Ý Ò ÔØ Ú ÓÒØÖÓÐ ÔÔÖÓ ØÓ Ö Ñ Ò Ñ ÒØ Ò ÒÓÑ Ð ÑÓ Ð ÏÓÐ Ò Âº ÊÙÒ Ð Ö Ö Ö ÌÖ Ú ÐÐ ØÓ Ò Ì Þ ÒÓ Î Ö ÓÐÙ ØÖ Øº Ï ÓÒ Ö Ø ÔÖÓ Ð Ñ Ó ÓÖØ ÐÐ Ö Ñ Ò Ñ Þ Ø ÓÒ Û Ò Ø

16  Download (0)

Full text

(1)

Wolfgang J.Runggaldier, Barbara Trivellatoand TizianoVargiolu

Abstra t. We onsidertheproblemofshortfallriskminimizationwhenthere

isun ertaintyabouttheexa tsto hasti dynami softheunderlying.Starting

from the generaldis rete time model and theapproa hdes ribedin

Rung-galdierandZa aria(1999),wederiveexpli itanalyti solutionsforthe

par-ti ular ase ofabinomialmodelwhenthereis un ertaintyaboutthe

proba-bility ofan\up-movement".Thesolution turnsout tobe aratherintuitive

extensionofthatforthe lassi alCox-Ross-Rubinsteinmodel.

1. Introdu tion

Inin ompletemarketsthesuperhedging riterionallowsonetoeliminatetherisk

ompletely, but it requires in general too mu h initial apital; it orresponds in

fa t to amin-max-type riterion. Onemaythen ask byhowmu h one anlower

the initial ost if one is willing to a ept some risk or,dually, what is the risk

orresponding to an initial apital less than what is required for superhedging.

Theshortfallriskminimization approa hallowsonetodealwiththeseissues.

Given a market with a non-risky and a ertain number of risky assets, let

H

T

bealiabilityto behedgedatsome xedfuturetimeT.DenotebyV

T (') the

valueat T ofaportfolio orrespondingto aself- nan inginvestmentstrategy',

possiblysatisfyingsomeadditional onstraintssu h asashortsellingprohibition.

Theproblemisto nd J 0 (S 0 ;V 0 ):=inf ' E P S0;V0  ` [H T V T (')℄ +  (1)

foragiveninitialvalueS

0

oftheasset(s)intheportfolio,foragiveninitial apital

V

0

andwhere`()isasuitablein reasingfun tionsu hthat`(0)=0and`(x)>0

for all x > 0. For `(z) = 1

fz>0g

, the right hand side in (1) orresponds to the

smallest shortfall probability. Problems of the type (1) have re ently attra ted

1991Mathemati sSubje tClassi ation. 91B28,93E35.

Keywordsand phrases. Riskmanagement,shortfallriskminimization,restri tedinformation,

(2)

onsiderableattention(seee.g.[2,3,4,5,7,8,9,10,16℄).Let,foragivenS 0 , V  0 (S 0 ):=inffV 0 jJ 0 (S 0 ;V 0 )=0g: (2) Sin e J 0 (S 0 ;V 0 )=0meansthat H T V T (' 

) P-almost surely(where '  is the optimal strategy in (1)), V  0 (S 0

) is the minimal initial apital needed to

super-hedge the laim. It follows that if V

0  V  0 (S 0 ) then V T ('  )  H T , P-a.s. For

superhedging,the hoi eoftheunderlyingprobabilisti modelfortheevolutionof

therisky assetsisthus irrelevant aslongasit indu es probabilitymeasures that

areequivalent.Howeverforthemoregeneralproblem ofriskminimization in(1)

theprobabilisti stru tureoftheunderlyingmodelmatters,but thetruemodelis

almostneverknownexa tly.Apossibilityisthentouseamin-max-type riterion

asin [4℄ tryingto nd inf ' sup P2P E P S0;V0  ` [H T V T (')℄ +  ;

whereP isafamilyof"realworldprobabilitymeasures".However,su ha riterion

doesnotallowonetoin orporateadditionalinformationontheunderlyingmodel

asitbe omessu essivelyavailable.

Thus,anadaptiveapproa h, orrespondingtoaBayesian-type riterion,

ap-pearsmoreappropriate. Su h adaptiveapproa heshavealreadybeendealt with

intheliterature(seee.g. [2,3,4,5℄and, inthe ontextofportfoliooptimization,

in [1,11, 12, 13℄).In allthese paperstheun ertaintyis onlyin the sto k

appre- iationrate.Thetoolsaremainly probabilisti innature (involvingalsomeasure

transformation)andarebasedon onvexduality.Anexpli itsolutionisessentially

possibleonlyinsimpler asesandtransa tion ostsarenottakenintoa ount.

