically eco e impac - and ic ionless
Hedging wi h ansien p ice impac 291
Black–Scholes p ices; see Co olla y 5.12 and Rema k 3.4 and likewise in [12] (wi h
espec o he Bachelie model). This indica es ha he supe hedging p ice wde ined
la e in (4.4) is obus in he sense ha i does no appea o depend on pa icula i-
ies o he said se . To explain, secondly, why such a obus ness holds o he almos
su e supe hedging p oblem, an almos su e uni o m app oxima ion esul (in e ms
o physical asse and cash holdings) o mo e gene al ading s a egies by a sui able
se o mo e elemen a y ones would be desi able in p inciple. P oposi ion 3.12 in [9]
con ibu es such a esul o he se o con inuous ini e- a ia ion s a egies; bu his
does no qui e i wi h he se up o he hedging p oblem in Sec . 4, as he espec i e
se (4.3) he e is di e en .
The p oceeds om a block ade o selling  sha es a ime a e
−¯
S 
0 (Y
−+x)dx, (2.7)
showing ha he p ice pe sha e ha he la ge ade pays ( esp. ob ains) o a block
buy ( esp. sell) o de is be ween he p ice (Y
−)¯
S be o e he ade and he p ice
(Y
)¯
S a e he ade. The o m o p oceeds and p ice impac om block ades can
be in e p e ed om he pe spec i e o a la en limi o de book, whe e a block ade
is execu ed agains a ailable o de s in he o de book o p ices be ween (Y
−)¯
S
and (Y
−+ )¯
S , see Beche e e al. [7, Sec . 2.1], and Ycan be unde s ood as a
olume e ec p ocess in he spi i o P edoiu e al. [27].
Fo a sel - inancing s a egy (B, ) in which he dynamic holdings in cash ( he
iskless asse , sa ings accoun ) and in he s ock ( he isky asse ) e ol e as Band ,
he sel - inancing condi ion is
B=B0−+L().
In o de o de ine a weal h dynamics o he la ge ade ’s s a egy, i emains o spec-
i y he alue o he isky asse posi ion in he po olio in a sui able way. I he la ge
ade is o ced o liquida e he posi ion o  s ocks immedia ely by a hypo he ical
single block ade a ma ke p ices, he liquida ion weal h Vliq
=Vliq
() a ime
≥0 (be o e which he ma ke impac is a Y
)is
Vliq
() := B +¯
S 
0 (Y
−x)dx
=B0−+L() +¯
S 
0 (Y
−x)dx. (2.8)
This weal h p ocess is ma hema ically con enien ly ac able, e ol ing con inuously
wi h
dVliq
=F(Y
−)−F(Y
−− −)d¯
S −¯
S  (Y
−)− (Y
−− −)h(Y )d (2.9)
and Vliq
0=B0−, and i inhe i s om he p oceeds (2.5) he con inuous dependence
p ope ies (on ) men ioned abo e. The no ion o liquida ion weal h Vliq() is el-
e an o he hedging applica ion o Sec . 3, and is di e en om he so-called book
292 D. Beche e , T. Bila e
weal h p ocess
Vbook() := B+S =B0−+L() +S, (2.10)
in which isky asse s a e e alua ed a he cu en ma ginal ma ke p ice S. Because
o p ice impac (mono onici y o , posi i i y o ,¯
S,S), clea ly Vliq
≤Vbook
.In
he e minology o Ja ow [24, Sec . IV], Vliq is eal weal h whe eas Vbook is pape
weal h. Recen ly, Kolm and Webs e [25] ha e gi en heo e ical and p ac ical easons
why accoun ing o he alue ( espec i ely he P&L, i.e., he changes in alue) o a
isky asse posi ion based on cu en ma ke p ices Sas in (2.10) can be misleading
and needs o be adjus ed o p ice impac ; in hei e minology, Vliq co esponds o
undamen al weal h whe eas Vbook is accoun ing weal h, also e e ed o as ma k- o-
ma ke weal h.
F om (2.9), we ob ain absence o a bi age wi hin he se o admissible s a egies
ANA := {( ) ≥0:is a bounded semima ingale wi h 0−=0
and  =0on ∈[T,∞) o some T∈(0,∞)}.
P oposi ion2.3 The ma ke is ee o a bi age up o any ini e ho izon T∈(0,∞)in
he sense ha he e exis s no ∈ANA wi h  =0on ∈[T,∞)such ha o he
sel - inancing s a egy (B, ) wi h Vliq
0−:= B0−≤0, we ha e P[Vliq
T≥0]=1and
P[Vliq
T>0]>0. Mo eo e , o any such (B, ), he e exis s a p obabili y measu e
Qequi alen o P(on FT)such ha Vliq is a Q-ma ingale.
In he e minology o [24, Sec . IV, Eq. (13)], he no-a bi age esul o P opo-
si ion 2.3 s a es ha he e exis no ma ke manipula ion ading s a egies.No e
ha in con as , he e is no eason o expec a no-a bi age esul in e ms o book
weal h Vbook; he e a e simple coun e examples, see Example 2.5 o implica ions
on (supe -)hedging p ices.
Rema k 2.4 In he seminal a icle by Hube man and S anzl [23], a no ion o no p o -
i able ound- ips (s onge han no-a bi age) is de ined, which (in ou no a ion) e-
qui es ha he e exis s no (sel - inancing) s a egy gi en by (B0−,)as in P oposi-
ion 2.3 wi h Vliq
0=0 and E[Vliq
T]>0. This means ha he e is no such s a egy
om ze o ini ial holdings (wi h Vliq
0=0) ha achie es a e minal liquida ion weal h
which is posi i e in expec a ion, E[Vliq
T]>0, wi hin a compac ime in e al [0,T].
By de ini ion, he liquida ion weal h Vliq is he alue o a cash-only posi ion held
a e all s ock holdings a e liquida ed.
A much ci ed esul om [23] s a es ha p ice impac needs o be linea o ex-
clude p o i able ound- ips. This is no in con lic wi h ou modelling, as he p oo
in [23] elies o cou se on some assump ions. These include pe manen and addi i e
impac . Fo compa ison, unde mul iplica i e pe manen p ice impac , a linea log-
p ice impac unc ion log is su icien o conclude ha Vliq is a ma ingale unde
P(by (3.2) in Rema k 3.4)i ¯
Sis a P-ma ingale (e.g. geome ic B ownian mo-
ion, like in he Black–Scholes model unde he isk-neu al measu e). This implies
E[Vliq
T]=E[Vliq
0−], excluding p o i able ound- ips.
Hedging wi h ansien p ice impac 299
s udying he p oblem in sui ably educed coo dina es which can be in e p e ed as
quan i ies ( o p ices and impac y) a liquida ion (o θ), and wi h espec o which a
DPP and a iscosi y cha ac e isa ion a e p o ed o he unc ion w. While his idea is
o iginal and helps o make he analysis mo e anspa en , in o he aspec s we can and
do adap echniques om Boucha d e al. [12].
To de i e a dynamic p og amming p inciple o he unc ion w, we wan o com-
pa e i (e alua ed a sui able coo dina e p ocesses) o e ime wi h he weal h p ocess.
Since by de ini ion, wassumes ze o ini ial isky asse holdings, i is na u al o con-
side he ( ic i ious) s a e p ocesses ha would p e ail i he ade we e o ced o liq-
uida e he posi ion in he isky asse immedia ely (wi h a block ade). To his end, le
S(S ,Y
,
):= ¯
S (Y
− )=S (Y
− )/ (Y 
),
Y(Y
,
):=Y
− .(4.6)
The p ocess S(s,y,θ) is in e p e ed as he p ice o he asse ha would p e ail
a e θasse s we e liquida ed, when sand ya e he p ice o he isky asse and
he ma ke impac jus be o e he ade, while Y(y, θ) would be he le el o he
ma ke impac a e his ade. In his sense, we e e o he p ocesses S(S, Y ,)
and Y(Y,) as he e ec i e p ice and impac p ocesses, espec i ely, o a
sel - inancing ading s a egy . Obse e ha bo h p ocesses a e con inuous e en
hough he ading s a egy mayha ejumps.
Fo he dynamic p og amming p inciple in Theo em 4.1, we compa e he liqui-
da ion weal h Vliq de ined in (2.8) wi h he alue unc ion walong e olu ions o he
e ec i e p ice and e ec i e impac p ocesses (S(S, Y ,),Y(Y,)).
