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A Conic Programming Approach to Generalized Tchebycheff Inequalities Luis F. Zuluaga∗ Javier F. Pe˜ na† Tepper School of Business Carnegie Mellon University January 10, 2003, Revised May 7, 2004

Abstract Consider the problem of finding optimal bounds on the expected value of piece-wise polynomials over all measures with a given set of moments. This is a special class of Generalized Tchebycheff Inequalities in probability theory. We study this problem within the framework of conic programming. Relying on a general approximation scheme for conic programming, we show that these bounds can be numerically computed or approximated via semidefinite programming. We also describe some applications of this class of Generalized Tchebycheff Inequalities in probability, finance, and inventory theory.

GSIA Working Paper 2003-01

∗ Supported † Supported

by NSF grants CCR-0092655 and DMI-0098427. ([email protected]) by NSF grant CCR-0092655. ([email protected])

1

1

Introduction

The theory and algorithms developed for conic programming provide a very general and sleek framework to study a wide variety of optimization problems. We use this framework to study the problem of finding optimal bounds on the expected value of piece-wise polynomials over all measures with a given set of moments. This problem is a special instance of a class of problems known as Generalized Tchebycheff Inequalities (for an historical recount on the development of this problem see [2]). There are important cases in which Tchebycheff Inequalities on piece-wise polynomials can be solved analytically (see, e.g., [2] or [9]); examples of these are the well-known Tchebycheff and Markov Inequalities. Here, we focus on numerical solutions for general instances of this problem. For this purpose, we take a two-step approach: First, we show that Generalized Tchebycheff Inequalities can be cast as conic programs. Next, we provide a generic approximation scheme for conic programs (see Theorem 2) that is of independent interest. These two steps yield our main result; namely, that under suitable conditions, we can either numerically compute or approximate the desired bounds via semidefinite programming. The area of semidefinite programming has been a subject of intense research activity in the last decade. One of its key features is the availability of effective algorithms and code to solve moderately sized problems (see, e.g. [7], [27], and [28]). For a general survey on the main developments of semidefinite programming see [29, 30]. The recent results by Lasserre ([12] and [13]), and Bertsimas and Popescu ([1] and [2]) are along the lines of our work. They provide semidefinite programming formulations to solve or approximate particular instances of the problem studied here; namely, when the piece-wise polynomial is either the indicator function of a set (in [2] and [12]), or the univariate function f (s) = max{s − K, 0} (in [1]). The first case encompasses many of the Tchebycheff type inequalities considered in probability theory (e.g., Tchebycheff and Markov Inequalities). The second case is studied in [1] to generalize Lo’s bound [14] on the payoff of a European call option when the mean and the variance of the underlying stock price at maturity are known. Other related results are the following. In [1], Bertismas and Popescu study other bounds on the univariate function f (s) = max{s − K, 0}, where instead of moment constraints, they impose various first order constraints (i.e., constraints of the form Eµ (max{s − K 0 , 0}) = q 0 ). Popescu [19] gives semidefinite programming formulations to obtain bounds on the indicator function of a subset of IR over measures that in addition to having given moments, possess certain structural properties such as symmetry, unimodality, convexity or smoothness. Boyle and Lin [4] give valid semi-parametric bounds for a European call on the maximum of any number of stocks. Our approach based on conic programming duality allows us to unify and extend several results derived in [1], [2] and [12]. For instance, we show that in most of the cases considered in [2] and [12], strong duality readily follows from the structure of the problem, and therefore, no extra assumption beyond feasibility is needed to ensure that the problem can be solved or approximated 2

via semidefinite programming. We also introduce some new results that enable us to consider Tchebychefftype inequalities in IRn and IRn+ . In particular, this allows us to consider bounds on the indicator function of a set, without making any assumption on the support of the optimal measure, and still guarantee convergence of the numerical schemes presented to solve this problem (see Example 6). Moreover, we can generalize Lo’s type of financial analysis (see [6] or [14]), to European rainbow options; that is, options whose payoff depends on the value of several stock prices. Given the complexity or strong assumptions necessary to obtain accurate prices for European rainbow options, these so called semi-parametric bounds provide an interesting application of our results (see Examples 3 and 5). These same results can also be applied to similar problems in inventory theory. The paper is organized as follows. In § 2 we recall the basic notation and duality results of conic programming. In § 3 we present the problem of Generalized Tchebycheff Inequalities and show that it can be studied within the framework of conic programming. We extend these ideas to special instances over piece-wise polynomials in § 4, and show how in many cases strong duality immediately follows from the structure of the problem. In § 5, we discuss valid relaxations for two important instances of the piece-wise polynomial problem. Finally, § 6 shows how these problems can be solved numerically. For this purpose, we introduce a key result on approximating conic programs, from which our main results follow. Throughout, we present examples to illustrate our results and different applications of Generalized Tchebycheff Inequalities.