Inthe presentpaperwebase ourselveson[17℄ followinganapproa h along

the lines of dis rete time sto hasti adaptive ontrol. In that work the authors

give ageneraldes riptionof this approa h and theyapply it, in parti ular, to a

multinomial modelfortheriskyassetswheretheprobabilitiesarenotknown; for

the spe i aseof abinomialmodel somenumeri al resultsare also presented.

Herewefo usourattentiononthebinomialmodel.Bynegle tingtransa tion osts

(the portfolioisrebalan edonlyatdis retedates, whi hlimitstheirimpa t)and

imposing theself- nan ing requirementastheonly onstrainton theinvestment

strategies, we su eed in deriving expli it solutions for the optimal investment

strategyandfor the orrespondingminimalvalueoftheshortfallriskin the ase

`(x)=x.Wedothisbothforthe asewhentheprobabilitypofan\up-movement"

isknownaswellaswhenitisunknownand,a ordingtotheBayesianapproa h,

treatedasarandomvariablewithaBeta-typedistribution.Weobtainananalyti

solution, that turns outto be aninteresting variantof the Cox-Ross-Rubinstein

(CRR) solution(seee.g. [15℄), whentheinitial apitalisinsuÆ ientforaperfe t

hedge.

The paper is organisedasfollows.In Se tion 2 we brie y re all somefa ts

(3)

Programming (DP).InSe tion4,byassuming`(x)=x,we omputethe

minimiz-ing admissiblestrategyaswell asanexpli it evaluation formula fortheminimal

dis ountedshortfallrisk in the ase whenp isknown.In Se tion 5,again by

as-suming `(x) = x, we ompute the minimizing strategy as well as the minimal

dis ountedshortfallriskinthe asewhenpisunknown.

2. The Cox-Ross-Rubinstein binomial model

We onsider adis rete-time market modelwith the set of dates 0;1;:::;N,and

withtwoprimarytradedse urities:ariskyasset(asto k)S andarisk-free

invest-ment(a bond)B. Weassumethatthevalueof thebond is onstantlyequalto 1

throughtime,andthatthesto kpri epro essS satis es

S n+1 =S n ! n ; n=0;:::;N; (3) where S 0

>0isagiven onstantandf!

n g

n=0;:::;N

isasequen eofi.i.d. random

variables de ned on aprobability spa e (;F;P),taking only tworeal values d

andusatisfying0<d<1<u,withprobabilitylaw

p:=Pf!

n

=ug=1 Pf!

n

=dg; n=0;:::;N:

We anassumewithoutlossofgeneralityB

n

1bylettingS

n

bethedis ounted

pri esof theasset.Letus thendenoteby'

n =( n ; n ),n=0;:::;N,an

invest-mentstrategy,where

n

standsfortheamountofthenonriskyassetand

n stands

forthenumberofunitsoftheriskyassetthatareheldintheportfolioinperiodn.

Weassumethat'

n

isadaptedtotheobservation-algebraF S

n

:=fS

m

; mng,

foralln=0;:::;N,andthat'=f'

n g

n=0;:::;N

satis estheself- nan ingproperty

V 0 =  0 + 0 S 0 V n+1 :=  n+1 + n+1 S n+1 = n + n S n+1 ; n=0;:::;N 1; whereV 0

isagiven onstant,representingtheinitialvalueoftheportfolio.Weshall

denote by A

ad

the set of all self- nan ing strategies (the admissible investment

strategies).

Itisarather lassi alresultthatV followsthedynami s

V n+1 =V n + n S n (! n 1)=:V n+1 (V n ;S n ;! n ; n );

sothat one anrestri toneselftojust thede isionvariable

n .

ConsideraEuropean ontingent laimH(S

N

)andletP 

bethemartingale

measureforourmodel.Itis wellknownthat P  orrespondsto P  f! n =ug=p  := 1 d u d ; P  f! n =dg=1 p  = u 1 u d ; n=0;:::;N;

(see e.g. [15, Ch. 2℄). The arbitrage free pri e V 

n

of H(S

N

) at time n, where

n=0;:::;N 1,isgivenbytheCox-Ross-Rubinstein(CRR) evaluationformula

(4)

wherebyE 

wedenotetheexpe tationwithrespe ttoP  .Inparti ular,V  0 (S 0 )

isthe minimalinitial valueof theportfolioneededto repli atethe laimde ned

in(2). Attimen,n=0;:::;N 1,therepli atingstrategy

n isgivenby n = V  n+1 (S n u) V  n+1 (S n d) S n (u d) : (5) IfV 0 <V  0 (S 0

),thentherepli ationoftheterminalpayo isnotpossible.In

thissituation,aninvestormaybeinterestedinanalyzingtheshortfallrisk de ned

astheexpe tation of theterminalde it weighted by somelossfun tion.Letus

thenintrodu ethis probleminmoredetail.

