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Sublinear price functionals under portfolio constraints
Journal of Mathematical Economics 33 Ž2000. 339–351 www.elsevier.comrlocaterjmateco
Sublinear price functionals under portfolio constraints Pierre-F. Koehl b
a,1
, Huyen ˆ Pham
b,)
a CREST and ENSAE, Laboratoire de Finance-Assurance, France Equipe d’Analyse et de Mathematiques Appliquees ´ ´ UniÕersite´ de Marne-la-Vallee ´ and CREST, Cite Descartes, Champs-sur-Marne, 5 BouleÕard Descartes, 77454 Marne-la-Vallee, Cedex 2, France
Received 17 December 1997; received in revised form 5 May 1999; accepted 9 June 1999
Abstract We consider a financial market model in discrete time with convex constraints on portfolios. We adopt an axiomatic approach of admissible price functionals which generalizes the familiar linear pricing rules for frictionless markets. We provide a dual representation formula of any admissible price functional. This formula is expressed as a supremum of expectation under a suitable family of probability measures. This result is applied to restrict the Žusually too large. super-replication bid-ask spread when the super-replication cost functional is not sublinear and otherwise to derive a dual characterization of the super-replication cost. q 2000 Elsevier Science S.A. All rights reserved. JEL classification: G12 Keywords: Price functional; Portfolio constraints; Dual representation; Bid-ask spread
1. Introduction The modern theory of asset pricing originates with the papers of Harrison and Kreps Ž1979., Harrison and Pliska Ž1981. and Kreps Ž1981. who define price ) Corresponding author. Tel.: q33-160-95-7537; fax: q33-160-95-7545; E-mail: [email protected] 1 E-mail: [email protected].
0304-4068r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 9 . 0 0 0 2 4 - 5
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P.-F. Koehl, H. Phamr Journal of Mathematical Economics 33 (2000) 339–351
functionals according to no-arbitrage principle. Using also arbitrage arguments, Bensaid et al. Ž1992. obtain a determination of the bid-ask spread, i.e., the spread between the upper and lower bounds of the price functionals, in terms of the super-replication cost, representing the minimal initial wealth that finances at least the payoff of a given contingent claim: The upper bound of the no-arbitrage pricing functional is equal to the super-replication cost. In frictionless markets, a no-arbitrage price functional is an expected value with respect to a martingale measure and the super-replication cost is equal to the supremum of expected values over all martingale measures, and is so sublinear. In this paper, we consider a financial market model in discrete time with convex constraints on portfolios as in the studies of Cvitanic´ and Karatzas Ž1993. and Karatzas and Kou Ž1996.. In such markets with frictions, the super-replication cost may not be sublinear Žsee Brannath, 1997; Follmer and Kramkov, 1997.. ¨ Moreover, the super-replication cost of a contingent claim may be no less than the price of a buy-and-hold strategy and hence may be too large, see e.g., Soner et al. Ž1995. and Cvitanic´ et al. Ž1999.. Therefore, one major question in finance is to find an interval as small as possible in which an admissible price must lie. We adopt an axiomatic approach by imposing some reasonable conditions in order for a price functional to be admissible. In the super-replication cost approach, it is implicitly required that a price functional must be nondecreasing. This is a natural economic assumption since no one will accept to buy a contingent claim if it is possible to obtain a better payoff for a lower price. This nondecreasing property is not a consequence of the no arbitrage condition and is an additional axiom. In our approach, we require furthermore the sublinearity of price functionals. Sublinearity assumption is economically justified since no one will accept to pay more for holding contingent claims together than separately. We mention that Jouini Ž1997. also uses an axiomatic approach in a transaction costs framework without constraints. He proves that any price functional satisfying his axioms lies between the supremum and infimum of expected values with respect to a family of probability measures. In a transaction costs model, the super-replication cost is equal to this supremum since it is a sublinear function. In contrast, in a model with portfolio constraints, the super-replication cost function is not always sublinear and therefore by requiring a sublinearity condition on price functionals, one can actually obtain a restriction of the bid-ask spread. The outline of the paper is organized as follows. Section 2 describes the financial market model with portfolio constraints. In Section 3, we define the set of axioms on price functionals and we provide a dual representation of any admissible price functional in terms of mathematical expectation under a suitable family of probability measures satisfying a generalization of the martingale property of securities prices. This set of probability measures is closely related to a no-arbitrage condition, see e.g., Jouini and Kallal Ž1995a., Pham and Touzi Ž1999. and Carassus et al. Ž1998.. The key tool for deriving the main result ŽTheorem 2.1. is the use of the Fenchel–Moreau theorem which appears to be very well-suited in
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this axiomatic approach. Section 4 is devoted to the applications. First, we show that if the super-replication cost function is not sublinear, our axiomatic approach leads to a reduction of the bid-ask spread with regard to the usually too large super-replication bid-ask spread. We illustrate with an example. On the other hand, if the super-replication cost is sublinear, then we state a discrete time version of Cvitanic´ and Karatzas Ž1993. and Karatzas and Kou Ž1996. results on the dual probabilistic representation of the super-replication cost.
2. The financial market model F, P . be a complete probability space equipped with a filtration Let Ž V ,F F s Ft ,t s 0, . . . ,T 4 where T ) 0 is a finite horizon. For convenience, we assume as usual that F0 is trivial and FT s F. The space L2 is the set of all square integrable random variables. The financial market model consists in one riskless asset of price equal to one and n risky assets of price process St s Ž St1, . . . ,Stn .X , t s 0, . . . ,T 4 assumed to be F-adapted. Here the notation X is for the transposition. For each t s 0, . . . ,T, we denote by diagŽ St . the n = n diagonal matrix whose ith diagonal term is Sti. We shall assume that for each t s 0, . . . ,T, diagŽ St . is nonsingular almost surely and we denote diagŽ St .y1 its inverse matrix. We denote the increment process of S by D St s St y Sty1 , t s 1, . . . ,T 4 and the return process of S by R t s diagŽ Sty1 .y1 D St , t s 1, . . . ,T 4 . We assume that R t g L2 for all t s 1, . . . ,T. We denote by Q the set of all R n-valued Ft , t s 0, . . . ,T y 14 adapted processes such that u tX D Stq1 g L2 for all t s 0, . . . ,T y 1. An element u t s Ž u t1, . . . , u tn .X , t s 0, . . . ,T y 14 of Q is a trading strategy where u ti is interpreted as the number of units invested in the ith risky asset at date t. Remark 2.1. Ž1. The nonsingularity on the process diag Ž S . means that for each t s 0, . . . ,T and i s 1, . . . ,n, P w Sti s 0x s 0. This is satisfied, for instance, in the usual case where S is a strictly positive process. Ž2. The assumption that R t g L2 can be weakened in R t g L1 , t s 1, . . . ,T, by imposing that trading strategy u satisfy u t g L1 and that probability measures introduced in Section 3.2 have a density in L` Ždual of L1 .. Given an initial wealth x g R and a trading strategy u g Q , the Žself-financed. wealth process evolves according to: ty1
X tx , u s x q Ý uuX D Suq1 , us0
t s 0, . . . ,T .
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The wealth proportion R n-valued process is then defined by:
p tx , u s
diag Ž St . u t X tx , u
if X tx , u / 0, t s 0, . . . ,T .
Ž 1.1 .
