AND
INTERTEMPORAL PRICE FOR RISK
JOHANNES LEITNER
JOHANNES.LEITNER@UNI-KONSTANZ.DE
CENTER OF FINANCE AND ECONOMETRICS (COFE)
UNIVERSITY OF KONSTANZ
Abstract. In a continuous time, arbitrage free, non-complete
market withazerobond, wend theintertemporalpriceforrisk
to equalthe standarddeviation of the discounted variance opti-
malmartingalemeasuredividedbythezerobondprice. Weshow
theHedgingNumerairetoequaltheMarketPortfolioandndthe
mean-varianceeÆcientportfolios.
Keywords: CAPM,MarketPortfolio, Sharpe-Ratio,Pricefor
Risk,Mean-VarianceEÆcientFrontier,HedgingNumeraire.
AMS91Classications: 90A09,90A10
JELClassications: G11
IwouldliketothankProfessorM.Kohlmannforhissuggestionsandsupport.
Research supported by the Center of Finance and Econometrics, ProjectMathe-
maticalFinance.
Introduction
Usingtheequivalentmartingalemeasureapproach,seeHarrisonand
Pliska (1981),Delbaen and Schachermayer (1994),westudy the prob-
lem of nding mean-variance eÆcient portfolios which is closely re-
latedto thenotion ofa pricefor risk. Originally,therst probabilistic
(single-period) market models (CAPM) were based on the idea of a
priceforriskandthe notionofmean-varianceeÆciency, seeMarkowitz
(1952, 1987),Sharpe(1964), Lintner (1965),Jensen (1972), Elton and
Gruber (1979). See also Li and Ng (2000) for a multi-period model.
The notion of a price for risk often appears in connection with the
equivalentmartingale measure.
In the most generalcase, where we consider a (not necessarily con-
tinuous but locallyS 2
)semimartingalemarketmodel, the centralidea
for solving this problemis just the notionof orthogonality,whereas in
the case of a continuous semimartingalemarket model, where we will
derive stronger results, the centralidea comes from stochastic duality
theory,whichgoesbacktoBismut(1973,1975). Foranalternativeap-
proach based onBSDEtheory,see Zhouand Li (2000),Limand Zhou
(2000).
After sometechnical preparationsinSection 1,we introducein Sec-
tion 2 dierent spaces of equivalent local martingale measures and
several spaces of self-nancing hedging strategies. In Section 3 we
provetheexistenceofthediscounted variance-optimalmartingalemea-
sureandthe hedgingnumeraireforalocallyS 2
-semimartingalemarket
model, see Gourieroux, Laurent and Pham (1998), (GLP98), for this
result inthe continuous case.
In Section 4 we generalize the classical single-period results about
mean-variance eÆcient portfolios to the case of a general semimartin-
galemarketmodel. Assumingtheexistenceofzerobondsinthemarket,
the mean-varianceeÆcient portfolios are shown to be linear combina-
tions of the hedgingnumeraire and the zero bond. We nd the mean-
variance eÆcient market linefor a xed time horizon to be a straight
line with a slope equal to the standard deviation of the discounted
variance-optimal martingale measure divided by the zero bond price.
This quantity can be interpreted as a price for intertemporalrisk. In
Section 5 we develop a conditional version of these results in the case
of a continuous semimartingale market model. This allows to dene
the term structure of the intertemporal price for risk and itsdynamic.
In Section6 we discussan applicationof our results tothe problemof
pricingnon-attainableclaims and the relationof the term structure of
interestrates tothetermstructure ofthe intertemporalprice forrisk.
1. Self-financing Hedging Strategies
Let a ltered probability space
1
:= (;F;(F
s )
s0
;P), satisfying
theusualconditionsbegiven. ForsimplicityweassumeF
0
tobetrivial
up to sets of measure 0 with respect to P and F
1
= F
1
:=F. For
a process X and a map : !
R
+
, denote the stopped process at
time by X
. We will often restrict a semimartingale X on
1 to
an interval [t;T]; 0 t T < 1, resp. to [t;1). Therefore we
introduce the following lteredprobability space (again satisfying the
usual conditions),
[t;T] :=
;F
T
;
F [t;T]
s
s0
;P
jF
T
for all 0 t
T 1, t < 1, where F [t;T]
s
:= F
t_s^T
for 0 s < 1. The process
X [t;T]
s
:=X
t_s^T
is then a semimartingaleon
[t;T]
. However, on[t;T]
we oftenwrite X instead of X [t;T]
. Set
T :=
[0;T] .
Dene L 2
(
[t;T]
), resp. L 2
t (
[t;T]
), as the set of F
T
-measurable ran-
dom variables X, such that E[X] < 1 a.s., resp. E
t
[X] < 1 a.s.,
whereE
t
[]:=E[jF
t
] denotes the generalized conditionalexpectation.
The conditional variance isdenoted by Var
t ().
The stochastic exponential of a semimartingale X is denoted as
E(X). As ageneralreferencewe citeJacod andShiryaev (1987),(J&S
87) and Jacod (1979). Denote the set of predictable processes which
are locally integrable, resp. locally Riemann-Stieltjes integrable, with
respect to a local martingale M, resp. with respect to a process A of
nite variation,by L 1
l oc
(M),resp. by L 1
l oc
(A). If thesemimartingaleX
admitsadecompositionX =X
0
+A+M,whereM isalocalmartingale
andAisaprocessofnitevariationthenL 1
l oc
(X):=L 1
l oc
(M)\L 1
l oc (A).
We can now dene the market model: Let S = (S
t )
0t<1
be a R d
-
valued semimartingale. M := (
1
;S) = ((;F;(F
s )
s0
;P);S) is a
model for a market, where S describesthe price processes of d assets.
Wewilloftenconsider suchamarketonaninterval[t;T]; 0t<T <
1. This isequivalentto work with the following marketmodel M
[t;T]
dened by M
[t;T]
:=
[t;T]
;S [t;T]
. Set M
T
:=M
[0;T] .
We want to model the economic activity of investing money into a
portfolio of assets and changing the number of assets held over time
according to a certain strategy. This is achieved with the following
denition:
Denition 1.1. AhedgingstrategyinthemarketMisaH 2L 1
l oc (S).
H H
The gains process of H is dened as the semimartingaleG :=HS.