Denote by V N (') = N + N S N

the value of the portfolio at time N

or-responding toanadmissibleinvestmentstrategy'.Theminimal shortfall risk is

de nedas J 0 (S 0 ;V 0 ):= min '2Aad E P S0;V0  ` [H(S N ) V N (')℄ +  ; (6)

for agiveninitial valueS

0

of the riskyasset in the portfolio and agiven initial

apitalV 0 <V  0 (S 0

),where`()0isasuitablelossfun tion,thatisanin reasing

fun tion su hthat `(0)=0and`(x)>0forallx>0.

Inthis paperwe onsider the optimization problem (6) byassuming either

thattheprobabilitypisknownorthat itisnot.Forthe asewhenpisunknown,

we adopt a Bayesian-type approa h whi h allows us to in orporate additional

informationontheunderlying modelasitbe omessu essivelyavailable.

3. The dynami programmingalgorithm

Inthis se tionweprovideaDPalgorithmto omputeasolutiontoourproblem

(6)bothforthe asewhenpisknownandwhenitisnot(see[6℄forananalogous

algorithmforthe aseofsuperhedgingwith transa tion osts).Inthe asewhere

p is unknown, adopting the Bayesian point of view, we use the ordinary Bayes

formulatosu essivelyupdate theinitial(prior)densityh(p)of ponthebasisof

fF S

n g

n=0;:::;N

. This leads to what is alled the Bayesian DPalgorithm (see e.g.

[14,18℄).

3.1. DPalgorithmwhenpisknown

TheDPalgorithmpro eedsba kwardsa ordingtothefollowingsteps:

(5)

3.2. DPalgorithmwhenpisunknown

TheBayesianDPalgorithmpro eedssimilarlytotheDPalgorithm:

J N (s;v) = `((H(s) v) + ); J n 1 (S n 1 ;V n 1 ) = inf n 1 E P n 1 S n 1 ;V n 1 fJ n (S n ;V n )g: (8)

Nowptooisarandomvariable,anditsdistributiondependsontheinformationF S

n

uptotimen.Wein orporatethisinformationintheprobabilitymeasureP

n ,that

dependsalsoonthedistributionofp.Sin ePf!

n

jpgistheBinomialdistribution,

a onjugatefamily ofdistributions of pis that ofthe Betadistributions.With a

priordensity h 0 (p)/p 0 (1 p) 0 ; with 0 ; 0

0,theposteriordensityinperiodnbe omes

h n (p)/p n (1 p) n ; where, denotingbyu n

thetotalnumberof \up-movements"(u

0 :=0) umulated uptotimen, n = 0 +u n ; n = 0 +n u n :

In parti ular, for n = 0 and

0 =

0

= 0 the prior density h

0

(p) be omes the

uniformdensity.

Sin e the values of p enter the DP re ursions linearly, by the \smoothing

property" of onditional expe tations it is easily seen that, also in the present

ase,the DPre ursions aregivenbythe previoussteps,ex ept that phasto be

repla ed inJ n 1 (S n 1 ;V n 1 ), n=N;:::;1, by E n 1; n 1 [p℄:=E  pjF S n 1  and 1 pby1 E n 1; n 1 [p℄,where E ; [p℄= +1 + +2 :

4. Expli it solutions when p is known

In this se tion we are on erned with the evaluation of the minimal dis ounted

shortfall risk (6)and the orrespondingstrategyin the asewhen `(x)=x.Due

to the possibility of making dire t al ulations on the DP algorithm steps, we

anderiveexpli itevaluationformulas.These evaluationformulasaresimpleand

meaningful,showingexpli itlywhatwastobeexpe ted:thatis,theshortfallriskis

de reasingwithrespe tto theinitial apital (wewillshowthatsu hdependen e

is linear), and it is always equal to zero when the level of the initial apital is

greaterthanorequaltoV 

0 (S

0

).We rst onsiderthe asewhenthereis omplete

informationonthe underlyingmarketandwe ompareourresultswiththe

well-knownresultsonperfe t hedging ofaEuropean ontingent laim.

(6)

Theorem4.1. Consider a European ontingent laim H on a sto k whose pri e

S is assumed to follow the CRR binomial model (3). Let V 

n (S

n

), where n =

0;:::;N 1, be the arbitrage freepri e attimen de ned by the CRRevaluation

formula(4).Assumethe lossfun tion `()in(6)isthe identityfun tion`(x)=x.