Theo em 4.1 Fo he geome ic DPP,we ix ( ,s,y, )∈[0,T]×R++ ×R×R.
(i) I >w( ,s,y), hen he e exis γ∈and θ∈Ksuch ha
Vliq, ,z,γ
τ≥wτ,S(S ,z,γ
τ,Y ,z,γ
τ,
,z,γ
τ), Y ,z,γ
τ− ,z,γ
τ
o all s opping imes τ≥ ,whe e z=(S(s, y, −θ),y +θ,θ, ).
(ii) Le k≥1. I <w
2k+2( ,s,y), hen o e e y γ∈k,θ∈K∩[−k,k]and
s opping ime τ≥ ,we ha e wi h z=(S(s, y, −θ),y +θ,θ, ) ha
PVliq, ,z,γ
τ>w
kτ,S(S ,z,γ
τ,Y ,z,γ
τ,
,z,γ
τ), Y ,z,γ
τ− ,z,γ
τ<1.
P oo The e a e simila i ies and di e ences o Boucha d e al. [12, p oo o P oposi-
ion 3.3] who ea he case o pe manen addi i e impac ; so we p esen he p oo in
ull de ail. As explained in Rema k 3.4, he assump ions in [12] do no allow co e -
ing mul iplica i e p ice impac , and ansience o impac na u ally equi es a u he
dimension in he DPP. The p oo uses gene al ideas on dynamic p og amming o
s ochas ic a ge p oblems and geome ic lows; see Sone and Touzi [29]. We em-
phasise ha o showing he DPP, ou p oo de elops ma hema ical a gumen s in
e ms o e ec i e coo dina es and liquida ion weal h Vliq, which simpli ies he ma h-
ema ical analysis and makes i mo e anspa en . This also shows up in he possible
ex ensions desc ibed in Sec . 7.

300 D. Beche e , T. Bila e
I is easy o see ha o k≥2 and ( ,s,y,θ)∈[0,T]×R++×R×(K∩[−k, k]),
¯wk( ,s,y,θ)≥wk+1 ,S(s,y,θ),Y(y, θ),(4.7)
wk−1 ,S(s,y,θ),Y(y, θ)≥¯wk( ,s,y,θ). (4.8)
Now suppose ha >w( ,s,y). Then by he de ini ion o w, he e exis θ∈Kand
some γ∈G( , z) o z=(S(s, y, −θ),y+θ,θ, ).Asin[29, p oo o Theo em 3.1,
S ep 1], we ha e o all s opping imes τ≥ ( he i s pa o ) he DPP o ¯w,
namely Vliq, ,z,γ
τ≥¯w(τ, S ,z,γ
τ,Y ,z,γ
τ,
,z,γ
τ). Then (i) ollows om (4.7) by aking
k→∞.
To p o e (ii), le <w
2k+2( ,s,y) and suppose ha he e exis γ∈k,some
∈K∩[−k,k]and a s opping ime τ≥ such ha
Vliq, ,z,γ
τ>w
kτ,S(S ,z,γ
τ,Y ,z,γ
τ,
,z,γ
τ), Y ,z,γ
τ− ,z,γ
τ
o z=(S(s, y, −θ),y +θ,θ, ). Then by (4.8), we ob ain
Vliq, ,z,γ
τ>¯wk+1(S ,z,γ
τ,Y ,z,γ
τ,
,z,γ
τ),
and hus we ge ≥¯w2k+1( , S(s, y, −θ),y +θ,θ) by [29, p oo o Theo em 3.1,
S ep 2]. In pa icula , we conclude om (4.7) ha ≥w2k+2( ,s,y), which is a
con adic ion. 
Rema k4.2 Pa (ii) o he heo em is s a ed in e ms o wkins ead o wbecause o a
measu able selec ion a gumen employed in he p oo ; c . [12, Rema k 3.2].
To de i e he p icing PDE om he dynamic p og amming p inciple in Theo-
em 4.1, we need he dynamics o he con inuous p ocesses
→ Vliq
−ϕ ,S(S ,Y
,
), Y(Y
,
)(4.9)
o su icien ly smoo h unc ions ϕ:[0,T]×R++ ×R,( ,s,y) → ϕ( ,s,y), ha
will la e se e as es unc ions when cha ac e ising alue unc ions by iscosi y
solu ions.
Lemma 4.3 Fo e e y γ=(a,b,ν) ∈and e e y ϕ∈C1,2,1([0,T]×R++ ×R),
we ha e, o =γ,
dVliq
−ϕ( ,S ,Y )
=S F(Y + )−F(Y )
(Y )−ϕs(μ −λ(Y )h(Y + )d +σdW 
+−ϕ −σ2S2
ϕss/2+h(Y + )ϕy+F(S ,Y ,
)d
wi h
F(s,y,θ)=sh(y+θ)λ(y)F(y +θ)−F(y)
(y) − (y+θ)− (y)
(y) ,
Hedging wi h ansien p ice impac 301
whe e S =S(S ,Y
,
),Y =Y(Y
,
)and he de i a i es o ϕa e e alua ed
a ( , S ,Y ).
P oo Since S =S(S ,Y
,
)equals ¯
S (Y
− ), he p oduc ule and
=λ imply
dS =S μ −λ(Y
− )h(Y
)d +σdW .(4.10)
By I ô’s o mula, we ob ain
dϕ( ,S ,Y
− )
=ϕ d +ϕsdS +ϕyd(Y
− )+ϕss/2d[S]
=ϕ −λ(Y
− )h(Y
)S ϕs−h(Y
)ϕy+σ2S2
ϕss/2d
+μ S ϕsd +σS ϕsdW .(4.11)
Wi h e e ence o (2.9), we ha e
dVliq
=−h(Y
)S
(Y
)− (Y
− )
(Y
− )d +μ S
F(Y
)−F(Y
− )
(Y
− )d
+σS
F(Y
)−F(Y
− )
(Y
− )dW .(4.12)
Combining (4.11) and (4.12) and ea anging e ms comple es he p oo . 
Rema k 4.4 Conside he case when λis cons an , i.e., (x) =exp(λx). Then we
ha e F≡0 and he dynamics o Vliq can be s a ed in a su p isingly simple o m,
namely
dVliq
=F( )dS ,
whe e S =S(S ,Y
,
)has he dynamics (4.10). As a consequence, he su-
pe hedging p ice (o he la ge in es o ) o an op ion wi h ma u i y Tand pu e cash
se lemen H(ST)is a leas he small in es o ’s p ice o H, in he absence o he
la ge ade , when he p ice p ocess is ¯
Sins ead. Indeed, o each (bounded) supe -
hedging s a egy (o he la ge in es o ) wi h ini ial capi al , he e exis s P≈P
(on FT) such ha S=S0−E(σ 
W)unde P o a P-B ownian mo ion 
W. Hence
Vliq() is a P-ma ingale and hus ≥EP[H(ST)]=EP[H(ST)]( ecall ha
T=0, implying ST=ST). On he o he hand, a Feynman–Kac a gumen shows
ha EP[H(ST)]is jus he classical Black–Scholes p ice o a small in es o in a
ic ionless ma ke wi h isky asse p ocess ¯
S.Aswas an a bi a y supe hedging
s a egy wi h ini ial capi al , aking he in imum yields he claim.
The abo e obse a ion shows a no able di e ence o he model in Bank and
Baum [3, Theo em 5.3], whe e he p ice o he la ge in es o is ypically smalle .
This is mainly due o a di e en speci ica ion o supe hedging s a egies wi h less
302 D. Beche e , T. Bila e
s ingen se lemen cons ain s, acco ding o which a la ge ade may be able o e-
duce a ma u i y he payo o he op ion o a la ge ex en by exploi ing he p ice
impac on he unde lying a ma u i y. In o he wo ds, she can a y a ma u i y he
isky asse posi ion in o de o minimise he payo wi h ewe cons ain s, and im-
media ely a e wa ds can unwind any esidual isky asse posi ion a no addi ional
cos (by he absence o a bid–ask sp ead). In con as , ou se up is mo e es ic i e
by imposing as se lemen cons ain on he s a egies ha hey ha e o eplica e he
physical deli e y pa exac ly, i.e., a e se lemen , he hedging s a egy has o hold a
nonnega i e cash posi ion wi hou esidual holdings in he isky asse .
We no e ha an a gumen as abo e does no apply in he gene al case wi h non-
cons an λ o ou p ice impac model. In ac , he examples in Sec . 6also e eal
si ua ions whe e supe hedging is cheape o he la ge ade ; c . Example 6.1.
