No-arbitrage in discrete-time under portfolio constraints1 Laurence Carassus

Huyˆen Pham

Nizar Touzi

Laboratoire de Probabilit´es et

Laboratoire de Probabilit´es et

CERMSEM

Mod`eles Al´eatoires

Mod`eles Al´eatoires

Universit´e Paris 1

Universit´e de Paris 7

Universit´e de Paris 7

[email protected]

and CREST

and CREST

[email protected]

[email protected]

December 1997 this version : October 1999

Abstract In frictionless securities markets, the characterization of the no arbitrage condition by the existence of equivalent martingale measures in discrete time is known as the Fundamental Theorem of Asset Pricing. In the presence of convex constraints on the trading strategies, we extend this theorem under a closedness condition and a nondegeneracy assumption. We then provide connections with the super-replication problem solved in F¨ollmer and Kramkov (1997).

Keywords : Fundamental theorem of asset pricing, arbitrage, super-replication cost, convex portfolio constraints, stochastic process.

1

We are grateful to Damien Lamberton for useful remarks and comments.

1

1

Introduction

We study a discrete-time financial market consisting of d risky assets, with the discounted price process, denoted S, and one riskless bond. In contrast with usual frictionless models, we consider the case where trading strategies are subject to portfolio constraints as in the framework of Cvitani´c and Karatzas (1992,1993) and Karatzas and Kou (1996); see also the textbook exposition by Pliska (1997). Namely, given two convex sets C + and C − containing the origin, the vector of wealth proportions invested in the risky assets is constrained to lie in C + or C − depending on the sign of the current wealth process. A precise description of the model is provided in Section 2. In this paper, we address the characterization of no arbitrage in such a financial market. Loosely speaking, an arbitrage opportunity is a way to produce nonnegative wealth with positive expected value out of nothing. Therefore, there should not exist such strategies in real markets. This problem has been studied in frictionless markets by Harrison and Kreps (1979), Harrison and Pliska (1981), Kreps (1981), Dalang, Morton and Willinger (1990), Schachermayer (1992) and Kabanov and Kramkov (1994). The Fundamental Theorem of Asset Pricing (FTAP) in this frictionless framework is the following : There is no arbitrage opportunity if and only of there exists an equivalent probability measure which turns the process S into a martingale. Jouini and Kallal (1995) and Sch¨ urger (1996) provided an extension of the FTAP to the case where trading strategies are subject to the no short sales condition. The case of (closed) cone constraints C on the amounts invested in the risky assets has been studied by Pham and Touzi (1999). Under a nondegeneracy assumption on the process S, they proved the following extension of the FTAP : There is no arbitrage opportunity if and only of there exists an equivalent probability measure Q (with bounded density) such that E Q [diag(St )−1 (St+1 − St )|Ft ] ∈ C o , for all t, where C o is the polar cone of C and diag(St ) is the diagonal matrix whose i-th diagonal element is Sti . This result has been extended by Brannath (1997) to the case of closed convex constraints on the invested amount. In our constrained proportion-portfolio framework, the self-financed wealth process is written in a multiplicative form (see relation (2.2)). Therefore, whenever the initial capital is 1

zero, the wealth process is zero for any proportion-portfolio choice. We introduce here a more general definition of arbitrage. Given an initial wealth x and a constrained portfolio strategy π, we define the excess wealth as the difference between the terminal wealth associated with (x, π), and the initial wealth x. We then define an arbitrage opportunity as a way to produce a nonnegative excess wealth with positive expected value. The main result of the paper is contained in Section 3 and proved in section 6. We show that there is no arbitrage if and only if there exists a pair of equivalent probability measures − + (Q+ , Q− ) such that E Q [diag(St )−1 (St+1 − St )|Ft ] ∈ Cˆ + and E Q [diag(St )−1 (St+1 − St )|Ft ]

∈ −Cˆ − , where Cˆ = {x ∈ IRd : < x, y > ≤ 0 for all y ∈ C}. The last result is established for convex sets C + and C − with closed generated cones and under a nondegeneracy assumption as in Pham and Touzi (1999). Section 4 contains relevant examples of application of the main result. In Section 5, we focus on the super-replication problem, i.e. the minimal initial wealth needed to hedge without risk some given contingent claim. F¨ollmer and Kramkov (1997) provided a dual formulation of the super-replication cost. We relate their assumption to the no-arbitrage concept, as in the usual literature on this problem.

2

The model

Let (Ω, F, P ) be a complete probability space equipped with a filtration IF = {Ft , t = 0, ..., T } where T > 0 is a finite time horizon. The set Ft represents the whole information available at time t. We make the usual assumption that F0 is trivial and FT = F. We first introduce some notation. The space L0,d is the space of all IRd -valued Ft t measurable functions. The space Lt0,1 will be simply denoted L0t . For 1 ≤ p ≤ ∞, the space Lp,d is the Banach space of all IRd -valued Ft measurable functions with finite Lp norm. As t before the superscript d will be omitted when d = 1. We also use the classical notation : IR∗ = IR \ {0}, Lp,d+ = {z ∈ Lp,d : z ≥ 0 P a.s }, for 0 ≤ p ≤ ∞, and |x| = t t

Pd

i=1

|xi |, for

x ∈ IRd . The financial market model consists of one riskless asset with price process normalized to one and d risky assets with price process S = {St = (St1 , ..., Std )∗ , t = 0, ..., T } valued in 2

(0, ∞)d and IF -adapted. Here the notation

∗

is for the transposition. We denote the return

process associated with S by {Rt = diag(St−1 )−1 (St − St−1 ), t = 1, . . . , T }. A trading strategy is a IRd -valued IF -adapted process π = {πt = (πt1 , . . . , πtd )∗ , t = 0, ..., T − 1}, where πti represents the proportion of wealth invested in the i-th risky asset at time t. Given an initial wealth x ∈ IR∗ and a trading strategy π, it follows from the self-financing condition that the wealth process X x,π is governed by : X0x,π = x x,π Xt+1 = Xtx,π (1 + πt∗ Rt+1 ) , for t = 0, . . . , T − 1

(2.1)

The induction equation (2.1) leads to Xtx,π = x +

t−1 X

Xux,π πu∗ Ru+1 ,

t = 0, . . . , T.