Then i) if p>p  ,then J n (S n ;V n )=  1 p 1 p   m [V  n (S n ) V n ℄ + ; (9)

for n = 0;:::;N 1. In parti ular, for n = 0 the minimal dis ounted

shortfall riskis J 0 (S 0 ;V 0 )=  1 p 1 p   N [V  0 (S 0 ) V 0 ℄ + ; whereV  0 (S 0 )=C  0 (S 0

). Theminimizing investment strategyis given by

1 n = V  n+1 (S n u) V n S n (u 1) ; (10) for n=0;:::;N 1; ii) if p<p  ,then J n (S n ;V n )=  p p   m [V  n (S n ) V n ℄ + ;

for n = 0;:::;N 1. In parti ular, for n = 0 the minimal dis ounted

shortfall riskis J 0 (S 0 ;V 0 )=  p p   N [V  0 (S 0 ) V 0 ℄ + ; whereV  0 (S 0 )=C  0 (S 0

). Theminimizing investment strategyis given by

2 n = V  n+1 (S n d) V n S n (d 1) ; (11) for n=0;:::;N 1.

Proof. We start from n = N 1 by onsidering expression (7) of

J N 1 (S N 1 ;V N 1

). Thefun tion to be minimizedin (7)is alinear ombination

ofthetwopie ewiseaÆnefun tions

[H(S N 1 u) V N 1 N 1 S N 1 (u 1)℄ + (12) and [H(S N 1 d) V N 1 N 1 S N 1 (d 1)℄ + (13)

Thefun tionin(12)isde reasingfor

N 1 lessthan 1 N 1 inEquation(10),sin e u>1andS N 1

is positive, andthereafteritis equaltozero,while thefun tion

in(13)isequaltozerofor

N 1

lessthan 2

N 1

inEquation(11),sin ed<1and

S

N 1

is positive,and from there on itis in reasing. Therefore,if 1

(7)

or,equivalently,ifV N 1 E  [H(S N

)℄, thenbothstrategies 1 N 1 and 2 N 1 are

optimalaswellasany admissiblestrategybetweenthem. IfV

N 1 <E  [H(S N )℄,

thenin orderto establishthein mumin (7)itsuÆ esto analyzethesign ofthe

slopeof p[H(S N 1 u) V N 1 N 1 S N 1 (u 1)℄+ +(1 p)[H(S N 1 d) V N 1 N 1 S N 1 (d 1)℄;

whi hisgivenbytheexpression

S

N 1

[p(d u)+1 d℄: (14)

If (14) is less than zero, or, equivalently, if p > p 

, then the in mum in (7) is

a hievedat 1

N 1

.Ifthisisthe ase,byputting 1 N 1 in(7)weobtain J N 1 (S N 1 ;V N 1 )= 1 p 1 p   C  N 1 (S N 1 ) V N 1  + : Conversely,ifp<p 

thenthein mumin (7)isa hievedat 2 N 1 ,andweobtain J N 1 (S N 1 ;V N 1 )= p p   C  N 1 (S N 1 ) V N 1  + :

Thisshowsthatformula(9)istrueforn=N 1.

We now pro eed byba kwardindu tion with respe tto n. Assume p>p 

fromnowon.Assumethatequality(9)holds forn,wheren=N 1;:::;1,with

theminimizingstrategygivenby(10).Weshowthat italso holdsforn 1,with

thesamestrategyas(10)forn 1.From(7)oftheDPalgorithmwehave

J n 1 (S n 1 ;V n 1 ) = inf n 1 fpJ n (S n 1 u;V n (V n 1 ;S n 1 ;u; n 1 ))+ +(1 p)J n (S n 1 d;V n (V n 1 ;S n 1 ;d; n 1 ))g;

whi himplies,byindu tion,

J n 1 (S n 1 ;V n 1 )=  1 p 1 p   m inf n 1 fp[V  n (S n 1 u) V n 1 (15) n 1 S n 1 (u 1)℄ + +(1 p) [V  n (S n 1 d) V n 1 n 1 S n 1 (d 1)℄ + o :

Using thesame argumentsas in the rst step, andtaking into a ount that the

slopeoftheexpression

p[V  n (S n 1 u) V n 1 n 1 S n 1 (u 1)℄+ +(1 p)[V  n (S n 1 d) V n 1 n 1 S n 1 (d 1)℄

is again given by (14) with S

N 1

repla ed by S

n 1

, we have that, under the

assumption p>p 

(8)

This ends the proof of i). The proof of ii) an be obtained by using arguments

similar to those of i). We only observethat, under the assumption p < p 

, the

in mum in the DP algorithm steps is a hieved at

n 1 , for n = 0;:::;N 1, satisfying V  n (S n 1 d) V n 1 n 1 S n 1 (d 1)=0: 