We now constrain the trading strategies as in Cvitanic´ et al. Ž1999.. Let us consider two nonempty convex sets Kq and Ky of R n containing the origin 0. We define the convex cones of R n : Kqs x g R n : ;p g Kq , p X x F 0 4 , Kys x g R n : ;p g Ky , p X x F 0 4 . For any x g R, the set of constrained trading strategies is defined by:
Q Ž x . s u g Q : p tx , u g Kq if X tx , u ) 0, p tx , u g Ky if X tx , u - 0, u t s 0 if X tx , u s 0 4 . In other words, Kq Žresp. Ky . represents constraints on proportions when the wealth is positive Žresp. negative..Remark 2.2. In Cvitanic´ and Karatzas Ž1993. or Karatzas and Kou Ž1996., the authors define the Žself-financed. wealth process from an initial wealth x and a process p interpreted as a proportion process by: x ,p s X tx ,p Ž 1 q p tX R tq1 . , X 0x ,p s x and X tq1
t s 0, . . . ,T y 1.
According to relation Ž1.1., this formulation of wealth process is equivalent to the one with the number of units u if and only if the wealth process X tx, u / 0, for all t, almost surely. In particular, with the formulation of the wealth process in terms x,p of proportion, we see that if X tx,p s 0 then X tq1 s 0 while this is not necessarily the case when the wealth process is expressed in function of the number of units. This is the reason why we add the constraint u t s 0 if X tx, u s 0 in order to ensure x, u that X tq1 s 0. Remark 2.3. Notice that for all l ) 0 and x g R, p tl x , lu s p tx, u . Moreover, since Q Ž0. s 04 , this implies that:
Q Ž l x . s lQ Ž x . , ;l G 0, ; x g R.
Ž 1.2 .
Here are some examples of convex constraints taken from Karatzas and Kou Ž1996.. Example 2.1. Ži. Unconstrained case: Kqs Kys R n. Then Kqs Kys 04 . Žii. Prohibition of short-selling of stocks: Kqs yKys w0,`. n. Then Kqs yKys n w Žy`,0x n. Žiii. Constraints on the short-selling of stocks: Kqs P is1 yk i ,`., n Kys P is1 Žy`,l i x with k i , l i ) 0 for i s 1, . . . ,n. Then Kqs Kys 04 . Živ. n n w wykXi ,lXi x with k i , kXi , l i , Rectangular constraints: Kqs P is1 yk i ,l i x, Kys P is1 X l i ) 0 for i s 1, . . . ,n. Then Kqs Kys 04 . Žv. Incomplete market, in which only
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the m Žwith m - n. first stocks can be traded: Kqs yKys K s p g R n : p i s 0, i s m q 1, . . . ,n4 . Then K s x g R n : x i s 0, i s 1, . . . ,m4 . Žvi. Incomplete and constrained market with prohibition of short-selling of the first n1 stocks, prohibition of being long in the next n 2 stocks and prohibition of investment in the next n 3 stocks Žwith n1 q n 2 q n 3 F n.: Kqs yKys K s p g R n :p i G 0, i s 1, . . . ,n1 , p i F 0, i s n1 q 1, . . . ,n1 q n 2 , p i s 0, i s n1 q n 2 q 1, . . . ,n1 q n 2 q n 34 . Then K s x g R n : x i F 0, i s 1, . . . ,n1 , x i G 0, i s n1 q 1, . . . ,n1 q n 2 , x i s 0, i s n1 q n 2 q n 3 , . . . ,n4 . Žvii. Both Kq and Ky are closed convex cones with vertex in zero: This clearly generalizes all the previous cases except Žiii. and Živ.. Žviii. Constraints on borrowing: Kqs p g R n : Ý nis1p i F k 4 and Kys p g R n : Ý nis1p i G l 4 for some k G 1 and l F 0. Then Kqs Kys 04 .
3. Admissible price functionals 3.1. Axiomatic of the price functionals A price functional is a function c defined on L2 and valued in R j q`4 , where c Ž X . represents the price at which the contingent claim X can be bought. We say that the price functional is admissible if it satisfies the following conditions: ; X , Y g L2 , X F Y c Ž X . F c Ž Y .
Ž A1.
;l G 0, ; X g L2 ,
Ž A2.
2
;X, YgL 2
;XgL
c Ž l X . s lc Ž X .
c Ž XqY . Fc Ž X . qc Ž Y .
c Ž X . Ff Ž X .
c is lower semi-continuous
Ž A3. Ž A4 . Ž A5.
where f is the super-replication cost function defined by:
f Ž X . s inf x g R: 'u g Q Ž x . , X Tx , u G X 4 , ; X g L2 , with the convention that infB s q`. Condition ŽA1. is a nondecreasing property of the price functional. In words, no one will accept to buy a contingent claim if it is possible to obtain a better payoff for a lower price. Condition ŽA3. means that it is cheaper to buy the sum X q Y of two contingent claims than to buy X and Y separately. The economic reason for this sublinearity condition is the following: if it was not true for some X and Y, then the cheapest way to buy X q Y is to buy them separately, for a cost c Ž X . q c Ž Y . - c Ž X q Y ., and then c Ž X q Y . cannot be an equilibrium price. The economic interpretation of ŽA4. is that if it is possible to obtain a better payoff than X at a cost f Ž X ., then no one will accept to pay more than f Ž X ..
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The positive homogeneity condition ŽA2. is less natural than the others. It is assumed in all the classical financial market models, since, for a mathematical viewpoint, it is required for applying the Riesz representation theorem on which are based the main theorems of asset pricing Žsee, e.g., Harrison and Kreps, 1979 in a frictionless market and Jouini and Kallal, 1995a in a no short-sales constraints model.. Condition ŽA2. implies that c Ž0. s 0, which joined with condition ŽA1. ensures that c Ž X . G 0 for all X G 0. In words, one cannot receive money for holding a nonnegative contingent claim; this can be interpreted as a no-arbitrage condition. The technical condition ŽA5. means that if X n converges to X in C , then lim inf c Ž X n . G c Ž X .. Notice that Jouini Ž1997. has introduced the same axioms except axiom ŽA1. which is replaced by: ; X g L2 , X ) 0, c Ž X . ) 0.
¨
Remark 3.1. The functional c is interpreted as the ask-price functional. Therefore, the function X g C yc ŽyX . is the bid-price functional. Remark 3.2. Using ŽA3., ŽA4. and c Ž0. s 0, we have: ; X g L2 , yf ŽyX . F yc ŽyX . F c Ž X . F f Ž X .. It implies in particular that the bid-ask spread c Ž X . q c ŽyX . lies in the interval w0, f Ž X . q f ŽyX .x. We say that f Ž X . q f ŽyX . is the super-replication bid-ask spread.
3.2. Dual representation of admissible price functionals We introduce the set P of probability measures Q absolutely continuous with respect to P, with density dQrd P g L2 and satisfying the following condition:
Ž P . E Q R tq1 < Ft g Kq F y Ky , t s 0, . . . ,T y 1, Q a.s. In the sequel, we shall identify probability measures and their densities; we say that Q n s Z n P converges in L2 to Q s ZP if Z n converges to Z in L2 . We denote by P e the elements of P which are equivalent to P. Remark 3.1. Suppose that P e / B. Then, it is easily checked that the closure of P e in L2 is equal to P. Indeed, take an arbitrary Q g P and let Q 0 g P e. The sequence Qn s 1y
ž
1 n
/
Qq
1 n
Q 0 , n g NU ,
obviously lies in P e and converges in L2 to Q. By noting that P is closed in L2 , this proves the result. The condition that P e is not empty is related to a no arbitrage condition: for more details, we refer to Jouini and Kallal Ž1995a. in a short-sales constraint model, Pham and Touzi Ž1999. for an extension to the case of cone constraints, and Carassus et al. Ž1998. in the case of convex constraints as considered here.