5 The p icing PDEs and main esul s
Nex , we de e mine he e minal alue o he unc ion wa ma u i y da e T ha will
se e as a bounda y condi ion o he p icing PDE. Recall ha Kis he (cons ain )
se in which ading s a egies ake alues and se Kn=K∩[−n, n] o n∈N.
Lemma 5.1 Fo he PDE bounda y condi ions, o n∈N,le
Hn(s, y) := in g0s (y+θ)
(y) ,y+θ+sF(y +θ)−F(y)
(y) :
θ∈Kn,θ =g1s (y+θ)
(y) ,y+θ.
Then we ha e wn(T, ·)=Hn(·)and w(T, ·)=H(·),whe e he unc ion His
gi en by
H:= in
n≥0Hn.(5.1)
P oo A he ma u i y ime T, he hedge o he op ion mus do a block ade o size θ
in o de o mee he physical-deli e y pa speci ied by g1, he eby mo ing he p ice
o he unde lying om s o s (y+θ)
(y) and he impac le el om y o y+θ. Such a
block ade incu s cos s o size sF(y+θ)−F(y)
(y) , and hence i supe eplica es he payo
(g0,g
1)i he hedge can co e hese cos s and he equi ed cash-deli e y pa , which
a e he block ade is g0(s (y+θ)
(y) ,y+θ).
Rema k 5.2 No e ha H(s,y) =+∞i he equa ion θ=g1(s (y+θ)
(y) ,y +θ) does
no ha e a solu ion θin K.
Hedging wi h ansien p ice impac 303
As we do no know a his poin whe he he alue unc ion wis con inuous, we
need o wo k wi h discon inuous iscosi y solu ions and hence conside he elaxed
semilimi s
w∗( ,s,y):= lim in
( ,s,y,k)→( ,s,y,∞)wk( ,s,y), (5.2)
w∗( ,s,y):= lim sup
( ,s,y,k)→( ,s,y,∞)
wk( ,s,y), (5.3)
whe e he limi s a e aken o e <T. Recall ha wis a (discon inuous) iscos-
i y solu ion (o ou p icing equa ions, see Sec s. 5.1 and 5.2)i w∗( esp. w∗)isa
supe solu ion ( esp. subsolu ion). Fo p o ing he iscosi y p ope y, we make he
ollowing assump ion.
Assump ion 5.3 The unc ions w∗and w∗a e bounded on [0,T]×R++×R, and he
payo unc ion H om (5.1) is egula in he sense ha i is con inuous, bounded,
and he mono one con e gence Hn↓Hholds uni o mly on compac s.
In pa icula , Assump ion 5.3 implies ha w(T, ·)is ini e. This means ha he
payo is well beha ed in e ms o he physical-deli e y pa , i.e., i he ade was
supposed o ul il he obliga ion om selling he op ion immedia ely, she would be
able o do so in any si ua ion (in any s a e (s, y)) wi h an admissible ade, p o ided
ha she has enough capi al.
5.1 Case s udy o a gene al bounded p ice impac unc ion
In his sec ion, he ollowing assump ion is supposed o hold.
Assump ion 5.4 The esilience unc ion his Lipschi z and bounded; he p ice impac
unc ion is bounded away om 0 and ∞, i.e., in R >0 and supR <∞;λ
is bounded and con inuously di e en iable wi h bounded de i a i e; and K=R(no
del a cons ain s).
Unde Assump ion 5.4, he an ide i a i e F om (2.6) and i s in e se F−1a e
bijec ions R→Rand Lipschi z-con inuous wi h Lipschi z cons an s supR <∞
and 1/in R , espec i ely.
To de i e he p icing PDE jus o mally (a i s , o be jus i ied la e ) in his case,
le ( ,s,y)∈[0,T)×R++×Rand o mally apply pa (i) o he DPP in Theo em 4.1
o =w( ,s, y) (assuming ha he in imum in he de ini ion o wis a ained) and
τ= +, oge he wi h Lemma 4.3 o ϕ=w, assuming ha wis smoo h enough.
Thus we ge he exis ence o θ∗such ha
0≤sF(y +θ∗)−F(y)
(y) −ws( ,s,y)
μ −λ(y)h(y +θ∗)d +σdW 
+−w ( ,s,y)−σ2s2wss( ,s,y)/2
+h(y +θ∗)wy( ,s,y)+F(s,y,θ∗)d .
304 D. Beche e , T. Bila e
S ill a guing a a o mal le el, his canno hold unless
F(y +θ∗)= (y)w
s( ,s,y)+F(y),
−w ( ,s,y)−σ2s2wss( ,s,y)/2+h(y +θ∗)wy( ,s,y)+F(s,y,θ∗)≥0.(5.4)
In pa icula , θ∗=θ∗( ,y,s)=F−1( (y)ws( ,s,y)+F(y))−y. The second pa
o he DPP in Theo em 4.1 will ac ually gi e ha he d i e m mus be 0, i.e., we
should ha e equali y in (5.4). This o mally mo i a es ha he o m o he p icing
PDE o wshould be
0=−w −1
2σ2s2wss +˜
h( , s, y)wy+sλ(y)ws
+s˜
h( , s, y)1−˜
( ,s,y)
(y) ,(5.5)
whe e o ( ,s,y)∈[0,T)×R++ ×R,wese
˜
h( , s, y) := h◦F−1 (y)w
s( ,s,y)+F(y)
,
˜
( ,s,y):= ◦F−1 (y)w
s( ,s,y)+F(y)
.
Obse e ha he PDE is semilinea and degene a e (since i does no con ain second-
o de de i a i es in ol ing he y- a iable). Ou main esul is as ollows.
Theo em 5.5 Unde Assump ions 5.3 and 5.4, he alue unc ion wo he supe hedg-
ing p oblem is con inuous and is he unique bounded iscosi y solu ion o (5.5)wi h
he bounda y condi ion w(T, ·)=H(·),whe e His de ined in (5.1).
P oo The iscosi y p ope y, i.e., ha w∗( espec i ely w∗) is a iscosi y supe so-
lu ion ( esp. subsolu ion), ollows by he dynamic p og amming p inciple in Theo-
em 4.1 oge he wi h Lemma 4.3. The key a gumen s a e p esen ed in he Appendix
in de ail o he case whe e λis cons an , which ac ually leads o a sligh ly mo e
in ol ed p icing PDE (5.11) (including g adien cons ain s) equi ing addi ional
jus i ica ions.
The compa ison esul o Theo em A.5 p o es uniqueness and con inui y; c . Re-
ma k A.7.
Le us conclude his sec ion by commen ing on some consequences om The-
o em 5.5 o he supe hedging p ice and he exis ence o a co esponding hedging
s a egy. A nume ical example is p esen ed in Sec . 6.
Rema k 5.6 Like in he classical case o liquid ma ke s (wi hou p ice impac ), he
supe hedging p ice does no depend on he d i in he unpe u bed p ice p ocess. This
may be seen mo e di ec ly by wo king unde he equi alen ma ingale measu e o ¯
S
om he beginning. On he o he hand, he supe hedging p ice depends non i ially
on he ini ial le el o impac yand he esilience unc ion h, and can do so e en
o op ion payo s o he o m (g0(s), 0), i.e., payo s no depending on he le el o
impac . So i u ns ou ha o p icing and hedging (c . Rema k 5.8), he de ia ion o
he ma ke p ice om he ‘una ec ed’ alue, de e mined by he impac le el y,isa
ele an s a e a iable.

Hedging wi h ansien p ice impac 305
Rema k 5.7 Obse e ha o only pe manen impac , i.e., h≡0, (5.5) simpli ies o
he classical ( ic ionless) Black–Scholes p icing equa ion. Hence he supe hedging
p ice o he la ge ade hen equals he Black–Scholes p ice o he op ion wi h
payo H.
Rema k 5.8 Unde su icien egula i y, i u ns ou ha a s a egy can be cons uc ed
ha pe ec ly eplica es he op ion payou om he (minimal) supe hedging p ice.
This means ha we ha e dynamic hedging in he sense o eplica ion, like in he
ic ionless comple e Black–Scholes model.