Remark4.2. Noti e that this approa h is linked to the CRR model: in fa t, by

al ulatingtheexpe tedshortfallriskunderthehistori alprobabilityP,wearrive

atanexpression ontainingtheexpe tedpri eofthe laimundertherisk-neutral

probabilityP 

.Moreover,thehedgingstrategyinthis aseissimilarto(5),whi h

istheoneoftheCRRmodel:infa t intheCRRmodelthehedgingstrategy is

equaltotheratiobetweenthedi eren eoftheexpe tedpri esofthe laiminthe

twopossiblefutureout omesandthedi erentpri esoftheunderlying;here is

equaltotheratiobetweenthedi eren eoftheexpe tedpri eofthe laiminone

ofthe possiblefuture out omesand thevaluetheportfoliowould haveif itwere

invested in the bond B, and the di eren e between the pri e of the underlying

in the same future out ome onsidered before and its present pri easif it were

investedinthebondB.Inotherwords,itisasifwewerehedginga laimhaving

a payo that in ea h state of nature ould be the one of the original laim or

the money orrespondingto the present valueof theportfolio; in the sameway,

the underlying ould either assume the value orresponding to the value of the

derivative,orthevalue orrespondingto arisklessinvestment.

Remark4.3. Di erently from [8℄, here we have not imposed that V  0. This

leadstodi erentresults:infa t,ifweimposedV 0,wewouldhaveobtained(as

theydo)anoptimalstrategyequaltotherepli atingstrategyofamodi ed laim

that is betweenzeroand theoriginal laim;wehaveadi erent strategy,that in

generalgivesanoptimalexpe tedshortfalllowerthantheyhave.Inparti ular,our

strategysu eedsin repli atingperfe tly the laiminallthestatesof naturebut

theleastprobableone(see[7℄foranexpli itproof),sothattheexpe tedshortfall

omesentirelyfromthisstateofnature.However,ifV

0

isnearV 

0

,thenV remains

positiveatalltimespriortothematurityN,sothetwostrategiesthatweobtain

byimposingornotthe onstraintV 0 oin ide.

5. Expli it solutions when p is unknown

Theformulasgivenin thefollowingtheoremforthe optimaldis ountedshortfall

risk are similar to those given in Theorem 4.1, and the minimizing investment

strategiesarethesame.However,whilein theprevious aseonlytwoalternatives

forthepossiblevaluesofp 

were onsidered,i.e.p 

>pandp 

<p,hereat ea h

step n we haveto onsider several alternatives a ording to the estimates of p,

(9)

Theorem5.1. Consider the assumptions of Theorem 4.1 for the ase when p is

unknownwitha priorh

0 (p)/p 0 (1 p) 0 ,with 0 ; 0 0. Then i) if p  <E n ; n +m 1 [p℄, then J n (S n ;V n ) = 0  m 1 Y j=0 1 E n ; n +j [p℄ 1 p  1 A [V  n (S n ) V n ℄ + : (16)

The minimizing investmentstrategyisgiven by(10).

ii) if E n+i; n+m 1 i [p℄<p  <E n+i+1; n+m 2 i [p℄, wherei=0;:::;m 2, then J n (S n ;V n )= 0  m 2 i Y j=0 1 E n; n+j [p℄ 1 p  1 A   0  i Y j=0 E n+j; n+m 1 i [p℄ p  1 A [V  n (S n ) V n ℄ + = (17) = 0  i Y j=0 E n +j; n [p℄ p  1 A 0  m 2 i Y j=0 1 E n +i+1; n +j [p℄ 1 p  1 A [V  n (S n ) V n ℄ + (18)

Both the strategies (10) and (11) are optimal as well as any admissible

strategybetween them.

iii) if p  >E n+m 1; n [p℄, then J n (S n ;V n ) = 0  m 1 Y j=0 E n+j; n [p℄ p  1 A [V  n (S n ) V n ℄ + : (19)

The minimizing investmentstrategyisgiven by(11).

Proof. Westartfromn=N 1by onsideringtheexpression

J N 1 (S N 1 ;V N 1 )= (20) = inf N 1 E N 1; N 1 [p℄[H(S N 1 u) V N 1 N 1 S N 1 (u 1)℄ + + + 1 E N 1; N 1 [p℄  [H(S N 1 d) V N 1 N 1 S N 1 (d 1)℄ + ;

whose in mum is a hieved, by using argumentssimilar to those in the proof of

Theorem 4.1, at 1 n 1 in Equation (10) if E N 1 ; N 1 [p℄ > p  and at 2 n 1 in Equation(11)ifE N 1 ; N 1 [p℄<p 

.Therefore, puttingtheminimizing strategy

(10)

ifE N 1; N 1 [p℄>p  ,and J N 1 (S N 1 ;V N 1 )= E N 1; N 1 [p℄ p   C  N 1 (S N 1 ) V N 1  + if E N 1 ; N 1 [p℄< p 

, showingthat formulas(16){(19) are true forn = N 1

(letus observethatthevalidityof ii)istrivialforn=N 1).