H is called self-nancing if V H
= V H
0 +G
H
, i.e. H
t S
t
= H
0 S
0 +
R
t
0 H
s dS
s
;8t0. Denote thespaceofallself-nancinghedgingstrate-
gies in Mby SF(M).
Note that for H 2SF(M), we have H [t;T]
2SF(M
[t;T]
). The idea
of a self-nancing hedging strategy is that the changes over time of
thecorrespondingvalue processaresolelycausedbythechangesofthe
value ofthe assetsheldintheportfolioandnot bywithdrawing money
fromor adding money tothe portfolio.
Denition 1.2. A semimartingaleB such that B and B are strictly
positiveiscalledanumeraireforthemarketM. Themarketdiscounted
withrespecttoB isthendened asM B
:=
1
;S B
,whereS B
:=
S
B .
For 0 t T < 1, the market restricted to the interval [t;T] is
dened asM B
[t;T]
:= M B
[t;T]
=
[t;T]
; S B
[t;T]
.
Note that for a numeraire B, B 1
is a numeraire too and S B
is
a semimartingale.The following result is well known, see Geman, El
Karui and Rochet (1995)and Goll and Kallsen(2000):
Proposition 1.3. Let B be a numeraire for the market M. Then
SF(M B
)=SF(M) holds.
2. Arbitrage-free Markets
We consider in this section the market
M:= (
1
;
S), where
S :=
(S;B) is R d
R-valued and B is a numeraire,with B
0
=1, which we
assume to be uniformly bounded and uniformly bounded away from
0 on nite intervals. For 0 t T 1;t < 1, denote the set
of uniformly integrable, resp. local martingales, living on by
L u
(
[t;T]
),resp. byL(
[t;T]
). Working inthemarket
M
[t;T]
oron
[t;T]
the process S
B
[t;T]
willbe denoted asS
. Dene the followingsets of
localmartingale measures:
D(
M
[t;T] ):=
Z 2L(
[t;T]
)jZ1
[0;t]
=1;Z >0;S
Z 2L(
[t;T] ) ; (2.1)
D e
(
M
[t;T] ):=
Z 2D(
M
[t;T]
)jZ uniformlyintegrablemartingale ; (2.2)
and
D 2
(
M
[0;T]
) :=
Z 2D(
M
[0;T] )jZ
T 2L
2
(
[0;T] ) ; (2.3)
D 2
t (
M
[t;T] ) :=
Z
t_
Z
t
jZ 2D 2
(
M
[0;T] )
: (2.4)
We willnow introduce a space of simple self-nancing hedgingstrate-
gies, following closely theideas of Delbaenand Schachermayer(1996a,
1996b), (DS96a, DS96b). Assume S
to be locally in L 2
(
[0;T] ) in
the sense, that there exists a sequence U
n
;n 2 N of localizing stop-
ping times increasing to innity such that for each n 2N, the family
fS
j stoppingtime; U
n
g is bounded in L 2
(
[0;T]
). This condi-
tion is certainly satised for locally bounded or continuous S. De-
note by H (
[t;T]
) the set of R d
-valued predictable processes which are
a linear combination of processes of the form H = h1
]
1
;
2 ]
, where
t
1
2
T are stopping times dominated by some U
n and
h1
f
1
<
2 g
is a bounded F
1
-measurable random variable. Dene the
followingspace of semimartingales:
K
(
M
[t;T] ):=
HS
jH2H (
[t;T] ) ; (2.5)
and the corresponding space of terminalvalues
K
T (
M
[t;T] ):=
V
T
jV 2K
(
M
[t;T]
) L 2
(
[t;T] ):
(2.6)
Every H 2 H (
[t;T]
) can be extended to an
^
H 2 SF(
M
[t;T] ) with
V
^
H
0
= 0, hence K
(
M
[t;T]
) is just the space of discounted gains pro-
cesses G
^
H
B [t;T]
for such
^
H. ForH =h1
]1;2]
2H (
[t;T]
)asabove, we nd
HS
=h(S
2
S
1
)and for astoppingtime wehave (HS
)
=
h(S
2
^
S
1
^
)=
^
HS
, where
^
H :=h1
]
1
^;
2
^]
2H (
[t;T]
), since
h1
f
1
^<
2
^g
=h1
f
1
<
2 g
1
f
1
<g isF
1
^
-measurable. HenceK
(
M
[t;T] )
is stable under stopping. Dene the followingspace of uniformlyinte-
grable martingales:
D s;2
(
M
[t;T]
):=fE[ZjF
]
Z 2 L 2
(
[t;T]
);E[Z]=1;
E[V Z]=0; 8V 2K
(
M ) ;
(2.7)
and the corresponding space of terminalvalues
D s;2
T (
M
[t;T] ):=
Z
T
jZ 2D s;2
(
M
[t;T]
) : (2.8)
NotethatD s;2
T (
M
[t;T]
)isclosed. Since K
(
M
[t;T]
)isstable understop-
ping, we nd for V 2K
(
M
[t;T]
) and Z 2D s;2
(
M
[t;T]
)and a stopping
time 0 < 1, E[V
T Z
T
] = 0 = E[V
T Z
T
] = E[E[V
Z
T jF
]] =
E[V
Z
], hence VZ is a uniformly integrable martingale, see J&S 87,
Lemma I.1.44. Let Z 2 L 2
(
[t;T]
) with E[Z] = 1 be such that S
Z
is a local martingale. We want to show that VZ is a uniformly in-
tegrable martingale for all V 2 K
(
M
[t;T]
). For this it suÆces to
show that S
1
Z is uniformly integrable for every stopping time
1
bounded by some U
n
. We show uniform integrability of the family
fS
1
Z
j stoppingtime g. Wehave
lim
K!1 E
h
1
fjS
1
Zj>Kg i
lim
K!1 E
h
1
fjS
1
j>
p
Kg i
+E h
1
fjZj>
p
Kg i
= lim
K!1 E
h
1
fjS
1
j
2
>Kg i
=0;
since S
1
= S
^
1
, ^
1
U
n
and by the assumed boundedness
of fS
j stopping time; U
n
g in L 2
(
[0;T]
). Conversely, for Z 2
D s;2
(
M
[t;T]
), itiseasilyseen thatS
Z is alocalmartingale. Therefore
D s;2
(
M
[t;T]
) equalsthe set of signed localmartingalemeasures for the
market
M
[t;T] .