To his end, suppose ha a unc ion w∈C1,3,1
b([0,T]×R++×R)sol es he p ic-
ing PDE (5.5) wi h he bounda y condi ion w(T, ·)=H(·). Then o any ε>0,
a supe hedging s a egy wi h an ini ial cos o w(0,s,y)+εcan be cons uc ed as
ollows. Conside he sel - inancing s a egy (B, ) wi h B0−=w(0,s,y) +ε,
0=F−1( (y)ws(0,s,y) +F(y)) −y, meaning ha a block ade o size
0=0is pe o med a ime 0, and
 =F−1 (Y
)ws ,S(S ,Y
,
), Y
+F(Y
)−Y
o ∈[0,T), (5.6)
T=0,i.e., T=−T−,(5.7)
whe e Y=Y−. Then by Lemma 4.3 oge he wi h (5.6) and (5.5), we conclude
ha
ε=Vliq
0() −w(0,s,y)
=Vliq
T() −wT,S(ST,Y
T,
T), Y
T
=Vliq
T() −HS(ST,Y
T,
T), Y
T
=Vliq
T() −H(ST,Y
T),
T=0,
whe e he las line ollows om (5.7). By he de ini ion o H,ha ingH+εin cash a
ime Tis enough o supe eplica e he Eu opean claim wi h payo (g0,g
1)by doing
a possible addi ional inal block ade o size ε. No e ha such a block ade does
no a ec Vliq
T. Hence he s a egy +1{T}εis supe eplica ing o he Eu opean
claim. No e ha one can ake ε=0 i he cons uc ed s a egy is bounded and he
in imum in he de ini ion o Hnis a ained (c . Lemma 5.1), i.e., we ge a eplica ing
s a egy in his case.
An applica ion o I ô’s o mula gi es ha a s a egy sa is ying he ixed-poin
p oblem (5.6) can be ob ained, unde sui able egula i y, by sol ing he sys em
306 D. Beche e , T. Bila e
o SDEs
dS =S μ −λ(Y
)h(Y
+ )d +σdW ,
d =a( ,S ,Y
,
)d +b( , S ,Y
)dW ,
dY
=−h(Y
+ )d , (5.8)
wi h ini ial condi ions S0=s,Y
0=yand 0=F−1( (y)ws(0,s,y)+F(y))−y,
whe e
a( ,s,y,θ) := h(y +θ)1−λ ws− −wsy −λswss
(F−1( ws+F)) 
+w s +sμ wss +1
2σ2s2wsss
(F−1( ws+F)) ,
b( , s, y) := σswss
(F−1( ws+F)),
and whe e we w i e = (y),λ=λ(y), e c., when a gumen s o unc ions ha e no
been speci ied, o ease he no a ion. Thus an op imal (i.e., cheapes ) supe hedging
s a egy accoun s o he ansien na u e o p ice impac , which shows up by he
p esence o he esilience unc ion ho he impac in he o mulas abo e.
Rema k5.9 To desc ibe how eplica ing hedging s a egies in ou model a e desc ibed
by coupled o wa d–backwa d SDEs, suppose ha is a eplica ing s a egy o an
op ion wi h cash-equi alen payo Hand le (Y,S)be he e ec i e impac and
p ice p ocesses. By a change o measu e a gumen , we can assume wi hou loss o
gene ali y ha μ=0. Se ing Z := σS F(
Y + )−F(Y )
(Y ),gi ing
 =F−1(σ−1S−1
(Y )Z +F(Y )) −Y ,
and using (4.12) leads o he coupled FBSDE
dY =−(h ◦F−1)σ−1S−1
(Y )Z +F(Y )d ,
dS =S −λ(Y )(h ◦F−1)σ−1S−1
(Y )Z +F(Y )d +σdW ,
dVliq
=g(Y ,S ,Z
)d +Z dW ,V
liq
T=H(ST,YT),
whe e he d i e g:R×R++ ×R→Ro he FBSDE is gi en by
g(y,s,z)=−s(h ◦F−1)σ−1s−1 (y)z+F(y)

×( ◦F−1)(σ−1s−1 (y)z+F(y))− (y)
(y) .
Example5.10 As ins uc i e example, conside an op ion wi h ma u i y T>0 whose
payou a ma u i y is he spo p ice o he asse , i.e., H(s,y) =s. In he ic ion-
less Black–Scholes model, i s a bi age- ee p ice is BS(s) =sand a (minimal, i.e.,
Hedging wi h ansien p ice impac 307
cheapes ) eplica ing s a egy is o buy one sha e a ini ia ion and hold i un il ma-
u i y, whe e i is liquida ed a he spo p ice. Fo he solu ion in ou p ice impac
model, le us conside he classical solu ion o (5.5) wi h he bounda y condi ion H
gi en by he unc ion
w( ,s, y) =F(y +c( , y)) −F(y)
(y) s, (5.9)
whe e c:[0,T]×R→Ris a solu ion o he backwa d anspo equa ion
−c +h(y +c)cy=0on[0,T)×R,
c(T, y) =F−1 (y)+F(y)
−yon R.
In pa icula , by he dynamics o c, i holds o any s a egy  ha
c( , Y
)=c(0,Y
0) o ∈[0,T], whe e Yis he e ec i e impac p ocess
co esponding o . In pa icula , by (5.6), a minimal eplica ing s a egy sa is ies
on [0,T) he equa ion
∗
=c( , Y∗
)=c(0,Y∗
0)=c(0,Y
0−).
Hence a buy-and-hold s a egy is also op imal o he la ge ade . We can obse e
he ollowing:
1) Pu ely pe manen impac (h≡0) yields he Black–Scholes p ice w( , s,y) =s
and he buy-and-hold s a egy wi h
c(0,y)=c(T, y) =F−1 (y)+F(y)
−y
sha es, which does no depend on he ma u i y T.
2) In compa ison, i he p ice impac is no pe manen bu ansien (h≡ 0), he
p ice (5.9) depends non i ially on he ma u i y T, in addi ion o he p ice impac and
esilience unc ions and h, espec i ely.
3) The la ge ade ’s p ice w( , s, y) domina es he Black–Scholes p ice BS( , s)
(which is equal o sin his example) i and only i c( , y) > c(T, y). Mo eo e ,
he e a e si ua ions whe e his condi ion holds and si ua ions whe e i is iola ed. The
in ui i e eason is ha he e a e wo coun e balancing e ec s: a ini ia ion, whe e he
la ge ade buys sha es o se up he ini ial del a hedge, he eby mo ing p ices in
an un a ou able di ec ion, and a ma u i y, when she liquida es he del a and mo es
p ices in a di ec ion a ou able o he . Which o hese wo e ec s domina es o e all
depends in a non i ial way on he le el o liquidi y a ini ia ion and a ma u i y, and
on he se lemen speci ica ions o he op ion; see he discussion in Example 6.1.
Le us commen he e on Assump ion 5.4 which implies bijec i i y o Fon R.
Obse e ha he in e se F−1is used o desc ibe he op imal con ol θ∗. Simila con-
di ions a e also c ucial o he esul s in Bank and Baum [3] and Boucha d e al. [12];
see he su jec i i y assump ion (A5) in [3] and he in e ibili y assump ion (H2) in
[12]. The nex sec ion shows how depa ing om his assump ion leads na u ally o
singula i ies in he p icing PDE wi h espec o he g adien . Indeed, he lack o in-
e ibili y o F equi es condi ions on wsso ha θ∗can be de i ed. The e o e, he
analysis he e will in ol e cons ain s on he ‘del a’, i.e., on he holdings in he isky
asse , which in PDE e ms ansla es o cons ain s on he spa ial g adien ws.
308 D. Beche e , T. Bila e
5.2 Case s udy o p ice impac o exponen ial o m
We ex end he analysis o a na u al case whe e he an ide i a i e o he p ice impac
unc ion is no assumed o be su jec i e. To his end, he p ice impac unc ion is aken
o be o exponen ial o m (x) =exp(λx) wi h λa cons an (i.e., log is linea ),
meaning ha he ela i e ma ginal p ice impac unc ion λ= / > 0 is cons an .
A dis inc i e ea u e o his case is ha a any ime , knowing he (ma ginal) s ock
p ice S is su icien o de e mine he impac om an ins an block ade, since a e
a block ade o size , he p ice is ¯
S (Y
+) =S exp(λ). Hence he ela i e
displacemen (Y)o S om he undamen al p ice ¯
Sis imma e ial o de e mine
he p ice impac om a block ade, in con as o he si ua ion o Sec . 5.1. Mo i a ed
by Rema k 3.4, we impose sho -selling cons ain s by equi ing ading s a egies o
e ol e in K=[−K,∞) o some K>0.