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We first prove a useful lemma. Lemma 3.1. Let Q g P. Then, for all x g R, for all u g Q Ž x ., the wealth process X tx, u , t s 0, . . . ,T 4 is a supermartingale under Q. Proof. Let t g 0, . . . ,T y 14 . By definition of the wealth process and the return process, we have: x ,u s X tx , u q u tX diag Ž St . R tq1 . X tq1
If X tx, u / 0, then we have by definition of the proportion process: X
x ,u X tq1 s X tx , u q X tx , u Ž p tx , u . R tq1
By property Ž P . of Q, we have X tx, u Žp tx, u .X E Q w R tq1
X tx, u s 0,
If result.
then
x, u X tq1 s0
Ž 2.1 . and Ž2.1. still holds, which proves the required I
We now state the main result of this section which provides a complete dual characterization of admissible price functionals. Theorem 3.1. c is an admissible price functional if and only if there exists a subset Q of P such that:
c Ž X . s sup E Q Ž X . , ; X g L2 .
Ž 2.2 .
QgQ
Proof. Let c be a functional given by Ž2.2.. It is clear that it satisfies the axioms ŽA1., ŽA2., ŽA3. and ŽA5.. Now, consider some X g L2 and x g R, u g Q Ž x ., such that X F X Tx, u . By Lemma 2.1, this implies that E Q w X x F x for all Q g P and therefore c Ž X . F f Ž X . which means that condition ŽA4. is satisfied and so c is an admissible price functional. Let c be an admissible price functional. Axioms ŽA2. and ŽA3. imply that c is a convex function on the Hilbert space L2 . We consider then the convex conjugate of c defined on L2 by:
c˜ Ž Y . s sup E Ž XY . y c Ž X . 4 XgL2
and valued on R j q`4 . The domain of c˜ is denoted Dc˜ :s Y g L2 : c˜ Ž Y . - `4 . From condition ŽA2., it is clear that: Dc˜ s Y g L2 : ; X g L2 , E Ž XY . F c Ž X . 4 and c˜ Ž Y . s 0,
;Y g Dc˜ .
P.-F. Koehl, H. Phamr Journal of Mathematical Economics 33 (2000) 339–351
346
Therefore, by the Fenchel–Moreau theorem Žsee, e.g., Theorem I.10 in Brezis, ´ 1983., we have:
c Ž X . s sup E Ž YX . , ; X g L2 . Yg Dc˜
Let us first prove that Y G 0 for all Y g Dc . Indeed, denoting N s v g V :Y Ž v . F 04 , we have by ŽA1. and ŽA2. that EŽyY 1 N . F c Žy1 N . F c Ž0. s 0. But from definition of N , we also have EŽyY 1 N . G 0 which implies that EŽyY 1 N . s 0 and then Y s 0 on N . We now check that EŽ Y . s 1 for all Y g Dc˜ . Indeed by ŽA4., we have: E Ž YX . F c Ž X . F f Ž X . , ; X g L2 , ;Y g Dc˜
Ž 2.3 .
Choose X s 1. Then it is clear that the initial wealth x s 1 associated to a zero strategy u s 0 g Q Ž x . leads to a terminal wealth X Tx, u s 1 and so f Ž1. F 1. By Eq. Ž2.3., we deduce that EŽ Y . F 1. For the choice of X s y1, we deduce by a similar argument that EŽyY . F y1 and therefore EŽ Y . s 1. We have then proved that each element Y g Dc˜ can be identified with a probability measure Q absolutely continuous with respect to P, with density in L2 , so that
c Ž X . s sup E Q Ž X . , ; X g L2 .
Ž 2.4 .
Qg Dc˜
Let us finally check that Dc˜ is a subset of P by showing property Ž P . for each Q g Dc˜ . From the representation Ž2.4. and condition ŽA4. applied to elements X s X Tx, u , x g R, u g Q Ž x ., we have: Ty1
EQ
žÝ
uuX D Suq1 F 0, ;Q g Dc˜ , ; x g R, ;u g Q Ž x . .
/
us0
Ž 2.5 .
Fix t g 0, . . . ,T y 14 . Let x ) 0 Žresp. x - 0. and choose an arbitrary p g Kq Žresp. Ky . and F g Ft . Consider the following trading strategy: uu s 0, u s 0, . . . ,t y 1, t q 1, . . . ,T y 1,
u t s diag Ž St .
y1
p x1 F .
Notice that u g Q since uuX D Suq1 s 0 for u s 0, . . . ,t y 1,t q 1, . . . ,T y 1 and u tX D Stq1 s x1 F p X R tq1 g L2 . The associated wealth process is X ux , u s x ,
u s 0, . . . ,t ,
X ux , u s x q u tX D Stq1 ,
u s t q 1, . . . ,T .
The associated wealth proportion process is
pux , u s 0, p tx , u s
u s 0, . . . ,t y 1,
diag Ž St . u t
pux , u s 0,
X tx , u
s p 1F ,
u s t q 1, . . . ,T y 1 if X ux , u / 0.
P.-F. Koehl, H. Phamr Journal of Mathematical Economics 33 (2000) 339–351
347
Hence u g Q Ž x . and by substituting this strategy into Eq. Ž2.5. and by the arbitrariness of F g Ft , we get: xp X E Q R tq1 < Ft F 0, Q a.s. for all t s 0, . . . ,T y 1, x ) 0 Žresp. x - 0. and p g Kq Žresp. p g Ky ., which is exactly the property Ž P . for Q g Dc˜ . I Remark 3.2. The key result for deriving the dual representation formula Ž2.2. is the Fenchel–Moreau theorem. The idea of using this theorem appears more and less in Jouini and Kallal Ž1995b. ŽTheorem 2.2., although these authors cannot directly apply it since they do not work on vector space.
4. Applications 4.1. Reduction of the bid-ask spread One of the main topic in finance is the pricing of contingent claims. In incomplete or imperfects markets, there is no unique arbitrage-free price. Since Harrison and Kreps Ž1979., an important literature is devoted to find an interval as small as possible, in which the ‘‘fair’’ price must lie. Until now, as originated by Bensaid et al. Ž1992., the bid-ask spread is usually obtained from the super-replication cost; they obtain an interval wyf ŽyX ., f Ž X .x for the ‘‘fair’’ price of a contingent claim X. However, it is known that the super-replication spread can be too large. Indeed, several authors proved that the super-replication cost of a contingent claim can be no less than the cost of a buy-and-hold strategy, for example the price of the underlying risky asset in the case of a call option, see e.g., Soner et al. Ž1995. and Cvitanic´ et al. Ž1999.. Due to our additional sublinearity condition ŽA3., we show that we can restrict this bid-ask spread. We define the functional
F Ž X . s sup E Q Ž X . , ; X g L2 . Qg P
Notice from Remark 2.1 Žii. that if P e is not empty, the supremum in the formula of F can be taken over P e :
F Ž X . s sup E Q Ž X . , ; X g L2 Qg P e
We also denote f U the largest lower semi-continuous function below f , i.e.:
f U Ž X . s inf lim inf f Ž X n . , X n
™X 4, ;XgL . 2
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Theorem 4.1. Ž i . F F f U . Ž ii . For all admissible pricing functional c , we get: yF Ž yX . s inf E Q Ž X . F c Ž X . F F Ž X . s sup E Q Ž X . , ; X g L2 . QgP
Qg P
Proof. Ži. From Theorem 2.1, F is an admissible price functional and then F F f by axiom ŽA4.. Now, since F is a lower semi-continuous function below f , we obtain the first assertion from the definition of f U . Žii. The second assertion is an immediate consequences of Theorem 2.1 and the definition of F . I The last theorem says that an admissible price functional lies between a supremum and an infinum of expected values under a family of probability measures satisfying property Ž P .. This is similar to Jouini Ž1997. result in a transaction costs framework. However, while this supremum is equal to the super-replication cost in a transaction costs model, we shall see in the following example that in a model with portfolio constraints, the bid-ask spread induced by our approach, F Ž X . q F ŽyX ., can be strictly smaller than the super-replication bid-ask spread for some contingent claims X. Example. We choose T s 1 and consider a finite probability space V s v 1 , v 2 4 , which implies in particular that f U s f . We have one riskless bond of price equal to one and one risky asset such that S0 s 1. At date t s 1, S1 can take the values S1Ž v 1 . s u and S1Ž v 2 . s d, with d - u. A contingent claim X is identified with the pair Ž a , b . s Ž X Ž v 1 ., X Ž v 2 .. g R 2 . We model the constraints by Kqs w0,1x and Kys wy1,0x. Then, it is easily checked that Q Ž x . s w0, < x
f Ž a , b . s inf x g R: 'u g w 0, < x < x , x q u Ž u y 1 . G a and x q u Ž d y 1 . G b 4 An easily calculation shows that
½
ž
f Ž a , b . s inf x g R: x G max b ,
a Ž 1 y d . q b Ž u y 1. uyd
/
and
x q < x < Ž u y 1. G a 4 . In the case u - 2, the last expression simplifies into:
°max
a Ž 1 y d . q b Ž u y 1. a , uyd u f Ž a ,b . s a Ž 1 y d . q b Ž u y 1. a max b , , uyd 2yu
~
¢
ž ž
b,
/
if a G 0,
/
if a - 0.