(2.2)

u=0

Remark 2.1 Our class of trading strategies based on proportion leaves out portfolio strategies starting from initial wealth x = 0. Indeed, in this case, the proportion π0 at initial date t = 0, is not defined from the amount θ0 by π0 = θ0 /x. Therefore, we have considered initial wealth x ∈ IR∗ . Remark 2.2 Given an initial wealth x ∈ IR∗ and a trading strategy π, if we define θt = πt Xtx,π t = 0, . . . , T, then θti , i = 1, . . . , d represents the amount held in the i-th risky asset and ηt = Xtx,π −

Pd

i i=1 θt

is the amount of wealth in the riskless asset. Moreover, we have the self-financing condition written in additive form : x,π Xt+1 = Xtx,π + θt∗ Rt+1 ,

t = 0, . . . , T − 1.

Remark 2.3 Let x ∈ IR∗ be some initial weath and consider a trading strategy π. Then, for all λ ≥ 0, we have : Xtλx,π = λXtx,π

for all t = 0, . . . , T . 3

We now impose some constraints on the trading strategies. Let us consider two nonempty convex sets C + and C − of IRd containing the origin 0, where C + 6= {0}. For any x ∈ IR∗ , we say that a trading strategy is admissible, and we denote π ∈ A(x), if for all t = 0, . . . , T − 1 : πt ∈ C + if Xtx,π > 0, πt ∈ C − if Xtx,π < 0. In other words, C + (resp. C − ) represents constraints on proportions when the wealth is positive (resp. negative). Such constraints cover the usual examples as described in Karatzas and Kou (1996), see section 4. Remark 2.4 From Remark 2.3, it is clear that A(x) depends only on the sign of x, i.e. A(x) = A(1)

and

A(−x) = A(−1) for all x > 0 .

In the sequel, we shall denote by A = {(x, π) : x ∈ IR∗ , π ∈ A(x)}. Definition 2.1 We say that there is no arbitrage opportunity if, for all trading strategies (x, π) ∈ A we have, XTx,π − x ≥ 0

P − a.s =⇒ XTx,π − x = 0

P − a.s.

Loosely speaking, an arbitrage opportunity is a way to produce nonnegative excess wealth with positive expected value out of nothing. Remark 2.5 Suppose that (x, π) ∈ A defines an arbitrage opportunity. Then for all λ > 0, it follows from Remark 2.3 that ((x/λ), π) also defines an arbitrage opportunity. In other words, if the financial market is not arbitrage-free then arbitrage opportunities can be realized starting from any small amount of initial wealth.

3

The main results

In this section, we provide a characterization of the no arbitrage condition. We first introduce some notation. We define the convex cones of IRd : Cˆ + =

n

o

Cˆ − =

n

o

x ∈ IRd : π ∗ x ≤ 0, ∀π ∈ C + ,

x ∈ IRd : π ∗ x ≤ 0, ∀π ∈ C − . 4

We also denote cone(C + ) = {λπ : λ > 0, π ∈ C + } and cone(C − ) = {λπ : λ > 0, π ∈ C − }. We shall need the following assumption. Assumption 3.1 The sets cone(C + ) and cone(C − ) are closed in IRd . Remark 3.1 The assumption that cone(C + ) and cone(C − ) are closed is quite strong because both C + and C − contain the origin. Here is an example of compact convex set K in IR2 which contains the origin and such that cone(K) is not closed : let K = {(x, y) : x2 + (y − 1)2 ≤ 1}, then cone(K) = IR × (0, ∞) ∪ {(0, 0)}. However, Assumption 3.1 is satisfied in all practical examples; see our examples section. We also need to introduce the sets P + and P − : )

(

P

+

P−

dQ ∈ L∞ , Rt ∈ L1 (Q) and E Q [Rt |Ft−1 ] ∈ Cˆ + , 1 ≤ t ≤ T P − a.s. = Q∼P : dP ( ) dQ ∞ 1 Q − ˆ = Q∼P : ∈ L , Rt ∈ L (Q) and E [Rt |Ft−1 ] ∈ −C , 1 ≤ t ≤ T P − a.s. dP

In order to establish the main result, we need the following non-degeneracy assumption. Assumption 3.2 Let t = 0, . . . , T − 1. Then for all Ft -measurable random variables ϕ valued in C + ∪ (−C − ), ϕ∗ Rt+1 (ω) = 0 =⇒ ϕ(ω) = 0

for a.e. ω ∈ Ω .

Example 3.1 The above condition is trivially satisfied in the one-dimensional Cox-RossRubinstein model with parameters d < 1 < u. Indeed in this case, Rt+1 is valued in {u − 1, d − 1}, and P [Rt+1 = u − 1|Ft ] > 0 and P [Rt+1 = d − 1|Ft ] > 0. Example 3.2 Let us check Assumption 3.2 in the case of the Black and Scholes model. Suppose that {St , t = 0, . . . , T } is a discrete sample extracted from the geometric Brownian motion dSt = St (µdt+σdWt ), where µ and σ > 0 are some constants. Then a straightforward calculation shows that : 

2



V [St+1 |Ft ] = St2 e2µ eσ − 1

> 0,

where V (.|Ft ) is the conditional variance operator. Now take some real-valued Ft -measurable random variable ϕ such that ϕRt+1 = 0. Then taking conditional variance we see that ϕ2 V (St+1 |Ft ) = 0 and thus ϕ = 0. 5