We willwork with the following No-Arbitragecondition:
D 2
(
M
T
)6=;; 8T <1:
(2.9)
It implies that
D 2
t (
M
[t;T]
)6=;; 80tT <1:
(2.10)
Note that for Z 2 D 2
t (
M
[t;T]
) and t t 0
T 0
T, we have Z
[t 0
;T 0
]
Z
t 0
2
D 2
t 0 (
M
[t 0
;T 0
]
). Note also that D 2
0 (
M
[0;T]
) = D(
M
[0;T]
), since F
0 was
assumed to be trivial.
Let B SF(
M
[t;T]
). We call a H 2 B an B-arbitrage, if V H
0
= 0,
V H
T
0 and V H
T
6= 0 almost surely. If there exists no B-arbitrage,
then B is called arbitrage-free. In all probabilistictheories of nancial
markets allowingto tradeat aninnitely large number of instances of
time one has to exclude certain self-nancing hedging strategies, e.g.
doublingstrategies, inorder to avoidarbitrage opportunities. We will
dene several arbitrage-free subsets of SF(
M
[t;T] ):
1.
SF b
(
M ):=
H 2SF(
M )j9K 2R :V H
K : (2.11)
Let H 2 SF b
(
M
[t;T] ), V
H
0
= 0 and V H
K.
V H
B [t;T]
Z is then
a supermartingale for all Z 2 D e
(
M
[t;T]
) and V H
T
0 implies
E
t h
V H
T
B
T Z
T i
V
H
t
B
t
=0, henceV H
T
=0.
2. ForD 2
t (
M
[t;T]
)6=;, (see DS96b):
SF 2
(
M
[t;T] ) :=
H 2SF(
M
[t;T]
)jV H
T 2L
2
(
[t;T] );
V H
B [t;T]
Z 2L u
(
[t;T]
);8Z 2D 2
(
M
[t;T] )
; (2.12)
resp.
SF 2
t (
M
[t;T] ) :=
H 2SF(
M
[t;T]
)jV H
T 2L
2
t (
[t;T] );
V H
B [t;T]
Z 2L u
(
[t;T]
);8Z 2D 2
t (
M
[t;T] )
: (2.13)
3.
SF s;2
(
M
[t;T] ) :=
H 2SF(
M
[0;T]
)jV H
T 2L
2
(
[t;T] );
V H
B [t;T]
Z 2L u
(
[t;T]
);8Z 2D s;2
(
M
[t;T] )
: (2.14)
4.
SF 0
(
M
[t;T]
):=SF 0
t (
M
[t;T] ):=
H 2SF(
M
[t;T] )jV
H
0 : (2.15)
5. ForD 2
(
M
[t;T] )6=;
SF sup;2
t (
M
[t;T] ) :=
H 2SF(
M
[t;T] )
E
s
V H
T
B
T Z
T
V
H
s
B
s Z
s
;
8ts T;Z 2D 2
t (
M
[t;T] )
: (2.16)
6. ForD s;2
(
M
[0;T] )6=;
SF sup;s;2
(
M
[t;T] ) :=
H 2SF(
M
[t;T] )
E
s
V H
T
B
T Z
T
V
H
s
B
s Z
s
;
8ts T;Z 2D s;2
(
M
[t;T] )
: (2.17)
7. ForS 2S
l oc (
[t;T] )
G 2
(
M
[t;T] ):=
H 2SF(
M
[t;T]
)jV H
2S 2
(
[t;T] ) ; (2.18)
where S 2
(
[t;T]
) denotes the space of L 2
-integrable semimartin-
gales,seeDelbaen,Monat,Schachermayer, SchweizerandStricker
(1997) (DMSSS97).
Dene for F
t
-measurable v
A 2
v (
M
[t;T] ):=
V H
T
B
T
H 2SF 2
(
M
[t;T] );
V H
t
B
t
=v
; (2.19)
and
K 2
v (
M
[t;T] ):=
V H
T
B
T
H 2SF s;2
(
M
[t;T] );
V H
t
B
t
=v
: (2.20)
Denoteby
K
T (
M
[t;T]
)theclosureofK
T (
M
[t;T] )inL
2
(
[0;T]
)anddene
A
T (
M
[t;T] ):=K
T (
M
[t;T] ) L
2
+ (
[t;T] )\K
T (
M
[t;T] )+L
2
+ (
[t;T] ):
(2.21)
InDS96b, Theorem 1.2,Theorem 2.2, thefollowingwasshown usinga
results fromYor(1978):
Theorem 2.1. Under the above assumptions we have
A
T (
M
[t;T]
) = A 2
0 (
M
[t;T] )
=
V 2L 2
(
[t;T]
)jE[VZ]=0; 8Z 2D 2
(
M
[t;T] ) :
Inparticular, A 2
0 (
M
[t;T]
)isclosedin L 2
(
[t;T]
). Furthermore, forcon-
tinuous
S, we have
A
T (
M
[t;T] )=
K
T (
M
[t;T] ).
We willprove the following corollary:
Corollary 2.2. Under the above assumptions we have
K
T (
M
[t;T]
) = K 2
0 (
M
[t;T] )
=
V 2L 2
( )jE[VZ ]=0; 8Z 2D s;2
(
M ) :
Proof. For V 2
K
T (
M
[t;T]
) choose a sequence V n
2 K
T (
M
[t;T] ) con-
verging to V in L 2
(
[t;T]
). Since E[V n
Z
T
] = 0; 8Z 2 D s;2
(
M
[t;T] )
and since the inclusion L 2
(
[t;T]
) ,! L 1
(
[t;T]
;Q Z
), where Z
T
= dQ
Z
dP ,
is continuous for Z 2 D s;2
(
M
[t;T]
) by Cauchy-Schwarz inequality, we
nd E[VZ
T
] = 0; 8Z 2 D s;2
(
M
[t;T]
). Conversely, if V 2 L 2
(
[t;T] )
and E[VZ
T
] = 0; 8Z 2 D s;2
(
M
[t;T]
) and V 62
K
T (
M
[t;T]
), then V 62
span
K
T (
M
[t;T]
);1 , hence by the Hahn-Banach theorem and since
1 62
K
T (
M
[t;T]
), we nd an Z 2 D s;2
(
M
[t;T]
) such that E[VZ
T ] 6= 0,
a contradiction. Since
K
T (
M
[t;T] )
A
T (
M
[t;T]
), we nd by Theorem
2.1for V 2
K
T (
M
[t;T]
)a H 2SF 2
(
M
[t;T]
) with V H
0
=0and V
H
T
B
T
=V.