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349
Now, by noting that Kqs yKys Ry and R 1 s Ž u y 1,d y 1., we easily check that 1yd 2 P s Ž q,1 y q . g w 0,1 x : 0 F q F . uyd
½
5
We then explicitly determine the functional F :
F Ž a ,b . s
Ž 1 y d . max Ž a , b . q Ž u y 1 . b
. uyd If we take, for example, u s 3r2, d s 1r2 and a call option of exercise price k s 1, i.e., X s Ž1r2,0., an immediate computation shows that f Ž X . s 1r3 and f ŽyX . s 0. Then, the super-replication spread for this contingent claim is 1r3. Moreover, F Ž X . s 1r4 and F ŽyX . s 0. Hence, this example shows that we can obtain a significant reduction of the bid-ask spread by requiring sublinearity of price functionals. 4.2. Dual representation of the super-replication cost function There is a large literature in finance devoted to the dual characterization of the super-replication cost. We mention, among others, in continuous-time: El Karoui and Quenez Ž1995. in incomplete markets, Cvitanic´ and Karatzas Ž1993. for constrained portfolios, and in discrete-time: Jouini and Kallal Ž1995a. for no short-sales constraint and short-selling cost. Our axiomatic approach allows us to prove the same kind of result. Theorem 4.2 f ) s F if and only if f U is sublinear. Proof. Assume that f U s F . Then f U is an admissible price functional and, from axiom ŽA3., it is sublinear. Conversely, assume that f U is sublinear, i.e., it satisfies condition ŽA3.. Since f satisfies obviously conditions ŽA1. and ŽA4., it goes also for f U . By Ž1.2., we have f U Ž l X . s l f U Ž X ., ;l ) 0. It remains to prove that f U Ž0. s 0. By ŽA1., we get f U Žyl. s lf U Žy1. F f U Ž0. for all l ) 0. Sending l to zero, we obtain f U Ž0. G 0. Since we obviously have f U Ž0. F f Ž0. F 0, this shows that f U Ž0. s 0. Therefore, f U is an admissible price functional, and from Theorem main, we obtain that f U F F . Using the fact that f U G F Žsee Theorem 3.1 Ži.., this ends the proof. I In the case where the super-replication cost is sublinear, the last theorem provides a dual characterization of f U in terms of the set of probability measures P:
f U Ž X . s sup E Q w X x , ; X g L2 . Qg P
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However, this result does not hold any more if f U is not sublinear as in the example described in the previous paragraph. In that case, it leads to a reduction of the bid-ask spread. Notice that this dual representation is not exactly stated for the super-replication cost function but for its lower-semicontinuous envelope f U . In general, f is not lower-semicontinuous and is then different from f U . In a continuous-time diffusion setting, Cvitanic´ and Karatzas Ž1993. and Karatzas and Kou Ž1996. obtained a dual representation of the ‘true’ super-replication f , which implies in particular that it is lower-semicontinuous. In a general discrete-time context, such a property would require a closedness property of the set of all dominated contingent claims, see Brannath Ž1997. for more details. We mention that Brannath Ž1997. has obtained a dual representation formula for the super-replication cost in a discrete-time model with convex constraints on the number of shares u and it appears that in this context, the super-replication cost is sublinear if and only if the so-called upper-variation process is zero. His result is a discrete-time version of the paper of Follmer and Kramkov Ž1997.. These last ¨ cited authors also obtained a representation formula for the super-replication cost in a model with convex constraints on proportions but for nonnegative claims and nonnegative wealth. It would be of interest to derive similar result in our discrete-time context with convex constraints on proportions and without requiring nonnegativity of wealth process.
References Bensaid, B., Lesne, J.P., Pages, ¨ H., Scheinkman, J., 1992. Derivative asset pricing with transaction costs. Mathematical Finance 2, 63–86. Brannath, W., 1997. No arbitrage and martingale measures in option pricing. Doctoral dissertation, Wien Universitat. ¨ Brezis, et applications. Masson. ´ H., 1983. Analyse fonctionnelle. Theorie ´ Carassus, L., Pham, H., Touzi, N., 1998. No arbitrage in discrete time under portfolio constraints, preprint. Cvitanic, ´ J., Karatzas, I., 1993. Hedging contingent claims with constrained portfolios. Annals of Applied Probability 3, 652–681. Cvitanic, ´ J., Pham, H., Touzi, N., 1999. Super-replication in stochastic volatility models under portfolio constraints. Journal of Applied Probability 36. El Karoui, N., Quenez, M.C., 1995. Dynamic programming and pricing of contingent claims in an incomplete market. SIAM Journal of Control and Optimization 33, 29–66. Follmer, H., Kramkov, D., 1997. Optional decomposition under constraints. Probability Theory and ¨ Related Fields 109, 1–25. Harrison, J.M., Kreps, D., 1979. Martingales and arbitrage in multiperiods securities markets. Journal of Economic Theory 20, 381–408. Harrison, J.M., Pliska, S., 1981. Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 11, 215–260. Jouini, E., 1997. Price functionals with bid-ask spreads: an axiomatic approach. Preprint CREST. Jouini, E., Kallal, H., 1995a. Arbitrage in securities markets with short sales constraints. Mathematical finance 5, 197–232.
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Jouini, E., Kallal, H., 1995b. Martingale and arbitrage in securities markets with transaction costs. Journal of Economic Theory 66, 178–197. Karatzas, I., Kou, S., 1996. On the pricing of contingent claims under constraints. Annals of Applied Probability 6, 321–369. Kreps, D., 1981. Arbitrage and equilibrium in economies with infinitely many commodities. Journal of Mathematical Economics 8, 15–35. Pham, H., Touzi, N., 1999. The fundamental theorem of asset pricing with cone constraints. Journal of Mathematical Economics 31, 265–279. Soner, M., Shreve, S., Cvitanic, ´ J., 1995. There is no nontrivial hedging portfolio for option pricing with transaction costs. Annals of Applied Probability 5, 327–355.