Example 3.3 Suppose that V [(Rt+1 /αt+1 )|Ft ] exists and is invertible P -a.s. for some Ft+1 measurable positive random variable αt+1 . Then, considering ϕ as in Assumption 3.2, we see that V [(ϕ∗ Rt+1 /αt+1 )|Ft ] = ϕ∗ V [Rt+1 /αt+1 )|Ft ]ϕ = 0. Therefore ϕ = 0 P -a.s. and Assumption 3.2 holds. We are now in a position to state the main result of this section. Theorem 3.1 Under Assumptions 3.1 and 3.2, the following assertions are equivalent : (i) there is no arbitrage opportunity (ii) P + 6= ∅ and P − 6= ∅. The proof of this result is reported in subsection 6.1. An important (and somewhat surprising) feature of Theorem 3.1 is that the arbitrage characterization (ii) involves the constraints sets C + and C − only through the associated cones Cˆ + and Cˆ − . Notice that d +) = C d −) = C ˆ + and cone(C ˆ − . Therefore, the no-arbitrage characterization with cone(C

constraints sets (C + , C − ) is identical to the no-arbitrage characterization with constraints sets (cone(C + ), cone(C − )). This provides the following connection with the no-arbitrage condition under cone constraints; see Pham and Touzi (1999). Corollary 3.1 Let Assumptions 3.1 and 3.2 hold. Then there is no arbitrage opportunity if and only if for all IF -adapted processes θ valued in cone(C + ) or in cone(−C − ) we have TX −1

θt∗ Rt+1 ≥ 0

P − a.s. =⇒

TX −1

θt∗ Rt+1 = 0

P − a.s.

t=0

t=0

d +) = C d −) = C ˆ + and cone(C ˆ − , the result follows directly from the dual Proof. Since cone(C

characterization of no-arbitrage under cone constraints in Pham and Touzi (1999).

2

Remark 3.2 Corollary 3.1 says that our notion of no-arbitrage is in fact equivalent to the no arbitrage condition in a financial market where the amounts of wealth invested in the risky assets are constrained to lie in cone(C + )∪cone(−C − ). In the case C + = −C − , this is the financial market considered in Pham and Touzi (1999).

6

We now provide conditions on the portfolio constraints in order to ensure that P + ∩ P − 6= ∅. This would guarantee the existence of some Q ∈ P + ∩ P − under which any wealth process is a supermartingale; see also Remark 6.1. Furthermore, it would strenghten characterization (ii) of the no arbitrage condition in Theorem 3.1. We denote for all t = 0, . . . , T − 1 : At =

n

o

(x, ν) ∈ IR × L0,d : ν ∈ C + (resp. C − ) if x > 0 (resp. < 0) . t

Assumption 3.3 For all t = 0, . . . , T − 1, for all (x1 , ν1 ) ∈ At , (x2 , ν2 ) ∈ At , there exists (x, ν) ∈ At such that : xν ∗ Rt ≥ x1 ν1∗ Rt + x2 ν2∗ Rt . Theorem 3.2 Under Assumptions 3.1, 3.2 and 3.3, the following assertions are equivalent : (i) there is no arbitrage opportunity (ii) P + ∩ P − 6= ∅. The proof of the above result is reported in subsection 6.2. Before concluding this section, we give sufficient conditions on the sets C + and C − , inspired from Karatzas and Kou (1996), in order for Assumption 3.3 to hold. Proposition 3.1 Assume that the convex sets C + and C − satisfy : ∀ν + ∈ C + , ∀ν − ∈ C − ,

λν + + (1 − λ)ν − ∈

C + ∪ −C − is convex.

  

C + if λ ≥ 1,

 

C − if λ ≤ 0,

(3.1)

(3.2)

Then Assumption 3.3 holds. Proof. Let (x1 , ν1 ) ∈ At and (x2 , ν2 ) ∈ At for some t ∈ {0, . . . , T − 1}. We show that we can find (x, ν) ∈ At such that x1 ν1∗ Rt + x2 ν2∗ Rt = xν ∗ Rt . 7

(3.3)

(1) We first consider the case x1 + x2 6= 0. We have then : x1 ν1∗ Rt + x2 ν2∗ Rt = xν ∗ Rt where x = x1 + x2 , ν = λν1 + (1 − λ)ν2 and λ = x1 /x. If x1 = 0 (resp. x2 = 0) then x = x2 (resp. x = x1 ) and ν = ν2 (resp. ν = ν1 ) so that (x, ν) ∈ At . It remains to see what happens when x1 x2 6= 0. If x1 x2 > 0 then λ ∈ (0, 1). Either x > 0 and so ν1 and ν2 are valued in C + or x < 0 and so ν1 and ν2 are valued in C − . By the convexity of C + and C − , we deduce that (x, ν) ∈ At . Now assume that x1 > 0 and x2 < 0. We have then ν1 ∈ C + and ν2 ∈ C − . Either x > 0 and so λ > 1 or x < 0 and so λ < 0. It follows from (3.1) that (x, ν) ∈ At . By analogy, we show that (x, ν) ∈ At if x1 < 0 and x2 > 0. (2) Consider finally the case x1 + x2 = 0. Assume first x1 > 0. Then ν1 ∈ C + , ν2 ∈ C − and by 21 ν1 − 21 ν2 ∈ C + ∪ −C − by (3.2). If 21 ν1 − 12 ν2 ∈ C + , then we can write x1 ν1∗ Rt + x2 ν2∗ Rt = xν ∗ Rt with x = 2x1 > 0, ν = 21 ν1 − 21 ν2 ∈ C + and so (x, ν) ∈ At . Similarly, if 12 ν1 − 12 ν2 ∈ −C − , we can write x1 ν1∗ Rt + x2 ν2∗ Rt = xν ∗ Rt with x = 2x2 < 0, ν = 12 ν2 − 12 ν1 ∈ C − and so (x, ν) ∈ At . By analogy, we show that if x1 < 0 then we have a decomposition of the form (3.3) with (x, ν) ∈ At . Finally assume that x1 = 0 and so x2 = 0. Therefore (3.3) is satisfied with (0, ν) ∈ At for any ν ∈ L0,d t .