Wealsonda sequence
~
V n
2K
T (
M
[t;T]
)converging toV in L 2
(
[t;T] )
and a sequence V n
2K
(
M
[t;T]
) with V n
T
=
~
V n
. For Z 2 D s;2
(
M
[t;T] )
and 0sT we have for n;m!1
E[ jV n
s Z
s V
m
s Z
s
j] = E
E[V n
T Z
T jF
s
] E[V m
T Z
T jF
s ]
E
E
jV n
T Z
T V
m
T Z
T j
F
s
= E[jV n
T Z
T V
m
T Z
T
j]!0;
hence V n
s Z
s
is a Cauchy-sequence in L 1
(
[t;T]
). For
~
Z 2 D 2
(
M
[t;T] )
and 0 s T, we therefore know that V n
s
is a Cauchy-sequence
in L 1
(
[t;T]
;Q
~
Z
) converging to V H
and doing so Q
~
Z
-a.s. and P-a.s.
pointwise fora subsequence. Since for n!1
E[jV n
s Z
s
E[VZ
T jF
s
] j] = E[jE[V n
T Z
T jF
s
] E[VZ
T jF
s ] j]
E
E
jV n
T Z
T
VZ
T j
F
s
= E[jV n
T Z
T
VZ
T
j]!0;
we nd V n
s Z
s
! V H
s Z
s
, hence E[VZ
T jF
s ] = E
V H
T Z
T jF
s
= V H
s Z
s .
This proves
K
T (
M
[t;T] )K
2
0 (
M
[t;T]
). The corollaryfollowsnowfrom
the obvious inclusion
K 2
0 (
M
[t;T] )
V 2L 2
(
[t;T]
)jE[VZ
T
]=0; 8Z 2D s;2
(
M
[t;T] ) ;
and the rst part of the proof.
Wehave the following easy result:
Lemma 2.3. Assume D 2
t (
M
[t;T]
) 6= ; and
S to be continuous. Then
G 2
(
M
[t;T]
)SF 2
t (
M
[t;T]
). In particular G 2
(
M
[t;T]
) is arbitrage-free.
3. The Discounted Variance-optimal Martingale Measure
We will need the discounted variance-optimal martingale measure,
introducedinGPL98forthecaseofacontinuouspriceprocess
S,inthe
locally S 2
-semimartingale setting. We generalize the proof of Lemma
2.1. part (c), in DS96ato our situation:
Lemma 3.1. Assume D s;2
(
M
[t;T]
) 6= ;. Then there exists a unique
element Z opt;t;T
2 D s;2
(
M
[t;T]
) such that Z
opt;t;T
T
B
2 B
T K
2
v t;T
(
M
[t;T] ),
where
v t;T
=E 2
4 Z
opt;t;T
T
B
T
!
2 3
5
= inf
Z2D s;2
(
M
[t;T] )
E
"
Z
T
B
T
2
#
>0:
(3.1)
Furthermore, there exists a H opt
2 SF s;2
(
M
[t;T]
) with corresponding
value process V opt;t;T
:=V H
opt
such that V opt;t;T
0
=1,
V opt;t;T
T
= Z
opt;t;T
T
v t;T
B
t B
T (3.2)
and
E
V opt;t;T
T
2
= inf
V2B
T K
2
B 1
t (
M
[t;T] )
E
V 2
: (3.3)
Proof. By the uniform boundedness of B and B 1
on [t;T], we nd
the sets D := B 1
T D
s;2
T (
M
[t;T]
) and K := B
T span
K 2
0 (
M
[t;T]
);1 to
be closed in L 2
(
[t;T]
). Therefore we nd a Z
min
B
T
2 D with minimal
norm and a representation Z
min
B
T
= Z
1 +Z
2
, where Z
1
2 K and Z
2 2
K
?
, since L 2
(
[t;T] )
=
K K
?
. Denote by < ; > the standard
linear product of the Hilbert-space L 2
(
[t;T]
). From < Z
2
;K >= 0 it
follows E[Z
1 B
T
] = 1 and < Z
1
;B
T K
2
0 (
M
[t;T]
) >= 0, thus Z
1 2 D.
Since
Z
min
B
T
2
=kZ
1 k
2
+kZ
2 k
2
was minimal it follows Z
2
= 0, hence
Z min
B
T
2 K , i.e. there exists a H 2 SF s;2
(
M
[t;T]
) such that V H
T
=
Z min
B
T
. Dene Z opt;t;T
:= E[Z min
jF
]. Uniqueness follows from the
strictconvexityofkk 2
. WehaveE
t
Z opt;t;T
T
B
T
2
=E
t h
V H
T
B
T Z
opt;t;T
T i
=
V H
t
B
t Z
opt;t;T
t
. By construction v t;T
:=
V H
t
B
t
is deterministic, hence (3.1)
follows. Set H opt
:=
H
v t;T
B
and V opt;t;T
:= V H
opt
. Since V H
0
= v t;T
B
t ,
we have V
0
= 1. Let H 2 SF (
M
[t;T]
) such that V
0
= 1.
Then E h
V opt;t;T
T
(V opt;t;T
T
V H
0
T )
i
= 0, since V
opt;t;T
T V
H 0
T
B
T
2 K 2
0 (
M
[t;T] )
and B
T V
opt;t;T
T
E
[ B
T V
opt;t;T
T ]
2D s;2
T (
M
[t;T]
). Therefore V opt;t;T
T
is the element with
minimalnorm in B
T K
2
B 1
t (
M
[t;T] ).
Remark 3.2. V opt;t;T
is known as the hedging numeraire, see Gourier-
oux, Laurent and Pham (1998),(GLP98).
4. Mean-Variance Efficiency
Inthissectionweintroducearstversionoftheconstraintoptimiza-
tion problem known as the Mean-Variance EÆciency problem for the
market
M
[t;T]
, where
S is only assumed to be locally in L 2
(
[t;T] ), as
described inSection 2.