Sublinear price functionals under portfolio constraints Pierre-F. Koehl b
a,1
, Huyen ˆ Pham
b,)
a CREST and ENSAE, Laboratoire de Finance-Assurance, France Equipe d’Analyse et de Mathematiques Appliquees ´ ´ UniÕersite´ de Marne-la-Vallee ´ and CREST, Cite Descartes, Champs-sur-Marne, 5 BouleÕard Descartes, 77454 Marne-la-Vallee, Cedex 2, France
Received 17 December 1997; received in revised form 5 May 1999; accepted 9 June 1999
Abstract We consider a financial market model in discrete time with convex constraints on portfolios. We adopt an axiomatic approach of admissible price functionals which generalizes the familiar linear pricing rules for frictionless markets. We provide a dual representation formula of any admissible price functional. This formula is expressed as a supremum of expectation under a suitable family of probability measures. This result is applied to restrict the Žusually too large. super-replication bid-ask spread when the super-replication cost functional is not sublinear and otherwise to derive a dual characterization of the super-replication cost. q 2000 Elsevier Science S.A. All rights reserved. JEL classification: G12 Keywords: Price functional; Portfolio constraints; Dual representation; Bid-ask spread
1. Introduction The modern theory of asset pricing originates with the papers of Harrison and Kreps Ž1979., Harrison and Pliska Ž1981. and Kreps Ž1981. who define price ) Corresponding author. Tel.: q33-160-95-7537; fax: q33-160-95-7545; E-mail: [email protected] 1 E-mail: [email protected].
0304-4068r00r$ - see front matter q 2000 Elsevier Science S.A. All rights reserved. PII: S 0 3 0 4 - 4 0 6 8 Ž 9 9 . 0 0 0 2 4 - 5
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functionals according to no-arbitrage principle. Using also arbitrage arguments, Bensaid et al. Ž1992. obtain a determination of the bid-ask spread, i.e., the spread between the upper and lower bounds of the price functionals, in terms of the super-replication cost, representing the minimal initial wealth that finances at least the payoff of a given contingent claim: The upper bound of the no-arbitrage pricing functional is equal to the super-replication cost. In frictionless markets, a no-arbitrage price functional is an expected value with respect to a martingale measure and the super-replication cost is equal to the supremum of expected values over all martingale measures, and is so sublinear. In this paper, we consider a financial market model in discrete time with convex constraints on portfolios as in the studies of Cvitanic´ and Karatzas Ž1993. and Karatzas and Kou Ž1996.. In such markets with frictions, the super-replication cost may not be sublinear Žsee Brannath, 1997; Follmer and Kramkov, 1997.. ¨ Moreover, the super-replication cost of a contingent claim may be no less than the price of a buy-and-hold strategy and hence may be too large, see e.g., Soner et al. Ž1995. and Cvitanic´ et al. Ž1999.. Therefore, one major question in finance is to find an interval as small as possible in which an admissible price must lie. We adopt an axiomatic approach by imposing some reasonable conditions in order for a price functional to be admissible. In the super-replication cost approach, it is implicitly required that a price functional must be nondecreasing. This is a natural economic assumption since no one will accept to buy a contingent claim if it is possible to obtain a better payoff for a lower price. This nondecreasing property is not a consequence of the no arbitrage condition and is an additional axiom. In our approach, we require furthermore the sublinearity of price functionals. Sublinearity assumption is economically justified since no one will accept to pay more for holding contingent claims together than separately. We mention that Jouini Ž1997. also uses an axiomatic approach in a transaction costs framework without constraints. He proves that any price functional satisfying his axioms lies between the supremum and infimum of expected values with respect to a family of probability measures. In a transaction costs model, the super-replication cost is equal to this supremum since it is a sublinear function. In contrast, in a model with portfolio constraints, the super-replication cost function is not always sublinear and therefore by requiring a sublinearity condition on price functionals, one can actually obtain a restriction of the bid-ask spread. The outline of the paper is organized as follows. Section 2 describes the financial market model with portfolio constraints. In Section 3, we define the set of axioms on price functionals and we provide a dual representation of any admissible price functional in terms of mathematical expectation under a suitable family of probability measures satisfying a generalization of the martingale property of securities prices. This set of probability measures is closely related to a no-arbitrage condition, see e.g., Jouini and Kallal Ž1995a., Pham and Touzi Ž1999. and Carassus et al. Ž1998.. The key tool for deriving the main result ŽTheorem 2.1. is the use of the Fenchel–Moreau theorem which appears to be very well-suited in
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this axiomatic approach. Section 4 is devoted to the applications. First, we show that if the super-replication cost function is not sublinear, our axiomatic approach leads to a reduction of the bid-ask spread with regard to the usually too large super-replication bid-ask spread. We illustrate with an example. On the other hand, if the super-replication cost is sublinear, then we state a discrete time version of Cvitanic´ and Karatzas Ž1993. and Karatzas and Kou Ž1996. results on the dual probabilistic representation of the super-replication cost.
2. The financial market model F, P . be a complete probability space equipped with a filtration Let Ž V ,F F s Ft ,t s 0, . . . ,T 4 where T ) 0 is a finite horizon. For convenience, we assume as usual that F0 is trivial and FT s F. The space L2 is the set of all square integrable random variables. The financial market model consists in one riskless asset of price equal to one and n risky assets of price process St s Ž St1, . . . ,Stn .X , t s 0, . . . ,T 4 assumed to be F-adapted. Here the notation X is for the transposition. For each t s 0, . . . ,T, we denote by diagŽ St . the n = n diagonal matrix whose ith diagonal term is Sti. We shall assume that for each t s 0, . . . ,T, diagŽ St . is nonsingular almost surely and we denote diagŽ St .y1 its inverse matrix. We denote the increment process of S by D St s St y Sty1 , t s 1, . . . ,T 4 and the return process of S by R t s diagŽ Sty1 .y1 D St , t s 1, . . . ,T 4 . We assume that R t g L2 for all t s 1, . . . ,T. We denote by Q the set of all R n-valued Ft , t s 0, . . . ,T y 14 adapted processes such that u tX D Stq1 g L2 for all t s 0, . . . ,T y 1. An element u t s Ž u t1, . . . , u tn .X , t s 0, . . . ,T y 14 of Q is a trading strategy where u ti is interpreted as the number of units invested in the ith risky asset at date t. Remark 2.1. Ž1. The nonsingularity on the process diag Ž S . means that for each t s 0, . . . ,T and i s 1, . . . ,n, P w Sti s 0x s 0. This is satisfied, for instance, in the usual case where S is a strictly positive process. Ž2. The assumption that R t g L2 can be weakened in R t g L1 , t s 1, . . . ,T, by imposing that trading strategy u satisfy u t g L1 and that probability measures introduced in Section 3.2 have a density in L` Ždual of L1 .. Given an initial wealth x g R and a trading strategy u g Q , the Žself-financed. wealth process evolves according to: ty1
X tx , u s x q Ý uuX D Suq1 , us0
t s 0, . . . ,T .
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The wealth proportion R n-valued process is then defined by:
p tx , u s
diag Ž St . u t X tx , u
if X tx , u / 0, t s 0, . . . ,T .
Ž 1.1 .