4

2

Examples

In all following examples, Assumption 3.1 is satisfied. 1. Unconstrained case : this corresponds to the classical incomplete market framework C + = C − = IRd . Then, cone(C + ) = cone(C − ) = IRd , and Cˆ + = Cˆ − = {0}. Theorem 3.1 reduces to the classical characterization of the no-arbitrage condition by the existence of an equivalent martingale measure for the discounted price process S. 8

2. Prohibition of short-selling of stocks : this corresponds to the case C + = −C − = [0, ∞)d . Then cone(C + ) = cone(−C − ) = [0, +∞)d and Cˆ + = −Cˆ − = (−∞, 0]d . Theorem 3.1 reduces to the characterization of the no-arbitrage condition by the existence of an equivalent probability measure under which the price process S is a supermartingale; see Jouni and Kallal (1995). 3. Constraints on the short-selling of stocks : C + =

Qd

i=1 [−ki , ∞),

C− =

Qd

i=1 (−∞, li ]

with ki , li > 0 for i = 1, . . . , d. Then cone(C + ) = cone(C − ) = IRd and Cˆ + = Cˆ − = {0}. In this context, Theorem 3.1 says that the no-arbitrage condition is equivalent to the existence of an equivalent martingale measure for the price process S. Surprisingly, we obtain the same characterization as in the unconstrained case. 4. Rectangular constraints : C + =

Qd

i=1 [−ki , li ],

C− =

Qd

0 0 i=1 [−ki , li ]

with ki , ki0 , li , li0 > 0

for i = 1, . . . , d. Then cone(C + ) = cone(C − ) = IRd and Cˆ + = Cˆ − = {0}. As in the previous example, we obtain the same characterization of no-arbitrage as in the unconstrained case. 5. Incomplete and constrained market with prohibition of short-selling of the first n1 stocks, prohibition of being long in the next n2 stocks and prohibition of investment in the next n3 stocks (n1 +n2 +n3 ≤ d) : C + = −C − = C = {π ∈ IRd : πi ≥ 0, i = 1, ..., n1 , πi ≤ 0, i = n1 + 1, ..., n1 + n2 , πi = 0, i = n1 + n2 + 1, ..., n1 + n2 + n3 }. Then cone(C) = C and Cˆ = {x ∈ IRd : xi ≤ 0, i = 1, ..., n1 , xi ≥ 0, i = n1 + 1, ..., n1 + n2 , xi = 0, i = n1 + n2 + n3 , ..., d}. From Theorem 3.1, the no-arbitrage condition is characterized by the existence of an equivalent probability measure under which : -the first n1 components of the price process (assets subject to no short-selling constraint) are supermartingale, -the next n2 components of the price process are submartingale, -the last d − n1 − n2 − n3 components of the price process (unconstrained assets) are martingale. 6. Both C + and C − are closed convex cones with vertex at zero : this clearly generalizes all the previous cases except 3 and 4. Then Theorem 3.1 provides an extension 9

to the result of Pham and Touzi (1999). 7. Constraints on borrowing : in this example, we put constraints on the proportion of wealth invested in the non-risky asset 1 − C − = {π ∈ IRd :

Pd

i=1

Pd

i=1

πi . Let C + = {π ∈ IRd :

Pd

i=1

πi ≤ k} and

πi ≥ l} for some k ≥ 1 and l ≤ 0. Then cone(C + ) = cone(C − ) = IRd

and Cˆ + = Cˆ − = {0}. We find again that the no-arbitrage characteriztion is the same as in the unconstrained case. Remark 4.1 Conditions (3.1) and (3.2) are satisfied in the context of the above examples for the cases 1, 2, 5 and 6 with C + = −C − . Then Assumption 3.3 holds.

5

Connection with the problem of super-replication

In this section, we study the special case C := C + = −C − . We also denote P := P + = P − . We intend to relate our result to the dual formulation of the super-replication cost of contingent claims derived by F¨ollmer and Kramkov (1997). As in F¨ollmer and Kramkov (1997), we consider a non-negativity constraint on the wealth process. We therefore define the set of strategies : A+ :=

n

=

n

π ∈ A(1) : X 1,π (.) ≥ 0 P − a.s.

o

π IF − adapted valued in C + : X 1,π (.) ≥ 0 P − a.s.

o

Let B be a contingent claim, i.e., a nonnegative FT -measurable random variable. The super-replication cost of B is defined by p(B) = inf{x ∈ IR : ∃ π ∈ A+ , XTx,π ≥ B P − a.s}, i.e., the minimal initial capital needed for hedging without risk the contingent claim B. We first recall the result of F¨ollmer and Kramkov (1997). For all π ∈ A+ , we denote by Y π the process Ytπ

=

t−1 X

πu∗ Ru+1 ,

u=0

10

t = 0, . . . , T ,

so that the wealth process X x,π = xE(Y π ), where E(.) is the Dol´eans-Dade exponential. Next, we introduce the sets Y :=

n

Y π : π ∈ A+

o

and P(Y) := {Q ∼ P : ∃ A ∈ I, Y − A is Q − supermartingale for all Y ∈ Y} , where I is the set of all adapted nondecreasing processes. For all measures Q ∈ P(Y) we denote by AY (Q) the upper variation process of Y under Q as defined in F¨ollmer and Kramkov (1997). Then we have Theorem 5.1 (F¨ollmer and Kramkov 1997). Assume that P(Y) 6= ∅. Then p(B) =

h

i

sup E Q BE(AY (Q))−1 . Q∈P(Y)

In the previous literature on the super-replication problem without constraints, the dual formulation is obtained under the no-arbitrage condition. Our purpose is to relate the condition P(Y) 6= ∅ to the no-arbitrage concept. Proposition 5.1 (i) P ⊂ P(Y). (ii) Suppose that there exists no arbitrage opportunity. Then P(Y) 6= ∅ and p(1) = 1. (iii) Suppose that P(Y) 6= ∅ and p(1) = 1. Then there exists no arbitrage opportunity with nonnegative wealth, i.e. XT1,π ≥ 1 P − a.s. for some π ∈ A+ =⇒ XT1,π = 1 P − a.s. Proof. (i) is trivial since for all Q ∈ P and π ∈ A+ , the process Y π is a supermartingale under Q. We now prove (ii). Suppose that there is no arbitrage opportunity. Then P 6= ∅ by Theorem 3.1 and therefore P(Y) 6= ∅ by (i). Next, since 0 ∈ A+ , we see that p(1) ≤ 1. On the other hand, let Q be any probability measure in P, then it is easily checked that X x,π is a Qsupermartingale for all (x, π) ∈ (0, ∞) × A+ , and therefore p(1) ≥ 1. 11

To see that (iii) holds, we argue by contradiction. Suppose that XT1,π ≥ 1

P − a.s.

for some π ∈ A+ . Set B := XT1,π − 1 and assume that P [B > 0] > 0. By definition of the super-replication of the contingent claim 1 + B and from Theorem 5.1, we have : 1 ≥ p(1 + B) =

h

sup E Q (1 + B)E(AY (Q))−1

i

Q∈P(Y)

h

= 1 + sup E Q BE(AY (Q))−1

i

Q∈P(Y)

> 1, where we used the fact that p(1) = 1, B ≥ 0 and P [B > 0] > 0. This provides the required 2

contradiction.