To de i e (only heu is ically a i s , we jus i y i igo ously la e ) he p icing PDE,
le us apply o mally Theo em 4.1 o =w( , s, y) a ,s,y,τ= +, p o ided ha
wis smoo h enough, o ge he exis ence o θ∗∈Ksuch ha using Lemma 4.3,we
ha e
Lθ∗w( ,s, y) d −sws( ,s,y)−eλθ∗/λ +1/λ(σ dW +η d ) ≥0,(5.10)
whe e η =μ −λh(y +θ∗)and
Lθ∗w( ,s, y) := −w ( ,s,y)+h(y +θ∗)wy( ,s,y)−1
2σ2s2wss( ,s,y).
As in Sec . 5.1, he di usion pa in (5.10) should anish, gi ing he op imal con ol
θ∗=1
λlogλws( ,s,y)+1,
and om he d i pa , we iden i y he p icing PDE Lθ∗w( , s, y) =0. The cons ain
θ∗∈Kis now equi alen o HKw( , s, y) ≥0, whe e o a smoo h unc ion ϕ,wese
HKϕ( ,s,y) := λϕs( ,s,y)+1−e−λK .
Thus we conclude o mally ha wshould be a solu ion o he a ia ional inequali y
FK[w]:=min{Lθ[w]w,HKw}=0on[0,T)×R++ ×R,(5.11)
whe e
θ[w]( ,s,y):= 1
λlogλws( ,s,y)+1.(5.12)
As usual, he g adien cons ain s p opaga e o he bounda y, meaning ha he bound-
a y condi ion o (5.11) should be
min{w(T, ·)−H,HKw}=0.(5.13)
A e his mo i a ion, we s a e he main esul o he exponen ial p ice impac
unc ion (x)=exp(λx).
Hedging wi h ansien p ice impac 315
o addi i e pe manen p ice impac , as shown in de ail in Beche e and Bila e [6,
Sec . 8].
In con as o he p oblem s udied in he main body o he p esen pape and in
Boucha d e al. [12] o non-co e ed op ions, he s ochas ic a ge p oblem o co -
e ed op ions is e y di e en in ha he e is no p ice impac a incep ion and a
ma u i y in he hedging p oblem o co e ed op ions. The eason (see [13]) is ha
he buye o a co e ed op ion has o p o ide (upon eques and a he disc e ion o
he hedge ) he equi ed ini ial (del a) hedging posi ion as a pa o he op ion p e-
mium, and accep s any mix o cash and s ocks (a a sui able book alue i e alua ed
a cu en ma ginal ma ke p ices S) as an op ion se lemen . In his way, he hedge is
no exposed o ini ial and e minal impac o mee ing se lemen speci ica ions when
o ming and unwinding he hedging posi ion o co e ed op ions. We men ion ha
simila assump ions a e made in he li e a u e by F ey [19], F ey and Pol e [20], Çe in
e al. [16], whe e he analysis is in e ms o book alue ins ead o liquida ion alue;
see also Bank and Baum [3] and Boucha d e al. [12].
In he p e ious sec ions, he supe hedging p ice o (non-co e ed) op ions unde
ansien mul iplica i e p ice impac was cha ac e ised by a degene a e semilinea
PDE, whose non-linea i y in ol es he esilience unc ion hand he p ice impac
unc ion . I can in ol e g adien cons ain s (i.e., del a cons ain s), educing o he
Black–Scholes equa ion wi h g adien cons ain s in he si ua ion o Co olla y 5.12.
In con as , o co e ed op ions, he co esponding p icing equa ion u ns ou o
be ully nonlinea and singula in he second-o de e m. This induces gamma con-
s ain s, whe eas o non-co e ed op ions, a singula i y a ises in he i s -o de de i a-
i e and induces del a cons ain s; see Sec . 5.2. Fo co e ed op ions, i can be shown
(see Beche e and Bila e [6, Sec . 8]) ha he esilience o he p ice impac is im-
ma e ial o he hedging p ice, i espec i ely o a pa icula o m o he esilience
unc ion, which has been obse ed likewise in [13, Sec . 4] o addi i e impac . We
emphasise ha his is e y di e en o Sec . 5.1 whe e he esilience unc ion en e s
he p icing equa ion in a non i ial way. I u ns ou ha he cu en de ia ion o he
asse p ice om he una ec ed p ice becomes a ele an s a e a iable o desc ibing
he solu ion. Mo eo e , one can show (see Beche e and Bila e [6, Rema k 8.2, 2)])
ha he supe hedging p ice is dec easing in he impac unc ion λin he sense ha i
λ≥˜
λ, hen he p ice wi h espec o λdomina es he one wi h espec o ˜
λ. Fo a dual
o mula ion o he hedging o co e ed op ions, we e e o Boucha d and Tan [14].
Rema k 7.2 As explained in Sec . 4, wo king in e ec i e coo dina es u he pe mi s
ex ending esul s abou ansien p ice impac , in addi i e o mul iplica i e o m, o
mul iple isky asse s wi h c oss-impac om ansac ions ac oss di e en asse s (de-
sc ibed in Bila e [11, Chap. 5, see Example 5.1.6]). To his end, a key idea is ha he
impac unc ion needs o be he g adien ield o a sui able po en ial in o de o a oid
a o m o ins an aneously p o i able ound- ips (see [11, Theo em 5.1.4]). The eby,
esul s like om p e ious sec ions (o Boucha d e al. [12] o pe manen impac )
can be ex ended o mul iple asse s in an addi i e ansien c oss-impac model. One
ob ains a geome ic DPP and a iscosi y PDE o cha ac e ise supe hedging p ices,
which in ol es he esilience unc ion ho he ansien impac (see [11, Sec . 5.3.2]).
Mo eo e , unde ce ain condi ions, one eco e s as ins uc i e e e ence case again

316 D. Beche e , T. Bila e
esul s as in a mul idimensional Bachelie model wi h i s na u al p icing o mula
([11, Rema k 5.3.8]) ha does no in ol e he p ice impac . This ex ends o mul-
iple dimensions he ins uc i e one-dimensional linea pe manen impac example
om [12, Sec . 2.4], which also yields he amilia Bachelie p icing o mula. No-
ice ha he hedging s a egy is a ec ed by he p ice impac , hough closely ela ed
o he usual Bachelie del a-hedging s a egy o mula, by being compu ed a liqui-
da ion magni udes o he s ock p ice (i.e., in e ec i e coo dina es, analogously o
hose in (4.6) wi h Sins ead o S). This is en i ely analogous o Black–Scholes o -
mula ela ed quan i ies ( o p icing and hedging) occu ing unde (pe manen ) mul i-
plica i e impac in he basic log-linea example o ou model (see Example 2.1 and
Rema k 2.6).
Appendix: P oo s
This sec ion p o ides he p oo s elega ed om Sec . 5, in pa icula he p oo o
Theo em 5.11. Recall ha in his case, (x) =exp(λx) o λ>0, and hus he
e ec i e p ice simpli ies o S(s,y,θ)=se−λθ =: S(s, θ), i.e., he le el o impac
is no needed in o de o de e mine he p ice change o a block ade, gi en he p ice
be o e he ade. We conside s a egies aking alues in K=[−K,∞) o K>0.
This yields a g adien cons ain o he PDE ha is needed because o a singula i y
in he PDE, o he exp ession (5.12) o he o m o he op imal s a egy o be ini ely
de ined.
Fi s , we e i y in Appendix A.1 ha i he p icing PDE (5.11) admi s a su icien ly
smoo h classical solu ion, a eplica ing s a egy in eedback o m can be cons uc ed.
Such a cons uc ion is also needed o he con adic ion a gumen in he p oo o
he subsolu ion p ope y in Sec . A.2 whe e, using smoo h es unc ions, one con-
s uc s locally s a egies which, oughly speaking, beha e like eplica ing s a egies.
The iscosi y p ope y p oo s a e collec ed in Appendix A.2, and in Appendix A.3,
we p o e compa ison esul s ha imply uniqueness o he iscosi y solu ions o he
p icing PDEs and con inui y o he alue unc ion o he supe hedging p oblem.
A.1 Ve ifica ion a gumen o exponen ial impac unc ion
Suppose ha he unc ion w∈C1,2,1([0,T]×R++ ×R)has he p ope y ha o
any ( ,s,y)∈[0,T]×R++ ×R,weha e
1) θ[w]( ,s,y)∈K, ecalling he de ini ion in (5.12);
2) Lθ[w]( ,s,y)w( ,s, y) =0 when <T;
3) w(T,s,y) =H(s,y).