Dene K:=B
T K
2
B 1
t (
M
[t;T]
) andconsider the optimizationproblem
V(t;T;e):= inf
H2SF s;2
(
M
[t;T]
)
V H
0
=1 E
h
V H
T
2 i
= inf
V2K E
V 2
; (4.1)
under the constraint
E
V H
T
=e;
(4.2)
for e 2 R. Since K is closed in L 2
(
[t;T]
) by Corollary 2.2 and by
strict convexity, there exists a unique V t;T;e
2 K with V(t;T;e) =
E
V t;T;e
T
2
and E h
V t;T;e
T i
= e i K\ff 2 L 2
(
[t;T]
)jE[f] = eg 6=
;. By Lemma 3.1, we have V t;T;^e
t;T
= V opt;t;T
, where ^e := e^ t;T
:=
E h
V opt;t;T
T i
. Wecall V t;T;e
;e2R the mean-variance eÆcient frontier.
Wewillprove the following
Proposition 4.1. Assume the existence of an V 2 K with E[V] 6=
^ e
t;T
. Then
t;T 2
for a constant c t;T
1. c t;T
= 1 implies e^ = 0 and Var(V t;T;e
) =
V(t;T;e)^ for all e2R.
Furthermore, givenV t;T;e
for somee6=e^we have
V t;T;e
=s(e)V t;T;^e
+ 1 s(e)
V t;T;e
; 8e2R;
(4.4)
where s(e) :=
e e
^ e e
is dened in such a way that s(e)^e+(1 s(e))e=e
holds and
c t;T
=
V(t;T;e) V(t;T;^e)
(^e e) 2 (4.5)
Proof. LetV 2KwithE[V]6=e^begiven. SincesV+(1 s)V opt;t;T
T
2K
for all s 2R, we nd K\ff 2 L 2
(
[t;T]
)jE[f]=eg 6=; for all e 2R,
hence V t;T;e
exists for all e2 R. Dene for e^6=e , and s 2R,
~
V
e
(s):=
E h
sV t;T;^e
+(1 s)V t;T;e
2 i
. Since E h
V t;T;^e
T V
i
= E
V t;T;^e
T
2
for
allV 2K we have
~
V
e
(s)=(1 (1 s) 2
)V(t;T;e )^ +(1 s) 2
V(t;T;e).
Set V
e
(e):=
~
V
e
(s(e)). Wend
V
e
(e)=V(t;T;e)^ +
V(t;T;e) V(t;T;^e)
(^e e) 2
(e e)^ 2
: (4.6)
V
e
(e)is clearly apolynomialof at most second order in e with a min-
imum of V
e
(^e)= V(t;T;^e) in e.^ The assertionfollows nowif we show
V
e
(e)= V(t;T;e) for all e 2 R. Since E h
s(e)V t;T;^e
T
+(1 s(e))V t;T;e
T i
= s(e)^e +(1 s(e))e = e we have V
e
(e) V(t;T;e) and V
e (e )
V(t;T;e). Byasimplecalculation,thesetwoinequalitiesimplyV
e
(e)=
V(t;T;e) for all e;e. Calculating Var(V t;T;e
) = V(t;T;e) e 2
, which
must be non-negative, we nd c t;T
:=
(^e e) 2
1 and c t;T
= 1
to imply^e=0.
This result also allows to calculate variance optimal portfolios. Con-
sider the optimization problem
~
V(t;T;e):= inf
H2SF s;2
(
M
[t;T]
)
V H
0
=1 E
Var V H
T
= inf
V2K
E[Var(V)]; (4.7)
under the constraint
E
V H
T
=e;
(4.8)
for e2R. Since
~
V(t;T;e)=V(t;T;e) e 2
, we nd for c t;T
6=1,
min
e2R
~
V(t;T;e)=
~
V
t;T; c
t;T
c t;T
1
^ e
=V(t;T;^e) c
t;T
c t;T
1
^ e 2
: (4.9)
c t;T
=1 implies
~
V(t;T;e)=
~
V(t;T;e)^ for alle2R.
Assume now the zero bond B T
with maturity T tobe attainable in
M
[0;T]
, i.e. there exists a H 2SF s;2
(
M
[0;T]
) such that for B T
:=V H
,
B T
T
= 1 holds. This is equivalent to the existence of an almost surely
deterministic element in K . Necessarily we have B T
> 0 and B T
is
uniformly bounded. The existence of B T
together with the existence
of a V 2Kwith E[V]6=e^implies c 0;T
>1,since
^ e=E
Z opt;0;T
v 0;T
B
T
= E
h
B T
T
B
T Z
opt;0;T i
v 0;T
= B
T
0
v 0;T
>0:
(4.10)
Henceequation(4.9)impliesV(0;T;^e) c
t;T
c t;T
1
^ e 2
=0whichisequivalent
to c 0;T
(V(0;T;e)^ ^e 2
)=V(0;T;e).^ SinceV(0;T;e)^ >0we nd c 0;T
=
V(0;T;^e)
Var
( V
0;T;^e
T )
. By (3.3), wehave
V(0;T;e)^ =E
V opt;0;T
T
2
= 1
v 0;T
; (4.11)
hence
c 0;T
= v
0;T
v 0;T
(B T
0 )
2 : (4.12)
The unique risk-free self-nancing hedging strategy with initialvalue
1 is just given by V :=
1
B T
0 B
T
and the risk-free return is V
T
= 1
B T
0 .
ratio of excess expected return e over the risk-free return 1
B T
0
and the
standard deviation of the return, fore 6=
1
B T
0 :
[0;T] (e):=
e 1
B T
0
r
Var
V 0;T;e
T
= max
V2K
E[V]=e
e 1
B T
0
p
Var(V) : (4.13)
Lemma 4.2. Under the assumption of Proposition 4.1 and assuming
the existence of B T
, wehave for all e6=
1
B T
0 :
[0;T] (e)=
r
Var
Z opt;0;T
B
T
B T
0
>0:
(4.14)
Proof. The assertionfollows froman elementarycalculation using for-
mulas (4.10), (4.11) and (4.12):
Var
V 0;T;e
T
= V(0;T;e) e 2
= V(0;T;^e)+c 0;T
(e e)^ 2
e 2
= 1
v 0;T
+ v
0;T
v 0;T
(B T
0 )
2
e B
T
0
v 0;T
2
e 2
= eB
T
0 1
2
v 0;T
(B T
0 )
2
= B
T
0
2
e 1
B T
0
2
Var
Z opt;0;T
B
T
:
Lemma 4.3. If the zero bond B T
exists in
M
[0;T]
and E[V]= 1
B T
0 for
all V 2K , then Var
Z opt;0;T
B
T
=0.