6 6.1

Proof of the main results Proof of Theorem 3.1

To prove Theorem 3.1, we need some preliminary results. Lemma 6.1 Assume that P + and P − are not empty. Then for all (x, π) ∈ A, there exists a probability measure Qx,π equivalent to P such that the wealth process {Xtx,π , t = 0, . . . , T } is a supermartingale under Qx,π . Proof. Consider some Q+ (resp. Q− ) ∈ P + (resp. P − ) and denote by Z + (resp. Z − ) their Radon-Nikodym density with respect to P. Let (x, π) ∈ A and X x,π the associated wealth process defined by X0x,π = x and for all t = 0, . . . , T − 1 : x,π Xt+1 = Xtx,π + Xtx,π πt∗ Rt+1 .

Define then the positive FT -measurable random variable Z x,π by : Z

x,π

=

TY −1 t=0

12

x,π Zt,t+1

(6.1)

where x,π Zt,t+1 =

E[Z − |Ft+1 ] E[Z + |Ft+1 ] x,π 1 + 1 x,π . E[Z + |Ft ] {Xt ≥0} E[Z − |Ft ] {Xt <0}

Notice that x,π E[Zt,t+1 |Ft ] = 1.

(6.2)

Let Qx,π be the probability measure equivalent to P with density Z x,π . We first check that EQ

x,π

[Rt+1 |Ft ] exists. To see this, denote by Ztx,π := E

Qx,π

Qt−1

u=0

x,π Zu,u+1 . Then, by Bayes rule,

E[Ztx,π (Z x,π /Ztx,π )|Rt+1 | |Ft ] [|Rt+1 | |Ft ] = E[Ztx,π (Z x,π /Ztx,π )|Ft ] E[(Z x,π /Ztx,π )|Rt+1 | |Ft ] = E[(Z x,π /Ztx,π )|Ft ]

since all random variables inside the expectation are nonnegative. By the definition of Z x,π and using Bayes rule and the law of iterated expectations, we see that : EQ

x,π

h

x,π |Rt+1 | |Ft [|Rt+1 | |Ft ] = E Zt,t+1

i −

+

= 1{Xtx,π ≥0} E Q [|Rt+1 | |Ft ] + 1{Xtx,π <0} E Q [|Rt+1 | |Ft ] < ∞ since Rt ∈ L1 (Q+ ) ∩ L1 (Q− ). Thus, we get : Xtx,π πt∗ E Q

x,π

+

−

[Rt+1 |Ft ] = Xtx,π 1{Xtx,π ≥0} πt∗ E Q [Rt+1 |Ft ] + Xtx,π 1{Xtx,π <0} πt∗ E Q [Rt+1 |Ft ] ≤ 0

by definition of the sets P + and P − . Plugging the last inequality in (6.1), we obtain that x,π

EQ

x,π | Ft ] ≤ Xtx,π , [Xt+1

for all t = 0, . . . , T − 1, which proves the supermartingale property of X x,π under Qx,π . 2 Remark 6.1 Assume that P + ∩ P − 6= ∅ and let Q ∈ P + ∩ P − and Z its Radon-Nikodym density with respect to P. Then in the proof of Lemma 6.1, we can choose Z x,π = Z for all (x, π) ∈ A. It follows that for all (x, π) ∈ A, the wealth process X x,π is a Q-supermartingale for any Q ∈ P + ∩ P − . 13

Let us now define for all t = 0, . . . , T − 1 : A+ = t

n

o

= A− t

n

o

ν ∈ L0,d : ν ∈ C+ , t

ν ∈ L0,d : ν ∈ C− . t

As in Kabanov and Kramkov (1994), we consider the bounded IRd -valued random variable δt = Rt /(1 + |Rt |), for all t = 0, . . . , T. Remark 6.2 Notice that V (δt+1 |Ft ) exists and, therefore, a sufficient condition for Assumption 3.2 to be satisfied is that V (δt+1 |Ft ) is invertible P -a.s. for all t = 0, . . . , T − 1; see Example 3.3. Next, for all t = 1, . . . , T, define the sets : Bt+ =

n

o

Bt− =

n

o

∗ U ∈ L0t : ∃x > 0 and ν ∈ A+ t−1 , xν δt ≥ U , ∗ U ∈ L0t : ∃x < 0 and ν ∈ A− t−1 , xν δt ≥ U .

Notice that Bt+ and Bt− contain the origin since 0 ∈ C + ∩ C − . It is easily checked that Bt+ and Bt− are two convex cones of L0t . The cone property is immediate. To check the convexity property, consider some U1 , U2 ∈ Bt+ (resp. Bt− ), then there exist some positive − ∗ (resp. negative) real x1 , x2 and ν1 , ν2 ∈ A+ t−1 (resp. At−1 ) such that xν δt ≥ U1 + U2 where

x = x1 + x2 , ν = λν1 + (1 − λ)ν2 and λ = x1 /x. The convexity of C + (resp. C − ) implies that U1 + U2 ∈ Bt+ (resp. Bt− ). Lemma 6.2 Suppose that there is no arbitrage opportunity. Then for all t = 1, . . . , T, we have Bt+ ∩ L0+ = {0} and Bt− ∩ L0+ = {0}. t t Proof. We argue by contradiction. Assume that Bt+ ∩ L0+ 6= {0} (resp. Bt− ∩ L0+ 6= {0}) t t for some t ∈ {1, . . . , T }. Then there exist U ∈ L0+ t , U 6= 0, x > 0 (resp. x < 0) and ν ∈ − ∗ A+ t−1 (resp. ν ∈ At−1 ) such that xν δt ≥ U P -a.s. and then

xν ∗ Rt ≥ 0, P − a.s. and P [xν ∗ Rt > 0] > 0.