2

Conic Programming

A conic program (CP), together with its dual, is an optimization problem of the form [21] zPCP = (PCP )

sup s.t.

hc, xi Ax = b x∈K

(DCP )

zDCP = inf s.t.

hb, yi A∗ y − c ∈ K ∗ ,

where c ∈ IRn , b ∈ IRm , A : IRn → IRm is a linear map, K is a closed convex cone, and h , i represent the canonical inner products in IRn and IRm , i.e., ha, bi = aT b. In the dual problem, A∗ denotes the adjoint of A; that is, the unique linear map A∗ : IRm → IRn such that for any x ∈ IRn and y ∈ IRm , hAx, yi = hx, A∗ yi. Also, K ∗ denotes the dual cone of K; that is, K ∗ = {s ∈ IRn : hx, si ≥ 0, ∀ x ∈ K}. By convention, zPCP := −∞ whenever (PCP ) is infeasible, and zDCP := ∞ whenever (DCP ) is infeasible. It is easy to see that weak duality holds between (PCP ) and (DCP ); that is, zPCP ≤ zDCP . Moreover, strong duality holds under the following assumptions (see Theorems 3.2.6 to 3.2.8 in [21]). Proposition 1 If there exists y ∈ IRm such that A∗ y − c ∈ Int(K ∗ ), then zPCP = zDCP . If in addition, (PCP ) is feasible, then (PCP ) has an optimal solution. Here, Int(S) denotes the interior of the set S. 3

Proposition 2 If b ∈ Int({Ax : x ∈ K}), then zPCP = zDCP . If in addition, (DCP ) is feasible, then (DCP ) has an optimal solution.

3

Generalized Tchebycheff Inequalities

The following problem is our central object of study: Assume I is a finite index set. Given σ ∈ IRI , a Borel measurable set D ⊆ IRn , and Borel measurable functions f : D → IR and f α : D → IR, α ∈ I, compute: Z zPGTI = sup Eµ (f ) ≡ f (s)dµ(s) IRn (PGTI ) s.t. Eµ (f α ) = σα , ∀ α ∈ I µ ∈ M(D), where M(D) is the set of finite positive Borel measures supported by D. The following example shows a popular result in probability that involves solving simple instances of (PGTI ) (see, e.g., p. 476 in [9]). Example 1 (One-Sided Tchebycheff Inequalities) Let a > 0. The sharp upper bound on the probability P(S ≤ τ − a), and the sharp lower bound on the probability P(S ≤ τ + a), for random variables S with mean τ and variance ρ2 ≥ 0 are respectively given by sup s.t.

Eµ (1(−∞,τ −a] )

=

ρ2 ρ2 +a2

inf

Eµ (1) = 1 Eµ (s) = τ Eµ (s2 ) = ρ2 + τ 2 µ ∈ M(IR),

s.t.

Eµ (1(−∞,τ +a] )

=1−

ρ2 ρ2 +a2

Eµ (1) = 1 Eµ (s) = τ Eµ (s2 ) = ρ2 + τ 2 µ ∈ M(IR).

(Here 1S : IR → {0, 1} is the indicator function of the set S ⊆ IR, that is, 1S (s) = 1 if s ∈ S and 1S (s) = 0 otherwise.) Our definition of (PGTI ) differs slightly from the classical definition of Generalized Tchebycheff Inequalities (GTI) (see, e.g., Chapter XII in [9]) in that instead of optimizing over finite regular measures, we do it over finite positive Borel measures. This however, is of little relevance, and is done only to concur with more recent results related to this subject (see, e.g., [2] and [12]). Since classical results for GTI readily extend to (PGTI ), we cite them omitting this minor detail. The dual problem of (PGTI ) is (see, e.g., Chapter XII in [9]): X zDGTI = inf σα θ α α∈I X (DGTI ) s.t. θα f α (s) ≥ f (s) ∀ s ∈ D. α∈I

4

It is easy to see that weak duality holds between (PGTI ) and (DGTI ); that is, zPGTI ≤ zDGTI . Let MI (D) = {y ∈ IRI : yα = Eµ (f α ), ∀ α ∈ I for some µ ∈ M(D)}. Isii [8] characterizes strong duality between (PGTI ) and (DGTI ) as follows. Theorem 1 (Isii [8]) If (DGTI ) is feasible and σ ∈ Int(MI (D)), then zPGTI = zDGTI and (DGTI ) has an optimal solution. Although not completely evident from their definitions, both (PGTI ) and (DGTI ) can be cast as conic programs. This fact becomes evident in Karlin’s proof of Theorem 1 (see Chapter XII in [9]). Provided (DGTI ) is feasible, let I o = I ∪ {αo }, put f αo := f , and define   yα = Eµ (f α ), ∀ α ∈ I and Io T T o MI (D) = (y , yαo ) ∈ IR : . yαo = Eµ (f αo ) for some µ ∈ M(D) Notice that (PGTI ) is equivalent to zPCP = sup GTI (PCP GTI )

yαo

s.t.