Suppose u he ha wis su icien ly egula (see Rema k A.1 below) so ha he e
exis s an admissible s a egy ∈o he o m
 =1
λlogλws ,S(S ,
), Y − +1 o ∈[0,T),
T=0,i.e., T=−T−.(A.1)
Hedging wi h ansien p ice impac 317
In pa icula , 0=log(λws(0,s,y)+1)/λ and T∈K. Conside he sel - inan-
cing po olio (β, ) wi h β0−=w(0,s,y). Then as in Rema k 5.8, we ge
Vliq
T() =H(ST,Y
T), T=0.
By he de ini ion o H, his shows ha Vliq
T() a ma u i y Tis enough capi al o
(supe -) eplica e he Eu opean claim wi h payo (g0,g
1)wi h a possible addi ional
block ade (p o ided ha he in ima in he de ini ion o H, see Lemma 5.1,a e
a ained). Hence (β, ) will be a (supe -) eplica ing s a egy o he Eu opean claim
(g0,g
1)wi h ini ial capi al w(0,s,y), which is equal o he supe eplica ion p ice
w(0,s,y), meaning ha a eplica ing hedging s a egy as desc ibed exis s and is an
op imal (i.e., cheapes ) supe eplica ion s a egy unde he gi en assump ions.
Rema kA.1 To cons uc a eplica ing s a egy as in (A.1), we suppose mo eo e ha
w∈C1,3,1([0,T]×R++ ×R)and apply I ô’s o mula, simila ly as in Rema k 5.8,
o ge o <T he equa ion
d =1
λ1
λws+1d(λws+1)−1
2(λws+1)2d[λws+1] 
=a( ,S ,Y
,
)d +b( , S ,Y
)dW ,
whe e o S :=S(S ,
)and Y
=Y
− ,wese
a( ,S ,Y
,
):= 1
λws+1w s +wssS μ −λh(Y 
)−wsyh(Y 
)
+1
2wsssσ2S2
−λ2σ2S2
wss
2(λws+1),
b( , S ,Y
):= σS wss
λws+1,
wi h all de i a i es o we alua ed a ( , S(S ,
), Y − ). Thus a eplica ing s a -
egy, which is supe hedging he payou a a minimal cos (see he a gumen s p eceding
he cu en ema k), can be cons uc ed as he ( ) ∈[0,T )-pa (plus a e minal block
ade) om a solu ion, i i exis s, o he SDE sys em, o ∈[0,T],
dS =S μ −λh(Y
+ )d +σdW ,
d =a( ,S ,Y
,
)d +b( , S ,Y
)dW ,
dY
=−h(Y
+ )d ,
(A.2)
wi h ini ial condi ion S0=s,Y
0=yand 0=log(λws(0,s,y)+1)/λ.
A.2 Viscosi y solu ion p ope y o w o exponen ial impac unc ion
Fo he esul s om Sec . 5.2, we now p o e he iscosi y p ope y.
318 D. Beche e , T. Bila e
Theo em A.2 The unc ion w∗ om (5.2)is a iscosi y supe solu ion o (5.11)on
[0,T)×R++ ×Rwi h he bounda y condi ion (5.13)on {T}×R++ ×R.
P oo Fi s , le ( 0,s
0,y
0)∈[0,T)×R++ ×Rand ϕ∈C∞
b([0,T]×R++ ×R)be
a smoo h unc ion such ha we ha e a s ic (meaning uniquely a ained) minimum,
(s ic ) min
[0,T ]×R++×R(w∗−ϕ) =(w∗−ϕ)( 0,s
0,y
0)=0.
Case 1: Suppose ha HKϕ( 0,s
0,y
0)<0. By he con inui y o he ope a o HK,
he e exis s an open neighbou hood Oo ( 0,s
0,y
0)whose closu e is con ained in
[0,T)×R++ ×Rsuch ha HKϕ( ,s,y) < −εin O o some ε>0. The e o e,
a e possibly sh inking he neighbou hood O, he e exis s a cons an kε>0 such
ha
s|ϕS( ,s,y)+1/λ −eλθ /λ|≥kε o all θ∈K,( ,s,y)∈O.(A.3)
Le ( n,s
n,y
n)n∈N⊆Obe a sequence con e ging o ( 0,s
0,y
0)wi h
w( n,s
n,y
n)−→ w∗( 0,s
0,y
0),
whe e w∗is he lowe -semicon inuous en elope o w. Se n:= w( n,s
n,y
n)+1/n.
Since n>w(
n,s
n,y
n), Theo em 4.1 implies he exis ence o θn∈Kand s a egies
γn∈such ha o s opping imes τn≥ n( o be sui ably chosen la e ), we ha e
P-a.s. o ∈[ n,T] ha
Vliq, n,zn,γn
∧τn≥w·,S(S n,zn,γn,
n,zn,γn), Y n,zn,γn− n,zn,γn ∧τn,(A.4)
whe e zn=(sneλθn,y
n+θn,θ
n,
n). To abb e ia e no a ion, we w i e in he sequel
nas supe sc ip ins ead o he ull a gumen ( n,z
n,γ
n), so ha
Sn:= S(S n,zn,γn,
n,zn,γn), Yn:= Y n,zn,γn− n,zn,γn.
Take τn=in { ≥ n:( , Sn
,Yn
)∈ O}, which is he i s en ance ime o he
pa abolic bounda y o he open egion O. In pa icula , τn<T. Since w≥w∗≥ϕ
and w∗−ϕhas a s ic local minimum a ( 0,s
0,y
0), he e exis s ι>0 such ha
(w −ϕ)(τn,Sn
τn,Yn
τn)≥ι.
Hence P-a.s., we ha e Vliq,n
τn−ϕ(τn,Sn
τn,Yn
τn)≥ι. Now Lemma 4.3 oge he wi h
he ac ha Sn
n=sn,Yn
n=yngi es ha P-a.s.,
ι≤ n−ϕ( n,s
n,y
n)
−τn
n
Sn
uϕS(u, Sn
u,Yn
u)+1/λ −eλn
u/λ(σ dWu+ζn
udu), (A.5)
whe e
ζn
:=ηn
−Ln
ϕ
Sn
(ϕS(u, Sn
,Yn
)+1/λ −eλn
/λ) o ∈[ n,τ
n]
Hedging wi h ansien p ice impac 319
wi h ηn
:= μ −λh(Yn
). No e ha ζn
is well de ined on [ n,τ
n]and uni o mly
bounded, no ing (A.3) and he ac ha Ynis bounded since nis. Hence by
Gi sano ’s heo em, he e exis s a measu e Pnequi alen o Psuch ha
 ∧τn
n
SuϕS(u, Sn
u,Yn
u)+1/λ −eλu/λ(σ dWu+ζn
udu), ≥ n,
is a squa e-in eg able ma ingale unde Pn, as he in eg and o he s ochas ic in eg al
is uni o mly bounded because o he de ini ion o τn, he con inui y o ϕSand he
boundedness o he ange o , no ing τn≤T. Taking expec a ions unde Pno
he igh -hand side o (A.5) leads o n−ϕ( n,s
n,y
n)≥ι>0, which yields a
con adic ion as by ou choice o nand o he sequence ( n,s
n,y
n)n∈N,weha e
n−ϕ( n,s
n,y
n)−→ w∗( 0,s
0,y
0)−ϕ( 0,s
0,y
0)=0.
Case 2: F om Case 1, we know ha HKϕ( 0,s
0,y
0)≥0. Hence
θ[ϕ]( 0,s
0,y
0)=1
λlogλϕS( 0,s
0,y
0)+1
is well de ined (also in a neighbou hood o ( 0,s
0,y
0)).
Le us suppose ha Lθ[ϕ]ϕ( 0,s
0,y
0)<0. By he con inui y o he ope a o L,
he e exis hen an open neighbou hood O⊆[0,T]×R++ ×Ro ( 0,s
0,y
0)and
some >0 and ε>0 such ha
Lθϕ( ,s,y) < −ε o ( ,s,y)∈O,θ ∈θ[ϕ]( ,s,y)− , θ[ϕ]( ,s,y)+ .
In pa icula , by he con inui y o he in ol ed unc ions, we ha e (a e possibly
sh inking he open se O) ha o e e y( ,s,y)∈Oand o some >0,
Lθϕ( ,s,y) < −εwhene e |ϕS( ,s,y)+1/λ −eλθ /λ|≤ .