14

(6.3)

Consider the trading strategy π defined by πu = 0 for u 6= t − 1 and πt−1 = ν. The associated wealth process is : Xux,π = x,

u = 0, . . . , t − 1,

Xux,π = x + xν ∗ Rt ,

u = t, . . . , T.

It follows that (x, π) ∈ A (recall that 0 ∈ C + ∩ C − ). Moreover by (6.3), we have : XTx,π ≥ x, P − a.s. and P [XTx,π > x] > 0, 2

which is an arbitrage opportunity.

Lemma 6.3 Under Assumptions 3.1 and 3.2, for all t = 1, . . . , T, the following implications hold + 1 1+ Bt+ ∩ L1+ t = {0} =⇒ cl(Bt ∩ Lt ) ∩ Lt = {0}, − 1 1+ Bt− ∩ L1+ t = {0} =⇒ cl(Bt ∩ Lt ) ∩ Lt = {0},

where cl is the closure in the sense of the L1 topology. Proof. We only prove the first implication since the second one is proved by the same way. + 1+ n Let U ∈ cl(Bt+ ∩ L1t ) ∩ L1+ converging to t . Then there exists a sequence (U )n ⊂ Bt ∩ Lt

U in the sense of the L1 topology, and therefore P -a.s, possibly along a subsequence. By definition of Bt+ , for each n ∈ IN, there exists xn > 0, ν n ∈ A+ t−1 such that : xn ν n∗ δt ≥ U n .

(6.4)

In order to prove the required result, we intend to show that (|xn ν n |)n converges to zero in probability. Then along a subsequence it also converges almost surely to zero. Taking almost sure limit in inequality (6.4), we find that U is nonpositive. Recalling that U is by assumption nonnegative, we conclude that U = 0.

15

We adapt the arguments of Kabanov and Kramkov (1994). Fix some ε > 0 and let ϕn = xn ν n 1 n n . |xn ν n | |x ν |≥ε

Let E = {−1, +1}d and for all e ∈ E, Ae = {z ∈ IRd : z i ≥ 0 iff ei = +1}.

We consider the sequence defined by ϕne = ϕn 1ϕn ∈Ae . " n n

n

# n

P [|x ν | ≥ ε] = E[1|xn ν n |≥ε ] = E|ϕ | = E |ϕ |

X

1ϕn ∈Ae

e∈E

"

= E

# X

|ϕne |

=

e∈E

X

E|ϕne |

(6.5)

e∈E

For each e ∈ E, the sequence (ϕne )n is bounded by 1 uniformly in ω ∈ Ω. Since L2,d t−1 is n the dual space of L2,d t−1 , which is a Banach separable space, (ϕe )n converges (possibly along 2,d 2,d a subsequence) in the sense of the weak topology σ(L2,d t−1 , Lt−1 ) to some ϕe ∈ Lt−1 . Next, 1,d we prove that (ϕne )n converges to ϕe in the sense of the weak* topology σ(L∞,d t−1 , Lt−1 ). Let 2,d n ψ ∈ L1,d t−1 , we want to show that E[ψϕe ] converges to E[ψϕe ]. Recalling that the space Lt−1 2,d p is dense in the space L1,d t−1 , there exists a sequence (ψ )p ⊂ Lt−1 converging to ψ in the sense ∗ ∗ p of the L1,d t−1 norm. Let ε > 0, then there existe p such that for all p ≥ p , E[|ψ − ψ|] < ε/3. ∗

2,d p ∈ L2,d Since (ϕne )n converges to ϕe in the sense of the weak topology σ(L2,d t−1 , t−1 , Lt−1 ) and ψ ∗

∗

there exists Np∗ , such that for all n ≥ Np∗ , |E[ψ p ϕne ] − E[ψ p ϕe ]| ≤ ε/3. Next, recalling that (ϕne )n is bounded by 1 uniformly in ω ∈ Ω, it is straightforward to see that ϕe is also bounded by 1 uniformly in ω ∈ Ω. Now, notice that, ∗

∗

∗

∗

|E[ψϕne ] − E[ψϕe ]| ≤ |E[ψϕne ] − E[ψ p ϕne ]| + |E[ψ p ϕne ] − E[ψ p ϕe ]| + |E[ψ p ϕe ] − E[ψϕe ]| ∗

∗

∗

≤ 2E[|ψ p − ψ|] + |E[ψ p ϕne ] − E[ψ p ϕe ]|.

It follows that for all n ≥ Np∗ , |E[ψϕne ] − E[ψϕe ]| ≤ ε. This prove that the sequence (ϕne )n 1,d ∞,d converges to ϕe in the sense of the weak* topology σ(L∞,d t−1 , Lt−1 ) and ϕe ∈ Lt−1 . Furthermore,

ϕe ∈ Ae ∪ {0}, P -a.s. Next, for all e ∈ Ae , E[e∗ ϕne ] converges towards E[e∗ ϕe ] as n goes to infinity. Recalling the definition of E, we see that e∗ ϕne = |ϕne | and e∗ ϕe = |ϕe |. Sending n to infinity in (6.5), we have : lim inf P [|xn ν n | ≥ ε] = n→∞

X e∈E

16

E[|ϕe |].