Proof. Observe that Var
Z opt;0;T
B
T
= v 0;T
B T
0
2
and that 1
B T
0
=
E[V opt;0;T
T
]= B
T
0
0;T
by (3.2).
Denition 4.4. The intertemporal price for risk for maturity time T
inthe market
M
[t;T]
isdened as
[t;T] :=
r
Var
t
BtZ opt;t;T
B
T
B T
t
: (4.15)
Wehave the following result:
Theorem 4.5. Assume the existence of the zero bond B T
in
M
[0;T] .
Then the following inequality holds for all H 2 SF s;2
(
M
[0;T]
) with
V H
0
=1:
1
B T
0
[0;T] q
Var(V H
T
)E
V H
T
1
B T
0 +
[0;T] q
Var(V H
T ):
(4.16)
Inparticular,
[0;T]
=0impliesE[V H
T ]=
1
B T
0
forallH 2SF s;2
(
M
[0;T] ),
V opt;0;T
= B
T
B T
0
and the so-called Return-to-Maturity Expectation Hy-
pothesis for the zero bond price in t=0 holds:
B T
0
= 1
E[B
T ]
: (4.17)
Furthermore, if
[0;T]
6=0, then
V 0;T;e
=s(e)V opt;0;T
+ 1 s(e)
B T
B T
0
; (4.18)
where nows(e):=
v 0;T
B T
0
v 0;T
( B
T
0 )
2
e 1
B T
0
, and if
E
V H
T
1
B T
0
=
[0;T] q
Var(V H
T );
(4.19)
for a H 2SF s;2
(
M
[0;T]
) with V H
=1, then V H
=V 0;T;E
[ V
H
T ]
.
Proof. Inequality (4.16) follows from Lemma 4.2 and Lemma 4.3. If
[0;T]
= 0 we have Z
opt;0;T
T
B
T
= B T
0
a.s., since Z
opt;0;T
T
B
T
is almost surely
deterministic and E h
V opt;0;T
T i
= E h
Z opt;0;T
T
v 0;T
B
T i
= B
T
0
v 0;T
= 1
B T
0
. Hence
B T
0
1
=E h
Z opt;0;T
T
B T
0 i
=E[B
T
]. The remainingassertions followfrom
Proposition 4.1 and from the uniqueness of V 0;T;e
T
, which implies the
uniqueness of V 0;T;e
.
Remark 4.6. The hedging numeraire has turned out tobe the market
portfolio, see Markowitz (1952, 1987). See Laurent and Pham (1999)
and Leitner (2000)for explicit formulas forthe hedgingnumeraire.
Corollary 4.7. Assume theexistence of thezerobondB T
in themar-
ket
M
[0;T]
. Then for all H 2 SF sup;s;2
(
M
[0;T]
) with V H
0
= 1 and
E
V H
T
1
B T
0
, the following inequality holds:
E
V H
T
1
B T
0 +
[0;T]
q
Var(V H
T ):
(4.20)
Proof. V H
equals V 0;T;E[V
H
T ]
+V H
0
for a H 0
2 SF sup;s;2
(
M
[0;T] ) with
V H
0
0
=0and E
V H
0
T
=0. Now calculate, using (4.18):
E h
V H
T
2 i
= E
V 0;T;E[V
H
T ]
T
+V H
0
T
2
= E
V 0;T;E[V
H
T ]
T
2
+2 1 s E
V H
T
E
1
B T
0 V
H 0
T
+2s E
V H
T
E h
V opt;0;T
T V
H 0
T i
+E
V H
0
T
2
E
V 0;T;E[V
H
T ]
2
;
where the lastinequality follows from
E h
V opt;0;T
T V
H 0
T i
=E
V H
0
T
v 0;T
B
T Z
opt;0;T
T
V
H 0
0
v 0;T
=0
and s E
V H
T
0 forE
V H
T
1
B T
0 .
Remark 4.8. ThelastresultholdsalsoforH 2SF(
M
[0;T]
)withV H
0
=
1, E
V H
T
1
B T
0
and such that V
H
B [0;T]
Z opt;0;T
is a supermartingale. In
particular, if Z opt;0;T
2D 2
(
M
[0;T]
),then the above result holds forall
H 2SF 2
(
M
[0;T]
) with V H
0
=1.
Remark 4.9. Theresults of thissection hold inparticularfor the orig-
inalone-step CAPMand its multi-periodgeneralizations.
Inthe next sectionwe willderivefor acontinuousprice process
S sim-
ilarresults forthe market
M
[t;T]
using astochastic duality approach.
5. The Conditional Price for Intertemporal Risk
Let
S becontinuous. Fix0tT <1,andassumethezerobond
B T
maturing at time T to be attainable in
M
[t;T]
, i.e. there exists a
H 2 SF 2
t (
M
[t;T]
) such that V H
T
= 1 almost surely. In this section we
want tosolvethe optimizationproblem
V(t;T;e;B):=essinf H2B
V H
0
=1 E
t h
V H
T
2 i
; (5.1)
where B 2
SF 2
t (
M
[t;T] );SF
sup;2
t (
M
[t;T] );G
2
(
M
[t;T]
) , under the con-
straint
E
t
V H
T
=e;
(5.2)
for anF
t
-measurable random variablee.
Since F
0
was assumed to be trivial,we known that Z opt0;T
0
= 1. In
the continuous case we also know that Z opt;0;T
>0, see GLP98. This
allows to dene Z opt;t;T
:=
Z opt;t;T
t_
Z opt;t;T
t
2 D 2
t (
M
[t;T]
). We have V opt;0;T
with V opt;0;T
0
= 1 and V opt;0;T
T
= Z
opt;0;T
T
v 0;T
B
> 0. Since V
opt;0;T
B [0;T]
Z opt;0;T
is a uniformly integrable martingale with V
opt;0;T
T
B
T Z
opt;0;T
T
> 0 we nd
V opt;0;T
> 0. This allows to dene V opt;t;T
:=
V opt;0;T
t_
V opt;0;T
t
2 SF 2
t (
M
[t;T] ).
Wethen have
V opt;t;T
T
= Z
opt;t;T
T
v t;T
B
T
; (5.3)
where v t;T
:=
V opt;0;T
t
Z opt;t;T
t v
0;T
is F
t
-measurable. Set
C
[t;T] :=E
t 2
4 B
t Z
opt;t;T
T
B
T
!