Wenowpro eed byindu tion withrespe t to n.Weassume thatequalities

(16)|(19) holdfor n, where n=1;:::;N 1,and we showthat they alsohold

forn 1.As regardsalternativeii), weshall onlyproveformula(17).Indeed,it

isnotdiÆ ultto he k(weomit al ulations)thevalidityoftheequality

0  m 2 i Y j=0 1 E n ; n +j [p℄ 1 p  1 A 0  i Y j=0 E n +j; n +m 1 i [p℄ p  1 A = = 0  i Y j=0 E n+j; n [p℄ p  1 A 0  m 2 i Y j=0 1 E n+i+1; n+j [p℄ 1 p  1 A ;

orrespondingto,respe tively,(17)and(18)ofii).Letusremarkthat,asweshall

seebelow,formula(17)(respe tively,(18))isobtainedbyalways hoosingstrategy

(10)(respe tively,(11))atea hstepnwhere,forsomei2f0;:::;N n 2g,we

have E n+i; n+N n 1 i [p℄<p  <E n+i+1; n+N n 2 i [p℄:

This hoi e will be possible sin e, under the above ondition for p 

, both the

strategies (10) and (11), as well as any admissible strategy between them, will

beoptimal.Infa t,otherrepresentationformulasfor J

n

, di erentfrom (17)and

(18),but equivalentto them, ouldbepossible,ea h ofthem orrespondingto a

di erentpro edureforsele tingaminimizing strategybetween(10)and(11).

From(8), wehave J n 1 (S n 1 ;V n 1 )= inf n 1 E n 1; n 1 [p℄J n (S n 1 u;V n (V n 1 ;S n 1 ;u; n 1 ))+ +(1 E n 1 ; n 1 [p℄)J n ( S n 1 d;V n (V n 1 ;S n 1 ;d; n 1 ));

whi himplies,byindu tion,

(11)

where( hoosing e.g.(17)whentakingintoa ountalternativeii)) A( ; )= 0  N n 1 Y j=0 1 E ; +j [p℄ 1 p  1 A 1 fp  <E ; +N n 1 [p℄g + + N n 2 X k =0 0  N n 2 k Y j=0 1 E ; +j [p℄ 1 p  1 A 0  k Y j=0 E +j; +N n 1 k [p℄ p  1 A  1 fE +k ; +N n 1 k[p℄<p  <E +k +1; +N n 2 k[p℄g + + 0  N n 1 Y j=0 E +j; [p℄ p  1 A 1 fp  >E +N n 1; [p℄g ;

Proof of part i). Ifp  <E n 1 ; n 1 +N n

[p℄then it isnot diÆ ultto he kthat

onlythe rstindi atorfun tionsinbothA(

n 1 +1; n 1 )andA( n 1 ; n 1 +1)

areequalto one,whilealltheothersareequaltozero.Therefore

J n 1 (S n 1 ;V n 1 )= inf n 1 E n 1 ; n 1 [p℄ 0  N n 1 Y j=0 1 E n 1+1; n 1+j [p℄ 1 p  1 A  [V  n (S n 1 u) V n 1 n 1 S n 1 (u 1)℄ + + + 1 E n 1; n 1 [p℄  0  N n 1 Y j=0 1 E n 1 ; n 1 +j+1 [ p℄ 1 p  1 A  [V  n (S n 1 d) V n 1 n 1 S n 1 (d 1)℄ + : (21)

Thesignoftheslopeofthislinearexpressionin

n 1 isgivenby E n 1 ; n 1 [p℄ 0  N n 1 Y j=0 1 E n 1 +1; n 1 +j [p℄ 1 p  1 A (1 u)+ (22) + 1 E n 1; n 1 [p℄  0  N n 1 Y j=0 1 E n 1 ; n 1 +j+1 [p℄ 1 p  1 A (1 d):

Aftermakingelementarymanipulations, one anrewrite(22)astheprodu tofa

suitable stri tly positiveterm and (p 

E

n 1; n 1+N n

[p℄),whi h is less than

zero by assumption. Therefore, using thesame arguments asthose in theproof

(12)

Proofof partiii). ItisnotdiÆ ultto arguethattheproofof iii)pro eeds along

thesamelinesastheproofofi).Weonlyobservethatifp  >E n 1 +N n; n 1 [p℄

(alternativeiii)forn 1)thenonlythelastindi atorfun tionsinbothA(

n 1 + 1; n 1 ) and A( n 1 ; n 1

+1) are equal to one, while all the others are equal

tozero.Moreover,thesignoftheslopeofthelinearexpressionwhi harisesfrom

the DP algorithm is positive,so that the in mum is a hieved at

n 1

given by

formula(11),i.e.satisfying

V  n (S n 1 d) V n 1 n 1 S n 1 (d 1)=0:

Proofofpartii).Nowweonlyhavetoproveii).Asweshallsee,thelinear

expres-sionsarising from theDP algorithm will haveangular oeÆ ientsequal to zero,

giving thepossibility to hoose, as minimizing strategy, any admissible strategy

between (10) and (11). In parti ular, hoosing strategy (10) we shall prove the

validityof formula(17).If E n 1 +i; n 1 +N n i [p℄<p  <E n 1 +i+1; n 1 +N n 1 i [p℄;

forsomei2f0;:::;N n 1g,thenthelastindi atorfun tioninA(

n 1 +1;

n 1 )

andthe rstindi atorfun tionin A(

n 1 ;

n 1

+1)areequalto zero,i.e.