We now constrain the trading strategies as in Cvitanic´ et al. Ž1999.. Let us consider two nonempty convex sets Kq and Ky of R n containing the origin 0. We define the convex cones of R n : Kqs x g R n : ;p g Kq , p X x F 0 4 , Kys x g R n : ;p g Ky , p X x F 0 4 . For any x g R, the set of constrained trading strategies is defined by:
Q Ž x . s u g Q : p tx , u g Kq if X tx , u ) 0, p tx , u g Ky if X tx , u - 0, u t s 0 if X tx , u s 0 4 . In other words, Kq Žresp. Ky . represents constraints on proportions when the wealth is positive Žresp. negative..Remark 2.2. In Cvitanic´ and Karatzas Ž1993. or Karatzas and Kou Ž1996., the authors define the Žself-financed. wealth process from an initial wealth x and a process p interpreted as a proportion process by: x ,p s X tx ,p Ž 1 q p tX R tq1 . , X 0x ,p s x and X tq1
t s 0, . . . ,T y 1.
According to relation Ž1.1., this formulation of wealth process is equivalent to the one with the number of units u if and only if the wealth process X tx, u / 0, for all t, almost surely. In particular, with the formulation of the wealth process in terms x,p of proportion, we see that if X tx,p s 0 then X tq1 s 0 while this is not necessarily the case when the wealth process is expressed in function of the number of units. This is the reason why we add the constraint u t s 0 if X tx, u s 0 in order to ensure x, u that X tq1 s 0. Remark 2.3. Notice that for all l ) 0 and x g R, p tl x , lu s p tx, u . Moreover, since Q Ž0. s 04 , this implies that:
Q Ž l x . s lQ Ž x . , ;l G 0, ; x g R.
Ž 1.2 .
Here are some examples of convex constraints taken from Karatzas and Kou Ž1996.. Example 2.1. Ži. Unconstrained case: Kqs Kys R n. Then Kqs Kys 04 . Žii. Prohibition of short-selling of stocks: Kqs yKys w0,`. n. Then Kqs yKys n w Žy`,0x n. Žiii. Constraints on the short-selling of stocks: Kqs P is1 yk i ,`., n Kys P is1 Žy`,l i x with k i , l i ) 0 for i s 1, . . . ,n. Then Kqs Kys 04 . Živ. n n w wykXi ,lXi x with k i , kXi , l i , Rectangular constraints: Kqs P is1 yk i ,l i x, Kys P is1 X l i ) 0 for i s 1, . . . ,n. Then Kqs Kys 04 . Žv. Incomplete market, in which only
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the m Žwith m - n. first stocks can be traded: Kqs yKys K s p g R n : p i s 0, i s m q 1, . . . ,n4 . Then K s x g R n : x i s 0, i s 1, . . . ,m4 . Žvi. Incomplete and constrained market with prohibition of short-selling of the first n1 stocks, prohibition of being long in the next n 2 stocks and prohibition of investment in the next n 3 stocks Žwith n1 q n 2 q n 3 F n.: Kqs yKys K s p g R n :p i G 0, i s 1, . . . ,n1 , p i F 0, i s n1 q 1, . . . ,n1 q n 2 , p i s 0, i s n1 q n 2 q 1, . . . ,n1 q n 2 q n 34 . Then K s x g R n : x i F 0, i s 1, . . . ,n1 , x i G 0, i s n1 q 1, . . . ,n1 q n 2 , x i s 0, i s n1 q n 2 q n 3 , . . . ,n4 . Žvii. Both Kq and Ky are closed convex cones with vertex in zero: This clearly generalizes all the previous cases except Žiii. and Živ.. Žviii. Constraints on borrowing: Kqs p g R n : Ý nis1p i F k 4 and Kys p g R n : Ý nis1p i G l 4 for some k G 1 and l F 0. Then Kqs Kys 04 .
3. Admissible price functionals 3.1. Axiomatic of the price functionals A price functional is a function c defined on L2 and valued in R j q`4 , where c Ž X . represents the price at which the contingent claim X can be bought. We say that the price functional is admissible if it satisfies the following conditions: ; X , Y g L2 , X F Y c Ž X . F c Ž Y .
Ž A1.
;l G 0, ; X g L2 ,
Ž A2.
2
;X, YgL 2
;XgL
c Ž l X . s lc Ž X .
c Ž XqY . Fc Ž X . qc Ž Y .
c Ž X . Ff Ž X .
c is lower semi-continuous
Ž A3. Ž A4 . Ž A5.
where f is the super-replication cost function defined by:
f Ž X . s inf x g R: 'u g Q Ž x . , X Tx , u G X 4 , ; X g L2 , with the convention that infB s q`. Condition ŽA1. is a nondecreasing property of the price functional. In words, no one will accept to buy a contingent claim if it is possible to obtain a better payoff for a lower price. Condition ŽA3. means that it is cheaper to buy the sum X q Y of two contingent claims than to buy X and Y separately. The economic reason for this sublinearity condition is the following: if it was not true for some X and Y, then the cheapest way to buy X q Y is to buy them separately, for a cost c Ž X . q c Ž Y . - c Ž X q Y ., and then c Ž X q Y . cannot be an equilibrium price. The economic interpretation of ŽA4. is that if it is possible to obtain a better payoff than X at a cost f Ž X ., then no one will accept to pay more than f Ž X ..
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The positive homogeneity condition ŽA2. is less natural than the others. It is assumed in all the classical financial market models, since, for a mathematical viewpoint, it is required for applying the Riesz representation theorem on which are based the main theorems of asset pricing Žsee, e.g., Harrison and Kreps, 1979 in a frictionless market and Jouini and Kallal, 1995a in a no short-sales constraints model.. Condition ŽA2. implies that c Ž0. s 0, which joined with condition ŽA1. ensures that c Ž X . G 0 for all X G 0. In words, one cannot receive money for holding a nonnegative contingent claim; this can be interpreted as a no-arbitrage condition. The technical condition ŽA5. means that if X n converges to X in C , then lim inf c Ž X n . G c Ž X .. Notice that Jouini Ž1997. has introduced the same axioms except axiom ŽA1. which is replaced by: ; X g L2 , X ) 0, c Ž X . ) 0.
¨
Remark 3.1. The functional c is interpreted as the ask-price functional. Therefore, the function X g C yc ŽyX . is the bid-price functional. Remark 3.2. Using ŽA3., ŽA4. and c Ž0. s 0, we have: ; X g L2 , yf ŽyX . F yc ŽyX . F c Ž X . F f Ž X .. It implies in particular that the bid-ask spread c Ž X . q c ŽyX . lies in the interval w0, f Ž X . q f ŽyX .x. We say that f Ž X . q f ŽyX . is the super-replication bid-ask spread.
3.2. Dual representation of admissible price functionals We introduce the set P of probability measures Q absolutely continuous with respect to P, with density dQrd P g L2 and satisfying the following condition:
Ž P . E Q R tq1 < Ft g Kq F y Ky , t s 0, . . . ,T y 1, Q a.s. In the sequel, we shall identify probability measures and their densities; we say that Q n s Z n P converges in L2 to Q s ZP if Z n converges to Z in L2 . We denote by P e the elements of P which are equivalent to P. Remark 3.1. Suppose that P e / B. Then, it is easily checked that the closure of P e in L2 is equal to P. Indeed, take an arbitrary Q g P and let Q 0 g P e. The sequence Qn s 1y
ž
1 n
/
1 n
Q 0 , n g NU ,
obviously lies in P e and converges in L2 to Q. By noting that P is closed in L2 , this proves the result. The condition that P e is not empty is related to a no arbitrage condition: for more details, we refer to Jouini and Kallal Ž1995a. in a short-sales constraint model, Pham and Touzi Ž1999. for an extension to the case of cone constraints, and Carassus et al. Ž1998. in the case of convex constraints as considered here.