2 3
5
=E
t
"
v t;T
V opt;t;T
T
B 2
t Z
opt;t;T
T
B
T
#
=B
t v
t;T
; (5.4)
and note that
E
t h
V opt;t;T
T i
= B
T
t
C
[t;T]
; (5.5)
E
t
V opt;t;T
T
2
= 1
C
[t;T]
; (5.6)
and f
[t;T]
=0g=fC
[t;T]
=(B T
t )
2
g.
Lemma 5.1. On f
[t;T]
= 0g, we have BtZ
opt;t;T
T
B
T
= B T
t
almost surely
and for all H 2 SF 2
t (
M
[t;T]
), resp. H 2 SF sup;2
t (
M
[t;T]
), with V H
0
=
1 we have E
t
V H
T
= 1
B T
t
, resp. E
t
V H
T
1
B T
t
. Furthermore, on
f
[t;T]
= 0g,
[t 0
;T]
= 0 holds for all t t 0
T and on f
[t 0
;T]
= 0g
holds B T
t 0
= B
t 0
E
t 0
[B
T ]
and V opt;t
0
;T
= B
T
B T
t 0
.
Proof. SinceE
t h
BtZ opt;t;T
T
B
T i
=B T
t
,wend BtZ
opt;t;T
T
B
T
tobeF
t
-measurable
on f
[t;T]
=0g, hence the rst assertionholds. For H 2 SF 2
t (
M
[t;T] ),
resp. H 2SF sup;2
t (
M
[t;T]
),with V H
0
=1 wend
E
t
V H
T
= 1
B T
E
t
"
V H
T B
t Z
opt;t;T
T
B
T
#
= 1
B T
;
resp. E
t V
T
B T
t
, on f
[t;T]
= 0g. By the denition of Z
we nd Z opt;t
0
;T
T
tobeF
t 0
-measurableon f
[t;T]
=0g, hence
[t 0
;T]
=0
there. Since Z opt;t;T
T
= B
T B
T
t
B
t
on f
[t;T]
=0g, we nd 1=E
t h
B
T B
T
t
B
t i
=
B T
t E
t [B
T ]
Bt
there. Now applying what we have proved so far tothe case
tt 0
T we nd the lastassertion.
Proposition 5.2. Lete beaF
t
-measurablerandomvariablesatisfying
e = (B T
t )
1
on f
[t;T]
= 0g. Dene
e :=
C
[t;T] B
T
t
C
[t;T] (B
T
t )
2
e 1
B T
t
on
f
[t;T]
6=0g, resp.
e
:=0on f
[t;T]
=0g. Then
V t;T;e
:=
e V
opt;t;T
+(1
e )
B T
B T
t (5.7)
isthe uniquesolution of theconstraintoptimization problem(5.1)with
respecttoSF 2
t (
M
[t;T]
),undertheconstrainte. On f
[t;T]
6=0gwehave
V t;T;e;SF 2
t (
M
[t;T] )
=
B T
t
2
C
[t;T] (B
T
t )
2
e 1
B T
t
2
+e 2
; (5.8)
resp. V t;T;e;SF 2
t (
M
[t;T] )
= B T
t
2
on f
[t;T]
=0g.
Proof. First, note that V t;T;e
0
= 1 and E
t h
V t;T;e
T i
= e, hence V t;T;e
is
admissiblefor the constraint optimizationproblem(5.1). Dene
F (t;T)
e
(x):=x 2
2
e
v t;T
Z opt;t;T
T
B
T
x B
1
t
!
2
1
e
B T
t
(x e): (5.9)
F (t;T)
e
isdened in such away that forH 2SF 2
t (
M
[t;T]
),withV H
0
=1
and E
t
V H
T
=e, we have
E
t h
F (t;T)
V
H
T
i
=E
t h
V H
T
2 i
: (5.10)
Furthermore, since
dF (t;T)
e
dx
(x)=2x 2
e
v t;T
Z opt;t;T
T
B
T
2
1
e
B T
t
;
and by (5.3)
dF (t;T)
e
dx
(x)=0,x=
e
v t;T
Z opt;t;T
T
B
T +
1
e
B T
t
=V t;T;e
T
; (5.11)
and d
2
F (t;T)
e
dx 2
>0,we nd
F (t;T)
e
V t;T;e
T
= inf
x2R F
(t;T)
e (x):
(5.12)
Therefore
E
t
V t;T;e
T
2
= E
t h
F (t;T)
e
V t;T;e
T i
E
t h
F (t;T)
e V
H
T
i
=E
t h
V H
T
2 i
Nowcalculateonf
[t;T]
6=0g,usinge= eB
T
t
C
[t;T] +
1 e
B T
t
ande 2
= eB
T
t
C
[t;T] e+
1 e
B T
t e:
V t;T;e;SF 2
t (
M
[t;T] )
=E
t
V t;T;e
T
2
=
= E
t
e V
opt;t;T
T
+
1
e
B T
t
V t;T;e
T
=
e E
t h
V opt;t;T
T V
t;T;e
T i
+
1
e
B T
t e
=
e E
t
V opt;t;T
T
2
+e 2
e B
T
t
C
[t;T] e
=
e B
T
t
C
[t;T]
e 1
B T
t
+e 2
=
B T
t
2
C
[t;T] (B
T
t )
2
e 1
B T
t
2
+e 2
:
On f
[t;T]
=0g we have V t;T;e;SF 2
t (
M
[t;T] )
= B T
t
2
by Lemma
t;T;e
Theorem 5.3. Assume the existence of the zero bond B in M
[t;T] .
ThenthefollowinginequalityholdsforallH 2SF 2
t (
M
[t;T]
)withV H
0
=
1:
1
B T
t
[t;T] q
Var
t (V
H
T
)E
t
V H
T
1
B T
t +
[t;T] q
Var
t (V
H
T ):
(5.13)
Furthermore,
E
t
V H
T
1
B T
t
=
[t;T] q
Var
t (V
H
T );
(5.14)
holds if and only if V H
= V t;T;Et[V
H
T ]
on f
[t;T]
6= 0g. On f
[t;T]
=0g
the Return-to-Maturity Expectation Hypothesis holds:
B T
t_
= B
t_
E[B
T jF
t_
] : (5.15)
Proof. ByProposition 5.2 we nd onf
[t;T] 6=0g
[t;T]
=
e 1
B T
t
r
Var
t
V t;T;e
T
= max
H2SF 2
t (
M
[t;T] )
V H
0
=1;E
[ V
H
T ]
=e
e 1
B T
t
p
Var
t (V
H
T )
: (5.16)
This and Lemma 5.1 imply the rst assertion. The second assertion
followsfromthe uniquenessof V t;T;e
onf
[t;T]
6=0g. The lastassertion
follows againfrom Lemma 5.1.