(6.6)

Now, suppose that lim inf n→∞ P [(|xn ν n | ≥ ε]) > 0, then by (6.6) there exists e ∈ E such that E|ϕe | 6= 0. By (6.4), we have : n n ϕn∗ e δt ≥ α U .

where αn =

1 1 n n 1 n . |xn ν n | {|x ν |≥ε} {ϕ ∈Ae }

(6.7)

The sequence (αn )n is bounded by 1/ε uniformly in

ω ∈ Ω. Using the same line of argument as before, it converges (possibly along a subsequence) ∞ 1 in the sense of the weak* topology σ(L∞ t−1 , Lt−1 ) to some α ∈ Lt−1 . Notice also that for all

n, αn is non-negative and therefore α is non-negative. Recalling that U n converges to U in the sense of the L1 -convergence, we deduce by Proposition III.12 (iv) of Br´ezis (1983) that for any bounded Ft -measurable random variable ξ ≥ 0, E[ξU n αn ] converges to E[ξU α]. On the other hand, since δt belongs to L1 and ϕne 1,d n∗ converges in the sense the weak* topology σ(L∞,d t−1 , Lt−1 ) to some ϕe , we have that E[ξϕe δt ]

converges towards E[ξϕ∗e δt ] as n goes to infinity for any bounded Ft -measurable random variable ξ ≥ 0. By sending n to infinity in (6.7), we have then : E[ξϕ∗e δt ] ≥ E[ξU α] ≥ 0. From the arbitrariness of the Ft -measurable random variable ξ ≥ 0, we deduce that ϕ∗e δt ≥ 0. Recalling that ϕne =

xn ν n 1 n n 1 n , |xn ν n | |x ν |≥ε ϕ ∈Ae

we have that ϕne belongs to cone(C + ). From

Assumption 3.1, cone(C + ) is a closed set in IRd and it follows that ϕe ∈ cone(C + ). Then there exists a positive random variable λe and νe ∈ A+ t−1 such that ϕe = λe νe . Now, let ye = max(λe , 1) and πe = min(λe , 1)νe = min(λe , 1)νe + (1−min(λe , 1))0 ∈ C + . Then, we have ϕe + = ye πe with ye ≥ 1 and πe ∈ L∞,d t−1 . Let xe = inf{λ ≥ 1 / (ϕe /λ) ∈ C }. From the existence

of ye , we see that xe is well-defined. Notice that for all a ≥ 1, {xe ≤ a} = {(ϕe /a) ∈ C + }. The first inclusion is from ϕe /a = (ϕe /xe )(xe /a) + 0(1 − xe /a) and 0 belongs to the convex set C + . The reverse inclusion is trivial from the definition of xe . Recalling that ϕe ∈ Ft−1 , this proves that xe is Ft−1 measurable. Let πe = (ϕe /xe ), then πe∗ δt ∈ Bt+ ∩ L1+ = {0} and therefore πe∗ Rt = 0. By Assumption t 3.1, this implies that πe = 0 and therefore ϕe = 0 which contradicts the fact that E|ϕe | = 6 2

0. 17

Lemma 6.4 Suppose that there is no arbitrage opportunity. Then under Assumptions 3.1 + − and 3.2, for all t = 1, . . . , T, there exist Zt+ and Zt− ∈ L∞ t with Zt > 0, Zt > 0 P -a.s such

that : h

i

h

i

E Zt+ Rt |Ft−1 ∈ Cˆ + and E Zt− Rt |Ft−1 ∈ −Cˆ − , P − a.s. − 1+ 1 Proof. From Lemmas 6.2 and 6.3, we have cl(Bt+ ∩ L1t ) ∩ L1+ t = {0} and cl(Bt ∩ Lt ) ∩ Lt

= {0}. Moreover, Bt+ and Bt− contain L1− because C + and C − contain zero. From Yan’s t ˆ+ ˆ− Theorem (see Yan 1980), we deduce that there exist Zˆt+ and Zˆt− ∈ L∞ t with Zt > 0, Zt > 0 P -a.s. such that : h

i

≤ 0, for all U ∈ Bt+ ∩ L1t ,

(6.8)

h

i

≤ 0, for all U ∈ Bt− ∩ L1t .

(6.9)

E Zˆt+ U E Zˆt− U

− Let z ∈ C + (resp. C − ), t ∈ {1, ..., T }, B ∈ Ft−1 and ν = z1B . Then ν ∈ A+ t−1 (resp. At−1 )

and U = ν ∗ δt (resp. −ν ∗ δt ) ∈ Bt+ ∩L1t (resp. Bt− ∩L1t ). The separation inequalities (6.8)-(6.9) and the arbitrariness of B ∈ Ft−1 imply therefore : h



i

≤ 0, ∀z ∈ C + ,

h



i

≤ 0, ∀z ∈ C − .

z ∗ E Zt+ Rt Ft−1 −z ∗ E Zt− Rt Ft−1

Let Zt+ = Zˆt+ /(1 + |Rt |) and Zt− = Zˆt− /(1 + |Rt |) are positive P -a.s. and lie in L∞ t . Then E[Zt+ Rt |Ft−1 ] = E[Zˆt+ δt |Ft−1 ] and E[Zt− Rt |Ft−1 ] = E[Zˆt− δt |Ft−1 ] are well-defined and finited, and E[Zt+ Rt |Ft−1 ] ∈ Cˆ + and E[Zt− Rt |Ft−1 ] ∈ −Cˆ − , P -a.s.

2

Proof of Theorem 3.1 We first prove the implication (ii) =⇒ (i). Let (x, π) ∈ A and X x,π its wealth process. By Lemma 6.1, there exists a probability measure Qx,π equivalent to P such that EQ

x,π

[XTx,π ] ≤ x.

Since Qx,π is equivalent to P, this last relation shows that (x, π) cannot be an arbitrage opportunity.