1 fp  >E n 1 +N n; n 1 [p℄g 0 and 1 fp  <E n 1 ; n 1 +N n [p℄g 0

Moreover,wehaveto distinguishbetweenthefollowingthreealternatives:

1. Ifi=0,thenwehave J n 1 (S n 1 ;V n 1 )= inf n 1 E n 1 ; n 1 [p℄ 0  N n 1 Y j=0 1 E n 1+1; n 1+j [p℄ 1 p  1 A  [V  n (S n 1 u) V n 1 n 1 S n 1 (u 1)℄ + + + 1 E n 1; n 1 [p℄  0  N n 2 Y j=0 1 E n 1 ; n 1 +1+j [p℄ 1 p  1 A   E n 1 ; n 1 +N n [p℄ p  [V  n (S n 1 d) V n 1 n 1 S n 1 (d 1)℄ + :

Sin etheslopeofthislinearexpressionin

n 1

isequaltozero(weomit

al ulations),thein mumisa hievedatboth(10)and(11)aswellasat

anyadmissiblestrategybetweenthem. Choosinge.g.(10)weobtain

(13)

whi hgivesformula(17)withnrepla edbyn 1andi=0. 2. Ifi2f1;:::;N n 2g,thenwehave J n 1 (S n 1 ;V n 1 )= inf n 1 E n 1; n 1 [p℄ 0  N n 1 i Y j=0 1 E n 1+1; n 1+j [p℄ 1 p  1 A  0  i 1 Y j=0 E n 1+1+j; n 1+N n i [p℄ p  1 A  [V  n (S n 1 u) V n 1 n 1 S n 1 (u 1)℄ + + + 1 E n 1 ; n 1 [p℄  0  N n 2 i Y j=0 1 E n 1 ; n 1 +1+j [ p℄ 1 p  1 A   0  i Y j=0 E n 1+j; n 1+N n i [p℄ p  1 A [V  n (S n 1 d) V n 1 n 1 S n 1 (d 1)℄ + :

Theslopeofthislinearexpressionin

n 1

isagainequaltozero.Choosing

e.g.(10)as n 1 weobtain J n 1 (S n 1 ;V n 1 )= 0  N n 1 i Y j=0 1 E n 1; n 1+j [p℄ 1 p  1 A   0  i Y j=0 E n 1+j; n 1+N n i [p℄ p  1 A  V  n 1 (S n 1 ) V n 1  + ;

whi hgivesformula(17)withnrepla edbyn 1andi2f1;:::;N n 2g.

(14)

Theslopeofthislinearexpressionin

n 1

isagainequaltozero.Choosing

e.g.(10)as n 1 weobtain J n 1 (S n 1 ;V n 1 )= 1 E n 1; n 1 [p℄ 1 p    0  N n 1 Y j=0 E n 1+j; n 1+1 [p℄ p  1 A  V  n 1 (S n 1 ) V n 1  + :

Theindu tivestepis omplete,andsoistheproof. 

Corollary5.2. Lettingn=0inTheorem5.1,theminimaldis ountedshortfallrisk

is i) if p  <E 0 ; 0 +N 1 [p℄, then J 0 (S 0 ;V 0 )= 0  N 1 Y j=0 1 E 0; 0+j [p℄ 1 p  1 A [ V  0 (S 0 ) V 0 ℄ + ; ii) ifE 0+i; 0+N 1 i [p℄<p  <E 0+i+1; 0+N 2 i [p℄,wherei=0;:::;N 2, then J 0 (S 0 ;V 0 )= 0  N 2 i Y j=0 1 E 0 ; 0 +j [p℄ 1 p  1 A   0  i Y j=0 E 0+j; 0+N 1 i [p℄ p  1 A [V  0 (S 0 ) V 0 ℄ + = = 0  i Y j=0 E 0+j; 0 [p℄ p  1 A 0  N 2 i Y j=0 1 E 0+i+1; 0+j [p℄ 1 p  1 A [V  0 (S 0 ) V 0 ℄ + ; iii) if p  >E 0+N 1; 0 [p℄, then J 0 (S 0 ;V 0 )= 0  N 1 Y j=0 E 0 +j; 0 [p℄ p  1 A [V  0 (S 0 ) V 0 ℄ + :

The minimizing investmentstrategiesarethose of Theorem 5.1.