As in Case 1, conside a sequence ( n,s
n,y
n)in Owhich con e ges o ( 0,s
0,y
0)and
such ha w( n,s
n,y
n)→w∗( 0,s
0,y
0). Se n:=w( n,s
n,y
n)+1/n and le θn∈K
and s a egies γn∈be such ha he dynamic p og amming p inciple (A.4) holds
o he s opping imes τn ha a e he i s exi imes o (·,Sn,Yn) om he se O.
Now a con adic ion ollows simila ly as in Case 1 wi h he ollowing adjus men :
We ha e
Vliq,n
∧τn−ϕ(·,Sn,Yn) ∧τn
= n−ϕ( n,s
n,y
n)
− ∧τn
n
Sn
u(ϕS+1/λ −eλn
u/λ)(σ dWu+ζn
udu)
+ ∧τn
nLn
uϕ(u,Sn
u,Yn
u)1{|ϕS+1/λ−eλn
u/λ|≤ }du
≤ n−ϕ( n,s
n,y
n)− ∧τn
n
Sn
u(ϕS+1/λ −eλn
u/λ)(σ dWu+ζn
udu),
320 D. Beche e , T. Bila e
whe e we se
ζn
:=ηn
−Ln
ϕ
Sn
(ϕS+1/λ −eλn
/λ)1{|ϕS+1/λ−eλn
/λ|≥ } o ∈[ n,τ
n],
wi h he unc ions ϕand ϕSabo e e alua ed a (·,Sn
·,Yn
·). The con adic ion now
ollows by aking expec a ions unde Pn≈Pand le ing n→∞.
Bounda y condi ion. Le (s0,y
0)∈R++ ×Rand ϕbe a smoo h unc ion wi h
(s ic ) min
[0,T ]×R++×R(w∗−ϕ) =(w∗−ϕ)(T , s0,y
0)=0.
Suppose ha
min{w∗(T, s0,y
0)−H(s0,y
0), HKϕ(T,s0,y
0)}<0.
The case HKϕ(T,s0,y
0)<0 leads o a con adic ion by he same a gumen s as in
Case 1 abo e, using ha HKϕ<0 in a small neighbou hood o (T, s0,y
0). Hence
we ob ain HKϕ(T,s0,y
0)≥0.
Now i w∗(T , s0,y
0)<H(s
0,y
0), hen also ϕ(T,s0,y
0)−H(s0,y
0)<0.
A e possibly modi ying he es unc ion ϕby ( ,s,y) → ϕ( ,s,y) −√T− ,
we can assume ha ∂ ϕ( ,s,y) →∞when →T, uni o mly on compac s.
Hence in an ε-neighbou hood [T−ε, T ) ×Bε(s0,y
0)a ound (T, s0,y
0),weha e
Lθ[ϕ]ϕ<0. Mo eo e , a e possibly dec easing ε,weha eϕ(T, ·)≤H(·)−ι1
on Bε(s0,y
0) o some ι1>0. We can a gue as in Cases 1 and 2 abo e, by s a -
ing om ( n,s
n,y
n)in [T−ε, T ) ×Bε(s0,y
0), wi h ( n,s
n,y
n)→(T, s0,y
0)and
w( n,s
n,y
n)→w∗(T, s0,y
0), s opping a he (pa abolic) bounda y a ime τnand
using w(T, ·)=H(·), o ge
Vliq,n
τn−ϕ·,S(Sn,
n), Yn−nτn≥ι1∧ι2,
whe e ι2:= in [T−ε,T )×∂Bε(s0,y0)(w∗−ϕ) > 0. A con adic ion ollows as in Case 2
abo e. 
Now we p o e he subsolu ion p ope y.
Theo em A.3 The unc ion w∗ om (5.3)is a iscosi y subsolu ion o (5.11)on
[0,T)×R++ ×Rwi h he bounda y condi ion (5.13)on {T}×R++ ×R.
P oo The p oo is simila o and inspi ed by he one o he subsolu ion p ope y in
[12, Theo em 3.7]. The eason is ha in his case, he g adien cons ain s ensu e ha
a es unc ion ϕ ha would possibly con adic he subsolu ion p ope y mus sa is y
HKϕ>0 locally and hence is su icien ly “nice” o de ine (locally) con ol p ocesses
(employing he e i ica ion a gumen in Rema k A.1) ha lead o a con adic ion like
in [12]. Fo comple eness, we ou line di e ences in he line o p oo and ske ch he
main s eps.
Le ϕ∈C∞
b([0,T],R++ ×R)be a es unc ion wi h he p ope y ha he poin
( 0,s
0,y
0)∈[0,T]×R++ ×Ris a s ic (local) maximum o w∗−ϕ, i.e.,
(s ic ) max
[0,T ]×R++×R(w∗−ϕ) =(w∗−ϕ)( 0,s
0,y
0)=0.

Hedging wi h ansien p ice impac 321
Fi s assume ha 0<T. To ease no a ion, we use he a iable x o deno e
he pai (s, y). Because o he special o m o he second pa o he DPP in
Theo em 4.1(ii), we need o employ wk(ins ead o was in he p oo o he
supe solu ion p ope y). By Ba les [5, Lemma 6.1], we can ake a sequence
(kn,
n,x
n)n∈Nsuch ha kn→∞,any( n,x
n)is a local maximum o w∗
kn−ϕand
( n,x
n,w
kn( n,x
n)) →( 0,x
0,w∗( 0,x
0)).
Assume ha FK[ϕ]( 0,x
0)>0 and le
ϕn( , x) =ϕ( ,x) +| − n|2+|y−yn|2+|s−sn|4.
Then FK[ϕn]>0 holds in a neighbou hood Bo ( 0,x
0) ha con ains ( n,x
n) o all
nla ge enough. Since we wo k on he local neighbou hood Bwhe e also HKϕn>0,
we can modi y (in a smoo h way) he unc ions hand ϕnou side o B o be suppo ed
on a sligh ly bigge compac se whe e HKϕn>0 holds. Thus a e possibly passing
o a sui able subsequence, he e exis γn∈knsuch ha
 n,zn,γn
=1
λlogλ∂ϕn
∂s ( , S n,zn,γn
,Y n,zn,γn
)+1, ≥ n,
whe e we se S n,zn,γn
=S(S n,zn,γn
,
n,zn,γn
)and Y n,zn,γn
=(Y −) n,zn,γn
o
zn=(sn,y
n,0,w
kn( n,x
n)−n−1);seeRema kA.1.Le τnbe he i s ime a e n
a which he p ocess (S n,zn,γn
,Y n,zn,γn
) ≥ nlea es B.Likein[12, p oo o Theo-
em 3.7], we conclude by applying I ô’s o mula, using Lemma 4.3 and FK[ϕn]>0
on B ha P-a.s.,
Vliq, n,zn,γn
τn≥ϕnτn,S n,zn,γn
τn,(Y −) n,zn,γn
τn+ n−ϕn( n,x
n).
Now a con adic ion ollows as in [12, p oo o Theo em 3.7, subsolu ion p ope y,
(a)].
Fo he bounda y condi ion, i.e., he case 0=T, he a gumen s a e exac ly he
same as in [12, p oo o Theo em 3.7, subsolu ion p ope y, (b)]. 
A.3 Compa ison esul s o iscosi y solu ions
Fi s we p o ide a compa ison esul o he p icing PDE (5.5), needed o he p oo
o Theo em 5.5. No e ha (5.5) has he s uc u e
0=−ϕ −σ2s2
2ϕss
−B1y, (y)ϕsϕy−sB2y, (y)ϕsϕs−sB3y, (y)ϕs,(A.6)
whe e Bi:R2→R,i=1,2,3, a e bounded and Lipschi z-con inuous unc ions.
By a change o coo dina es, one can ans o m he PDE as ollows.
Lemma A.4 Le ube a iscosi y subsolu ion ( esp.supe solu ion)o he PDE (A.6).