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We first prove a useful lemma. Lemma 3.1. Let Q g P. Then, for all x g R, for all u g Q Ž x ., the wealth process X tx, u , t s 0, . . . ,T 4 is a supermartingale under Q. Proof. Let t g 0, . . . ,T y 14 . By definition of the wealth process and the return process, we have: x ,u s X tx , u q u tX diag Ž St . R tq1 . X tq1
If X tx, u / 0, then we have by definition of the proportion process: X
x ,u X tq1 s X tx , u q X tx , u Ž p tx , u . R tq1
By property Ž P . of Q, we have X tx, u Žp tx, u .X E Q w R tq1
X tx, u s 0,
If result.
then
x, u X tq1 s0
Ž 2.1 . and Ž2.1. still holds, which proves the required I
We now state the main result of this section which provides a complete dual characterization of admissible price functionals. Theorem 3.1. c is an admissible price functional if and only if there exists a subset Q of P such that:
c Ž X . s sup E Q Ž X . , ; X g L2 .
Ž 2.2 .
QgQ
Proof. Let c be a functional given by Ž2.2.. It is clear that it satisfies the axioms ŽA1., ŽA2., ŽA3. and ŽA5.. Now, consider some X g L2 and x g R, u g Q Ž x ., such that X F X Tx, u . By Lemma 2.1, this implies that E Q w X x F x for all Q g P and therefore c Ž X . F f Ž X . which means that condition ŽA4. is satisfied and so c is an admissible price functional. Let c be an admissible price functional. Axioms ŽA2. and ŽA3. imply that c is a convex function on the Hilbert space L2 . We consider then the convex conjugate of c defined on L2 by:
c˜ Ž Y . s sup E Ž XY . y c Ž X . 4 XgL2
and valued on R j q`4 . The domain of c˜ is denoted Dc˜ :s Y g L2 : c˜ Ž Y . - `4 . From condition ŽA2., it is clear that: Dc˜ s Y g L2 : ; X g L2 , E Ž XY . F c Ž X . 4 and c˜ Ž Y . s 0,
;Y g Dc˜ .
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Therefore, by the Fenchel–Moreau theorem Žsee, e.g., Theorem I.10 in Brezis, ´ 1983., we have:
c Ž X . s sup E Ž YX . , ; X g L2 . Yg Dc˜
Let us first prove that Y G 0 for all Y g Dc . Indeed, denoting N s v g V :Y Ž v . F 04 , we have by ŽA1. and ŽA2. that EŽyY 1 N . F c Žy1 N . F c Ž0. s 0. But from definition of N , we also have EŽyY 1 N . G 0 which implies that EŽyY 1 N . s 0 and then Y s 0 on N . We now check that EŽ Y . s 1 for all Y g Dc˜ . Indeed by ŽA4., we have: E Ž YX . F c Ž X . F f Ž X . , ; X g L2 , ;Y g Dc˜
Ž 2.3 .
Choose X s 1. Then it is clear that the initial wealth x s 1 associated to a zero strategy u s 0 g Q Ž x . leads to a terminal wealth X Tx, u s 1 and so f Ž1. F 1. By Eq. Ž2.3., we deduce that EŽ Y . F 1. For the choice of X s y1, we deduce by a similar argument that EŽyY . F y1 and therefore EŽ Y . s 1. We have then proved that each element Y g Dc˜ can be identified with a probability measure Q absolutely continuous with respect to P, with density in L2 , so that
c Ž X . s sup E Q Ž X . , ; X g L2 .
Ž 2.4 .
Qg Dc˜
Let us finally check that Dc˜ is a subset of P by showing property Ž P . for each Q g Dc˜ . From the representation Ž2.4. and condition ŽA4. applied to elements X s X Tx, u , x g R, u g Q Ž x ., we have: Ty1
EQ
žÝ
uuX D Suq1 F 0, ;Q g Dc˜ , ; x g R, ;u g Q Ž x . .
/
us0
Ž 2.5 .
Fix t g 0, . . . ,T y 14 . Let x ) 0 Žresp. x - 0. and choose an arbitrary p g Kq Žresp. Ky . and F g Ft . Consider the following trading strategy: uu s 0, u s 0, . . . ,t y 1, t q 1, . . . ,T y 1,
u t s diag Ž St .
y1
p x1 F .
Notice that u g Q since uuX D Suq1 s 0 for u s 0, . . . ,t y 1,t q 1, . . . ,T y 1 and u tX D Stq1 s x1 F p X R tq1 g L2 . The associated wealth process is X ux , u s x ,
u s 0, . . . ,t ,
X ux , u s x q u tX D Stq1 ,
u s t q 1, . . . ,T .
The associated wealth proportion process is
pux , u s 0, p tx , u s
u s 0, . . . ,t y 1,
diag Ž St . u t
pux , u s 0,
X tx , u
s p 1F ,
u s t q 1, . . . ,T y 1 if X ux , u / 0.
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Hence u g Q Ž x . and by substituting this strategy into Eq. Ž2.5. and by the arbitrariness of F g Ft , we get: xp X E Q R tq1 < Ft F 0, Q a.s. for all t s 0, . . . ,T y 1, x ) 0 Žresp. x - 0. and p g Kq Žresp. p g Ky ., which is exactly the property Ž P . for Q g Dc˜ . I Remark 3.2. The key result for deriving the dual representation formula Ž2.2. is the Fenchel–Moreau theorem. The idea of using this theorem appears more and less in Jouini and Kallal Ž1995b. ŽTheorem 2.2., although these authors cannot directly apply it since they do not work on vector space.
4. Applications 4.1. Reduction of the bid-ask spread One of the main topic in finance is the pricing of contingent claims. In incomplete or imperfects markets, there is no unique arbitrage-free price. Since Harrison and Kreps Ž1979., an important literature is devoted to find an interval as small as possible, in which the ‘‘fair’’ price must lie. Until now, as originated by Bensaid et al. Ž1992., the bid-ask spread is usually obtained from the super-replication cost; they obtain an interval wyf ŽyX ., f Ž X .x for the ‘‘fair’’ price of a contingent claim X. However, it is known that the super-replication spread can be too large. Indeed, several authors proved that the super-replication cost of a contingent claim can be no less than the cost of a buy-and-hold strategy, for example the price of the underlying risky asset in the case of a call option, see e.g., Soner et al. Ž1995. and Cvitanic´ et al. Ž1999.. Due to our additional sublinearity condition ŽA3., we show that we can restrict this bid-ask spread. We define the functional
F Ž X . s sup E Q Ž X . , ; X g L2 . Qg P
Notice from Remark 2.1 Žii. that if P e is not empty, the supremum in the formula of F can be taken over P e :
F Ž X . s sup E Q Ž X . , ; X g L2 Qg P e
We also denote f U the largest lower semi-continuous function below f , i.e.:
f U Ž X . s inf lim inf f Ž X n . , X n
™X 4, ;XgL . 2
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Theorem 4.1. Ž i . F F f U . Ž ii . For all admissible pricing functional c , we get: yF Ž yX . s inf E Q Ž X . F c Ž X . F F Ž X . s sup E Q Ž X . , ; X g L2 . QgP
Qg P
Proof. Ži. From Theorem 2.1, F is an admissible price functional and then F F f by axiom ŽA4.. Now, since F is a lower semi-continuous function below f , we obtain the first assertion from the definition of f U . Žii. The second assertion is an immediate consequences of Theorem 2.1 and the definition of F . I The last theorem says that an admissible price functional lies between a supremum and an infinum of expected values under a family of probability measures satisfying property Ž P .. This is similar to Jouini Ž1997. result in a transaction costs framework. However, while this supremum is equal to the super-replication cost in a transaction costs model, we shall see in the following example that in a model with portfolio constraints, the bid-ask spread induced by our approach, F Ž X . q F ŽyX ., can be strictly smaller than the super-replication bid-ask spread for some contingent claims X. Example. We choose T s 1 and consider a finite probability space V s v 1 , v 2 4 , which implies in particular that f U s f . We have one riskless bond of price equal to one and one risky asset such that S0 s 1. At date t s 1, S1 can take the values S1Ž v 1 . s u and S1Ž v 2 . s d, with d - u. A contingent claim X is identified with the pair Ž a , b . s Ž X Ž v 1 ., X Ž v 2 .. g R 2 . We model the constraints by Kqs w0,1x and Kys wy1,0x. Then, it is easily checked that Q Ž x . s w0, < x
f Ž a , b . s inf x g R: 'u g w 0, < x < x , x q u Ž u y 1 . G a and x q u Ž d y 1 . G b 4 An easily calculation shows that
½
ž
f Ž a , b . s inf x g R: x G max b ,
a Ž 1 y d . q b Ž u y 1. uyd
/
and
x q < x < Ž u y 1. G a 4 . In the case u - 2, the last expression simplifies into:
°max
a Ž 1 y d . q b Ž u y 1. a , uyd u f Ž a ,b . s a Ž 1 y d . q b Ž u y 1. a max b , , uyd 2yu
~
¢
ž ž
b,
/
if a G 0,
/
if a - 0.