Remark5.3. Inthe asewhenpisunknownweobtainformulasthataresimilarto

thoseofthe asewhenpisknown,withthefollowingdi eren e:whileinthe ase

whenpis known weknowimmediately whether p>p 

ornot,and that relation

either holdsat alltimesn ordoesnothold, whenpisunknown we annot know

whetherp>p 

ornot,sowemustusetheBayesestimatorsofp.Obviouslythese

estimators hangeovertime,soweobtainprodu tsofdi erentfa torsdepending

onp 

andonthe Bayesestimatorsof p,whilein the asewhenpis knownthese

fa torsareallequaleithertop=p 

orto (1 p)=(1 p 

(15)

theBayesestimatorsof penter linearlyin the DPalgorithm anddo notmodify

theoptimum.

Remark5.4. Also when p is unknown, if we impose V  0,we obtain di erent

results:in fa t in this asetooour strategysu eeds in repli ating perfe tly the

laim in all thestates of nature but one (see [7℄);with the onstraintV 0,it

mayhappenthatthisisnotpossible,sothattheoptimalsolutiongivesashortfall

higherthaninour ase.However,alsointhis ase(asinthe asewhenpisknown),

if V

0

is near V 

0

, then V remains positiveat all times prior to the maturity N,

so the two strategies that we obtain by imposing or not the onstraint V  0

oin ide.

Referen es

[1℄ S.Browneand W.Whitt, Portfolio hoi e and the Bayesian Kelly riterion,Adv.

Appl.Prob.,28(1996),1145{1176.

[2℄ J. Cvitani , Methods of partial hedging, Asia-Pa i Finan ial Markets, 6 (1999),

7{35.

[3℄ J.Cvitani ,Minimizingexpe tedlossofhedginginin ompleteand onstrained

mar-kets,SIAMJ.Control Optimization,38(4)(2000),1050{1066.

[4℄ J.Cvitani andI.Karatzas,Ondynami measuresofrisk,Finan eandSto hasti s,

3(4)(1999),451{482.

[5℄ J.Cvitani andG.Spivak,Maximizingtheprobabilityofaperfe thedge,TheAnnals

ofAppliedProbability,9(4)(1999),1303{1328.

[6℄ C.Edirisinghe,V.Naik andR.Uppal,Optimal repli ationofoptions with

transa -tions osts and trading restri tions, J. of Finan ial and Quantitative Analysis, 28

(1993),117{138.

[7℄ G.Favero,Shortfallriskminimizationundermodelun ertaintyinthebinomial ase:

adaptiveandrobustapproa hes,Preprint,UniversityofPadova,DepartmentofPure

andAppliedMathemati s(2000)

[8℄ H.FollmerandP.Leukert,Quantile hedging,Finan eandSto hasti s,3(3)(1999),

251{273.

[9℄ H.FollmerandP.Leukert,EÆ ienthedging: ostversusshortfallrisk,Finan eand

Sto hasti s, 4(2)(2000),117{146

[10℄ P. Guasoni, Risk minimizationunder transa tion osts, Preprint, S uola Normale

SuperiorediPisa,(1999).

[11℄ I. Karatzas and X. Zhao, Bayesian adaptive portfolio optimization, to appear in

Advan esinMathemati alFinan e,CambridgeUniversityPress, (1999).

[12℄ P.Lakner, Utilitymaximization withpartial information,Sto hasti Pro essesand

theirAppli ations,56(1995),247{273.

[13℄ P.Lakner,Optimal trading strategyforaninvestor:the aseofpartialinformation,

Sto hasti Pro essesandtheirAppli ations,76(1998),77{97.

(16)

[15℄ M.MusielaandM.Rutkowski,Martingalemethodsin nan ialmodelling,Springer,

Berlin,1997.

[16℄ H. Pham, Dynami L p

-hedging in dis rete time under one onstraints, SIAM J.

ControlOptimization,38(3)(2000),665{682.

[17℄ W.J.RunggaldierandA.Za aria,Asto hasti ontrolapproa htoriskmanagement

underrestri ted information,Mathemati alFinan e,10(2000),277{288.

[18℄ K.M.VanHee,Bayesian ontrolofMarkov hains,Mathemati alCentreTra ts95,

Amsterdam,1978.

DepartmentofPureandAppliedMathemati s

UniversityofPadova

viaBelzoni7

I-35131Padova,Italy

E-mailaddress:runggalmath.unipd.it

barbaramath.unipd.it

vargiolumath.unipd.it

Figure

Updating...

References

Related subjects :