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,
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
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
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:
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.
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
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
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,
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
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,
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
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
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.
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
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.
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DepartmentofPureandAppliedMathemati s
UniversityofPadova
viaBelzoni7
I-35131Padova,Italy
E-mailaddress:runggalmath.unipd.it
barbaramath.unipd.it
vargiolumath.unipd.it