Corollary 5.4. Assume the existence of the zero bond B T
in
M
[t;T] .
Then the following inequality holds for all H 2 SF sup;2
t (
M
[t;T] ) with
V H
0
=1 and E
t
V H
T
1
B T
t :
E
t
V H
T
1
B T
+
[t;T] q
Var
t (V
H
T ):
(5.17)
Proof. ForH 2 SF sup;2
t (
M
[t;T]
), with V H
0
=1 and E
t
V H
T
=e 1
B T
t ,
we have, see (5.9),
E
t h
F (t;T)
e V
H
T
i
E
t h
V H
T
2 i
; (5.18)
since
e
0for e 1
B T
t
, hence
V(t;T;e;SF 2
t (
M
[t;T]
))E
t h
V H
T
2 i
; (5.19)
whichimplies the assertion.
In the special case of a deterministic B, or working with the dis-
counted market
M
[t;T]
:=
[t;T]
; S
B
;1) [t;T]
, where zero bonds triv-
ially exist for all maturity times, the intertemporalprice for risk, de-
noted as
[t;T]
in the market
M
[t;T]
, is relatedto resultsby DMSSS97,
especiallyTheoremB,wherefor S
B
[t;T]
2S 2
l oc (
[t;T]
),theclosednessof
G 2
(
M
[t;T]
) is shown to be equivalent to the (non-discounted) variance
optimal martingale measure in
M
[t;T]
, denoted as Z opt;t;T
, satisfying
the so-called reverse Holder inequality:
E
s 2
4 Z
opt;t;T
T
Z opt;t;T
s
!
2 3
5
K; 8t sT;
(5.20)
for aconstant K. This condition is equivalentto
[s;T]
p
K 1 8ts T; (5.21)
since Z opt;s;T
= Z
opt;t;T
s_
Z opt;t;T
s
for allt sT and
[s;T]
= v
u
u
u
tE
s 2
4 Z
opt;t;T
T
Z opt;t;T
s
!
2 3
5
1:
For an F
t
-measurable random variable e 1, such that e = 1 on
f
[t;T]
=0g, denote the solution for the constraint optimization prob-
lem(4.1)inthe discounted market
M
[t;T] by V
;t;T;e
. ForV :=V
;t;T;e
^T ,
whichcanbeseenasthevalueprocessofaself-nancinghedgingstrat-
egy in
M
[t;T 0
]
for T T 0
, we have E
t [V
T 0
] = 1 +
[t;T] p
Var
t (V
T 0).
Therefore
0
; 8T T 0
: (5.22)
Wesummarize these observations:
Theorem 5.5. For T > 0 let S
B
[0;T]
2 S 2
l oc (
[0;T]
). We then have
equivalence between
1. G 2
(
M
[0;T]
) is closed.
2. f
[t;T]
j0tTg is uniformly bounded.
3. f
[t;T 0
]
j0tT 0
Tg is uniformly bounded.
4. G 2
(
M
[t;T 0
]
) is closed for all 0t T 0
T.
Corollary 5.6. IfG 2
(
M
[t;T]
)isclosedandifB T
isattainablein
M
[t;T]
with a self-nancing hedging strategy in G 2
(
M
[t;T]
), then
V(t;T;e;SF 2
t (
M
[t;T]
))=V(t;T;e;G 2
(
M
[t;T] )):
(5.23)
6. Application
In an incomplete market with zero bond, one way to price non-
attainable claims is to price them with respect to an equivalent mar-
tingale measure that is in some sense optimal,e.g. minimal,variance-
optimal, L q
-optimal, entropy minimal. If the discounted variance-
optimalmeasureisanequivalentprobabilitymeasure,ithasthespecial
property that the intertemporal price for risk
[t;T]
for maturity time
T in the market
M
[t;T]
remains unchanged if new securities priced
according to it are introduced to the market: For a non-attainable
squareintegrableF
T
-measurablecontingentclaim
X wecan denethe
price process X
s :=
B [t;T]
s Es
h
X
B
T Z
opt;t;T
T i
Z opt;t;T
s
. X
B [t;T]
is a uniformlyintegrable
martingalewith respect tothe discounted varianceoptimalmartingale
measure of the market
M
[t;T]
dened by Z opt;t;T
T
. Therefore, for the
extended market
M
;X
[t;T]
:=
[t;T]
;(
S;X) [t;T]
with intertemporalprice
forriskdenotedas X
[t;T]
,wehaveZ opt;t;T
2D 2
t
M
;X
[t;T]
D 2
t (
M
[t;T] ),
opt;t;T
measure for the extended market
M
;X
[t;T]
and X
[t;T]
=
[t;T] . X
t
is also
knowntobetheinitialpriceofthemean-varianceoptimalself-nancing
hedgingstrategy approximating
X.
Ingeneralitisnot easytocalculateany ofthe quantities
[t;T]
;C
[t;T]
and B T
t
, whichare relatedin the following way:
[t;T]
= q
C
[t;T] (B
T
t )
2
B T
t
: (6.1)
(This equation follows immediately from the denition of
[t;T]
.) In
Leitner (2000), an example is given where C
[t;T]
can be calculated ex-
plicitly. InamarkoviansettingaPDE isderived, fromwhichC
[t;T]
can
be calculated.
Estimatingthefunctiont 7!C
t
0
;t
0 +t
, t
0 t
0
+tt
1
, fromhistorical
data and calculating
[t
0
;t
0 +t]
via equation (6.1) using historical zero
bond prices, one can trytond a modelforthe quantities C
t1;t1+t and
t1;t1+t
. Solving(6.1)forB t
1 +t
t1
wendamodelforthezerobondprices,
which can be compared to observed prices. Alternatively, one can
estimate
[t
1
;t
1 +t]
;t>0fromobserved zerobondpricesandamodelfor
C
t
1
;t
1 +t
and look for interesting patterns in the graph of t7!
[t
1
;t
1 +t]
.
7. Conclusions
Wehaveshownthattheterm-structureofinterestratesandtheterm
structure of intertemporalprices for risk are closely related.
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