18

We now prove the implication (i) =⇒ (ii). From Lemma 6.4, there exist ZT+ and ZT− ∈ + − + − ˆ+ L∞ T with ZT > 0 and ZT > 0, P -a.s such that E[ZT RT |FT −1 ] ∈ C and E[ZT RT |FT −1 ] ∈

−Cˆ − , P -a.s. Obviously, we can assume without loss of generality that E[ZT+ ] = E[ZT− ] = − + 1. Then, we define the probabilty measure Q+ T (resp. QT ) equivalent to P with density ZT

(resp. ZT− ). Notice that the statement of Lemma 6.4 is valid for any complete probability space (Ω, F, IF, Q) with a probability measure Q equivalent to P. Applying Lemma 6.4 to − + − + ∞ (Ω, F, IF, Q+ T ) and (Ω, F, IF, QT ), we see that there exist ZT −1 and ZT −1 ∈ LT −1 with ZT −1 − + > 0 and ZT−−1 > 0, P -a.s such that E QT [ZT+−1 RT −1 |FT −2 ] ∈ Cˆ + and E QT [ZT−−1 RT −1 |FT −2 ] ∈

−Cˆ − , P -a.s. This leads to E[ZT+−1 ZT+ RT −1 |FT −2 ] ∈ Cˆ + and E[ZT−−1 ZT− RT −1 |FT −2 ] ∈ −Cˆ − , P -a.s. Without loss of generality, we can assume that E[ZT+−1 ZT+ ] = 1 and E[ZT−−1 ZT− ] = 1. By induction, using Lemma 6.4, we construct for all t = 1, . . . , T, Zt+ and Zt− ∈ L∞ t with Zt+ > 0, Zt− > 0 P -a.s such that : E[Zt+ · · · ZT+ ] = 1 h

i

E Zt+ · · · ZT+ Rt |Ft−1 ∈ Cˆ +

and

E[Zt− · · · ZT− ] = 1

and

E Zt− · · · ZT− Rt |Ft−1 ∈ −Cˆ − , P -a.s. (6.10)

h

i

We then define the probability measures Q+ and Q− equivalent to P by their RadonNikodym densities : T T Y Y dQ+ dQ− Zt+ and Zt− . = = dP dP t=1 t=1

It follows that dQ+ /dP and dQ− /dP ∈ L∞ . By Bayes rule, we have for all t = 1, ..., T, +

h

i

h

h

+ E Q [|Rt |] = E Z1+ · · · ZT+ |Rt | = E Z1+ · · · Zt−1 E Zt+ · · · ZT+ |Rt ||Ft−1

ii

< ∞. By a similar +

argument we also prove that Rt ∈ L1 (Q− ). Finally, equation (6.10) provides E Q [Rt |Ft−1 ] − ∈ Cˆ + , E Q [Rt |Ft−1 ] ∈ −Cˆ − , P -a.s. and therefore Q+ ∈ P + and Q− ∈ P − .

6.2

Proof of Theorem 3.2

The proof is similar to that of Theorem 3.1; then we only highlight the main changes. The implication (ii) =⇒ (i) follows from the supermartingale property of any wealth process X x,π under some fixed Q ∈ P + ∩ P − (see Remark 6.1). To prove the converse implication, we consider for any t = 1, . . . , T the sets : Bt = Bt+ ∪ Bt− =

n

o

U ∈ L0t : ∃(x, ν) ∈ At−1 , xν ∗ δt ≥ U . 19

It is clear that Bt is a cone of L0t . By Assumption 3.3, it is easily checked that Bt is a convex set in L0t . By noting that cl(Bt ∩ L1t ) = cl(Bt+ ∩ L1t ) ∪ cl(Bt− ∩ L1t ), we deduce from Lemmas 6.2 and 6.3 that the no arbitrage condition implies cl(Bt ∩ L1t ) ∩ L1+ = {0} for all t = t ˆ 0, . . . , T. As in Lemma 6.4, applying Yan’s theorem, we obtain the existence of Zˆt ∈ L∞ t , Zt > 0, P -a.s., for all t = 0, . . . , T, such that : h

E Zˆt U

i

≤ 0, ∀U ∈ Bt ∩ L1t .

By similar arguments as in the end of the proof of Lemma 6.4, we can prove that E[Zt Rt |Ft−1 ] ∈ Cˆ + ∩ −Cˆ − , P -a.s. where Zt = Zˆt /(1 + |Rt |). By induction, we construct Zt0 , such that 0 0 for all t = 1, . . . , T, Zt0 ∈ L∞ t with Zt > 0 P -a.s. The process Z will also have the following

properties : E[Zt0 · · · ZT0 ] = 1 and E [Zt0 · · · ZT0 Rt |Ft−1 ] ∈ Cˆ + ∩ −Cˆ − P − a.s. Then, we construct a probability measure Q ∼ P defined by : T Y dQ Zt0 = dP t=1

and which satisfies therefore Q ∈ P + ∩ P − .

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[5] Dalang R.C., A. Morton and W. Willinger, 1990, Equivalent martingale measures and no-arbitrage in stochastic securities market models, Stochastics and Stochastic Reports 29, 185-201. [6] F¨ollmer H. and D. Kramkov 1997, Optional Decompositions under Constraints, Probability Theory and Related Fields 109, 1-25. [7] Harrison J.M. and D.M. Kreps 1979, Martingale and Arbitrage in Multiperiods Securities Markets, Journal of Economic Theory 20, 381-408. [8] Harrison J.M. and S. Pliska 1981, Martingales and Stochastic Integrals in the Theory of Continuous Trading, Stochastic Processes and their Applications 11, 215-260. [9] Jouini E. and H. Kallal, 1995, Arbitrage in securities markets with short-sales constraints, Mathematical Finance 3, 197-232. [10] Kabanov Yu. and D. Kramkov, 1994, No-arbitrage and equivalent martingale measures : An elementary proof of the Harrison-Pliska theorem, Theory of Probability and its Applications 39, 523-526. [11] Karatzas I. and S. Kou, 1996, On the pricing of contingent claims under constraints, Annals of Applied Probability 6, 321-369. [12] Kreps D., 1981, Arbitrage and equilibrium in economies with infinitely many commodities, Journal of Mathematical Economics 8, 15-35. [13] Pham H. and N. Touzi, 1999, The fundamental theorem of asset pricing with cone constraints, Journal of Mathematical Economics, 31, 265-279. [14] Pliska S., 1997, Introduction to Mathematical Finance : Discrete-Time Models, Blackwell Publishers. [15] Schachermayer W., 1992, A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time, Insurance : Mathematics and Economics 11, 249-257.

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[16] Sch¨ urger K., 1996, On the existence of equivalent τ -measures in finite discrete time, Stochastic Processes and their Applications 61, 109-128. [17] Yan J.A., 1980, Caract´erisation d’une classe d’ensembles convexes de L1 ou H 1 , S´eminaires de Probabilit´es XIV, Lect. Notes Mathematics 784, 220-222.

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