Fix κ>0. Then he unc ion ˜ude ined by
˜u( , s, y) =eκ u , s (y), y o ( ,s,y)∈[0,T]×R++ ×R
322 D. Beche e , T. Bila e
is a iscosi y subsolu ion ( esp.supe solu ion)o he PDE
0=κϕ −ϕ −σ2s2
2ϕss −B1(y, e−κ ϕs)ϕy+λ(y)B1(y, e−κ ϕs)ϕs
−sB2(y, e−κ ϕs)ϕs−eκ s (y)B3(y, e−κ ϕs). (A.7)
P oo To p o e he supe -( esp. sub-)solu ion p ope y, ake any poin ( 0,s
0,y
0)
in [0,T)×R++ ×Rand a es unc ion ˜ϕ∈C∞
b([0,T]×R++ ×R) o ˜ua
( 0,s
0,y
0), i.e.,
min
[0,T ]×R++×R( esp. max)( ˜u−˜ϕ) =˜u( 0,s
0,y
0)−˜ϕ( 0,s
0,y
0)=0.(A.8)
Conside ϕ( ,s,y) := e−κ ˜ϕ( ,s/ (y),y) o ( ,s,y) ∈[0,T]×R++ ××R.
We ha e by de ini ion ha eκ ϕ( , s (y), y) =˜ϕ( ,s,y) o
( ,s,y) ∈[0,T]×R++ ××R. In pa icula , ϕis a es unc ion o ua
( 0,s
0 (y
0), y0)since by (A.8), we ge
min
[0,T ]×R++×R( esp. max)(u −ϕ) =u 0,s
0 (y
0), y0−˜ϕ 0,s
0 (y
0), y0
=0.(A.9)
We also ha e
˜ϕs( ,s,y)=eκ (y)ϕ
s , s (y), y,
˜ϕss( ,s,y)=eκ 2(y)ϕss , s (y), y,
˜ϕy( ,s,y)=eκ λ(y) (y)ϕs , s (y), y+eκ ϕy , s (y), y
=λ(y) ˜ϕs( ,s,y)+eκ ϕy , s (y), y,
˜ϕ ( ,s,y)=eκ ϕ  , s (y), y+κeκ ϕ , s (y), y.
By di ec applica ion o hese iden i ies, we de i e om he igh -hand side o
(A.7) o ˜ϕe alua ed a ( 0,s
0,y
0)exac ly he igh -hand side o (A.6) o ϕa
( 0,s
0 (y
0), y0). By he iscosi y p ope y o uand (A.9), we hus conclude ha
(A.7) holds o ˜ϕa ( 0,s
0,y
0)wi h “≥” ( esp. “≤”). This p o es he claim. 
By Lemma A.4, i now su ices o p o e compa ison o (A.7) since his implies a
compa ison esul o (A.6). This is done in he ollowing esul .
Theo em A.5 Le u( espec i ely )be a bounded uppe -semicon inuous subsolu ion
( esp.lowe -semicon inuous supe solu ion)on [0,T)×R++ ×Ro (A.7). Suppose
ha u≤ on {T}×R++ ×R.Then u≤ on [0,T]×R++ ×R.
P oo To p o e he claim by con adic ion, le us suppose ha
sup
( ,s,y)∈[0,T ]×R++×R(u − )( ,s,y)>0.
Hedging wi h ansien p ice impac 323
Then we can ind R>1 such ha wi h OR:= (1/R, R),weha e
sup
( ,s,y)∈[0,T ]×OR×[−R,R]
(u − )( ,s,y)>0.
In pa icula , he e exis δ>0 and ( 0,s
0,y
0)∈OR×[−R,R]wi h he p ope y
ha (u − )( 0,s
0,y
0)=δ>0.
Now conside o n∈N he bounded uppe -semicon inuous unc ion
n( , s1,s
2,y
1,y
2):= u( , s1,y
1)− ( ,s2,y
2)−n
2(s1−s2)2−n
2(y1−y2)2.
I a ains i s maximum a some ( n,sn
1,sn
2,yn
1,yn
2)∈[0,T]×O2
R×[−R,R]2by
compac ness o ha se , and we clea ly ha e
n( n,sn
1,sn
2,yn
1,yn
2)≥δ o all n∈N.(A.10)
By a gumen s as in [12, p oo o Lemma 3.11], one ob ains (a e possibly passing o
a subsequence) ha
n(sn
1−sn
2)2+n(yn
1−yn
2)2−→ 0asn→∞.(A.11)
No e ha (A.11) also implies n(sn
1−sn
2)(yn
1−yn
2)→0asn→∞.
Now by Ishii’s lemma as s a ed in C andall e al. [18, Theo em 8.3], he e exis
(bn,Xn,Yn)in R×S2×S2such ha wi h pn=n(sn
1−sn
2)and qn=n(yn
1−yn
2),
we ha e
bn,(pn,qn), Xn∈¯
P2,+
Oau( n,sn
1,yn
1),
bn,(pn,qn), Y n∈¯
P2,−
Oa ( n,sn
2,yn
2),
whe e Xnand Ynsa is y
Xn0
0−Yn≤3nI2−I2
−I2I2.(A.12)
He e, S2deno es he se o 2 ×2 symme ic nonnega i e ma ices and I2∈S2is he
iden i y ma ix. Using he iscosi y p ope y o uand a ( n,sn
1,yn
1)and ( n,sn
2,yn
2),
espec i ely, we ha e
κu( n,sn
1,yn
1)−bn−1
2σ2(sn
1)2Xn
11 +L(sn
1,yn
1,pn,qn)≤0,
κ ( n,sn
2,yn
2)−bn−1
2σ2(sn
2)2Yn
11 +L(sn
2,yn
2,pn,qn)≥0,
whe e
L( , s, y, p, q) := −B1(y, e−κ p)q +λ(y)B1(y, e−κ p)p
−sB2(y, e−κ p)p −eκ s (y)B3(y, e−κ p).
324 D. Beche e , T. Bila e
As a consequence,
0<κδ<κ
u( n,sn
1,yn
1)− ( n,sn
2,yn
2)
≤−1
2σ2(sn
2)2Yn
11 +1
2σ2(sn
1)2Xn
11
+L( n,sn
2,yn
2,pn,qn)−L( n,sn
1,yn
1,pn,qn). (A.13)
On he o he hand, by (A.12), we ge ha
1
2σ2(sn
1)2Xn
11 −1
2σ2(sn
2)2Yn
11 ≤3
2σ2n(sn
1−sn
2)2,
which con e ges o 0 o n→∞due o (A.11). Le us now analyse he di e ence
L( n,sn
2,yn
2,pn,qn)−L( n,sn
1,yn
1,pn,qn). Wi h C( esp. CR) deno ing a Lipschi z
cons an (depending on R) ha may change om line o line, we ge es ima es o he
co esponding e ms ia
|B1(yn
1,e
−κ npn)qn−B1(yn
2,e
−κ pn)qn|≤C|yn
1−yn
2||qn|,
|λ(yn
1)B1(yn
1,e
−κ npn)pn
−λ(yn
2)B1(yn
2,e
−κ npn)pn|≤C|yn
1−yn
2||pn|,
|sn
1B2(yn
1,e
−κ npn)pn−sn
2B2(yn
2,e
−κ npn)pn|≤C|(sn
1−sn
2)pn|
+CR|(yn
1−yn
2)pn|,
|eκ nsn
1 (yn
1)B3(yn
1,e
−κ npn)
−eκ nsn
2 (yn
2)B3(yn
2,e
−κ npn)|≤CR(|sn
1−sn
2|+|yn
1−yn
2|).
As all es ima es om abo e anish o n→∞, he igh -hand side in (A.13)is
bounded by some hing ha con e ges o 0 as n→∞. Bu his yields a con adic ion
o la ge n.
Because o lack o a p ecise e e ence, we p o ide a compa ison esul also in he
case o del a cons ain s leading o he a ia ional inequali y (5.11).
Theo em A.6 Suppose ha he esilience unc ion his Lipschi z-con inuous and As-
sump ion 5.3 holds.Le u( esp. )be a bounded uppe -( esp.lowe -)semicon inuous
iscosi y subsolu ion ( esp.supe solu ion)o he a ia ional inequali y (5.11)wi h
he e minal condi ion (5.13). Then u≤ on [0,T]×R++ ×R.
P oo We a gue by con adic ion. Fo any a>0, se Oa:= [a,∞)×[−1/a, 1/a].
I sup[0,T ]×R++×R(u − ) > 0, he e exis s a>0 wi h sup[0,T ]×Oa(u − ) > 0.
Fo κ>0, conside ˜u:= eκ uand ˜ := eκ . Then ˜u( esp. ˜ ) is a iscosi y
sub-( esp. supe -)solu ion o
min{κϕ +˜
L[ϕ],HK, ϕ}=0