P.-F. Koehl, H. Phamr Journal of Mathematical Economics 33 (2000) 339–351
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Now, by noting that Kqs yKys Ry and R 1 s Ž u y 1,d y 1., we easily check that 1yd 2 P s Ž q,1 y q . g w 0,1 x : 0 F q F . uyd
½
5
We then explicitly determine the functional F :
F Ž a ,b . s
Ž 1 y d . max Ž a , b . q Ž u y 1 . b
. uyd If we take, for example, u s 3r2, d s 1r2 and a call option of exercise price k s 1, i.e., X s Ž1r2,0., an immediate computation shows that f Ž X . s 1r3 and f ŽyX . s 0. Then, the super-replication spread for this contingent claim is 1r3. Moreover, F Ž X . s 1r4 and F ŽyX . s 0. Hence, this example shows that we can obtain a significant reduction of the bid-ask spread by requiring sublinearity of price functionals. 4.2. Dual representation of the super-replication cost function There is a large literature in finance devoted to the dual characterization of the super-replication cost. We mention, among others, in continuous-time: El Karoui and Quenez Ž1995. in incomplete markets, Cvitanic´ and Karatzas Ž1993. for constrained portfolios, and in discrete-time: Jouini and Kallal Ž1995a. for no short-sales constraint and short-selling cost. Our axiomatic approach allows us to prove the same kind of result. Theorem 4.2 f ) s F if and only if f U is sublinear. Proof. Assume that f U s F . Then f U is an admissible price functional and, from axiom ŽA3., it is sublinear. Conversely, assume that f U is sublinear, i.e., it satisfies condition ŽA3.. Since f satisfies obviously conditions ŽA1. and ŽA4., it goes also for f U . By Ž1.2., we have f U Ž l X . s l f U Ž X ., ;l ) 0. It remains to prove that f U Ž0. s 0. By ŽA1., we get f U Žyl. s lf U Žy1. F f U Ž0. for all l ) 0. Sending l to zero, we obtain f U Ž0. G 0. Since we obviously have f U Ž0. F f Ž0. F 0, this shows that f U Ž0. s 0. Therefore, f U is an admissible price functional, and from Theorem main, we obtain that f U F F . Using the fact that f U G F Žsee Theorem 3.1 Ži.., this ends the proof. I In the case where the super-replication cost is sublinear, the last theorem provides a dual characterization of f U in terms of the set of probability measures P:
f U Ž X . s sup E Q w X x , ; X g L2 . Qg P
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However, this result does not hold any more if f U is not sublinear as in the example described in the previous paragraph. In that case, it leads to a reduction of the bid-ask spread. Notice that this dual representation is not exactly stated for the super-replication cost function but for its lower-semicontinuous envelope f U . In general, f is not lower-semicontinuous and is then different from f U . In a continuous-time diffusion setting, Cvitanic´ and Karatzas Ž1993. and Karatzas and Kou Ž1996. obtained a dual representation of the ‘true’ super-replication f , which implies in particular that it is lower-semicontinuous. In a general discrete-time context, such a property would require a closedness property of the set of all dominated contingent claims, see Brannath Ž1997. for more details. We mention that Brannath Ž1997. has obtained a dual representation formula for the super-replication cost in a discrete-time model with convex constraints on the number of shares u and it appears that in this context, the super-replication cost is sublinear if and only if the so-called upper-variation process is zero. His result is a discrete-time version of the paper of Follmer and Kramkov Ž1997.. These last ¨ cited authors also obtained a representation formula for the super-replication cost in a model with convex constraints on proportions but for nonnegative claims and nonnegative wealth. It would be of interest to derive similar result in our discrete-time context with convex constraints on proportions and without requiring nonnegativity of wealth process.
References Bensaid, B., Lesne, J.P., Pages, ¨ H., Scheinkman, J., 1992. Derivative asset pricing with transaction costs. Mathematical Finance 2, 63–86. Brannath, W., 1997. No arbitrage and martingale measures in option pricing. Doctoral dissertation, Wien Universitat. ¨ Brezis, et applications. Masson. ´ H., 1983. Analyse fonctionnelle. Theorie ´ Carassus, L., Pham, H., Touzi, N., 1998. No arbitrage in discrete time under portfolio constraints, preprint. Cvitanic, ´ J., Karatzas, I., 1993. Hedging contingent claims with constrained portfolios. Annals of Applied Probability 3, 652–681. Cvitanic, ´ J., Pham, H., Touzi, N., 1999. Super-replication in stochastic volatility models under portfolio constraints. Journal of Applied Probability 36. El Karoui, N., Quenez, M.C., 1995. Dynamic programming and pricing of contingent claims in an incomplete market. SIAM Journal of Control and Optimization 33, 29–66. Follmer, H., Kramkov, D., 1997. Optional decomposition under constraints. Probability Theory and ¨ Related Fields 109, 1–25. Harrison, J.M., Kreps, D., 1979. Martingales and arbitrage in multiperiods securities markets. Journal of Economic Theory 20, 381–408. Harrison, J.M., Pliska, S., 1981. Martingales and stochastic integrals in the theory of continuous trading. Stochastic Processes and their Applications 11, 215–260. Jouini, E., 1997. Price functionals with bid-ask spreads: an axiomatic approach. Preprint CREST. Jouini, E., Kallal, H., 1995a. Arbitrage in securities markets with short sales constraints. Mathematical finance 5, 197–232.
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Jouini, E., Kallal, H., 1995b. Martingale and arbitrage in securities markets with transaction costs. Journal of Economic Theory 66, 178–197. Karatzas, I., Kou, S., 1996. On the pricing of contingent claims under constraints. Annals of Applied Probability 6, 321–369. Kreps, D., 1981. Arbitrage and equilibrium in economies with infinitely many commodities. Journal of Mathematical Economics 8, 15–35. Pham, H., Touzi, N., 1999. The fundamental theorem of asset pricing with cone constraints. Journal of Mathematical Economics 31, 265–279. Soner, M., Shreve, S., Cvitanic, ´ J., 1995. There is no nontrivial hedging portfolio for option pricing with transaction costs. Annals of Applied Probability 5, 327–355.