y=σ (y T , yαo )T ∈ MI o (D),

where K denotes the closure of K. Since MI o (D) is a convex cone, it follows that (PCP GTI ) is a conic program. Similarly, if we let ( ) X Io T T α PI o (D) = (θ , θαo ) ∈ IR : θα f (s) + θαo f (s) ≥ 0 for all s ∈ D , α∈I

then (DGTI ) is equivalent to (DCP GTI )

zDCP = GTI

hσ, θi

inf s.t.

(θT , −1)T ∈ PI o (D). ∗

Since PI o (D) is a closed convex cone satisfying PI o (D) = MI o (D) , it folCP lows that (DCP GTI ) is the conic programming dual of (PGTI ). Thus, weak duality between (PGTI ) and (DGTI ) follows from zPGTI = zPCP ≤ zDCP = zDGTI , GTI GTI and Theorem 1 follows from Proposition 2. Furthermore, using Proposition 1 we get the following alternative sufficient condition for strong duality. Proposition 3 If σ ∈ MI (D) and there exists θ ∈ IRI such that (θT , −1)T ∈ Int(PI o (D)), then zPGTI = zDGTI and (PGTI ) has an optimal solution. Other sufficient conditions for strong duality between (PGTI ) and (DGTI ) have been proposed in the literature (for a recent survey see [24]). The type of sufficient condition derived in Proposition 3 above will be useful for the particular applications of GTI considered here. 5

4

(P ) for piece-wise polynomials GTI

From now on, we concentrate on a special instance of (PGTI ); namely, when f (s) is a piece-wise polynomial and {f α (s), α ∈ I} is the set of all the n-variate monomials of degree less than or equal to d. Formally, let Id = {α ∈ Nn : α1 + · · · + αn ≤ d}, and consider the following problem: Given σ ∈ IRId , disjoint Borel measurable sets Di ⊆ IRn , i = 1, . . . , p, and nvariate polynomials q i (s), i = 1, . . . , S p of degree less than or equal to d; define p the piece-wise polynomial f (s) : D = i=1 Di → IR as f (s) = q i (s) if s ∈ Di , i = 1, . . . , p. Then compute: zPpw = (Ppw )

sup s.t.

Eµ (f (s)) αn α 1 Eµ (sα 1 · · · sn ) ≡ Eµ (s ) = σα , ∀ α ∈ Id µ ∈ M(D).

The following example shows a result in finance that involves solving a simple instance of (Ppw ). Example 2 (Lo [14]) Let f (s) be the payoff of a European call option with strike K; that is,  0 if 0 ≤ s ≤ K f (s) = s − K if s > K, where s represents the random underlying stock price at maturity. The tight upper bound on the expected payoff of a European call option with strike K on a stock whose price at maturity has known mean τ and variance ρ2 ≥ 0 is given by  2 +τ 2  τ − τ2 2 K2 K ≤ ρ 2τ ρ +τ   p sup Eµ (f (s)) = 2 +τ 2  12 (τ − K) + (K − τ )2 + ρ2 K > ρ 2τ s.t.

Eµ (1) = 1 Eµ (s) = τ Eµ (s2 ) = ρ2 + τ 2 µ ∈ M(IR+ ).

Before addressing the problem (Ppw ), let us recall the following key multinomial notation. For a vector θ = (θα )α∈Id ∈ IRId , let θ(s) denote the n-variate polynomial of degree d given by X X αn 1 θ(s) = θα sα θα sα . 1 · · · sn ≡ α∈Id

α∈Id

6

P α Conversely, identify an n-variate polynomial q(s) = α∈Id qα s of degree d with its vector of coefficients q = (qα )α∈Id ∈ IRId . From the results of Section 3, it follows that both (Ppw ) and its dual (Dpw ) can be cast as conic programs. The aim of this section is to show that this can be done using two well-studied classes of cones; namely, the moment cones and the cones of positive semidefinite polynomials, whose definitions we recall next. For D ⊆ IRn , let the cone of moments supported in D be defined as Mn,d (D) = {y ∈ IRId : yα = Eµ (sα ), ∀ α ∈ Id for some µ ∈ M(D)}, and the cone of positive semidefinite polynomials in D be defined as n o Pn,d (D) = θ ∈ IRId : θ(s) ≥ 0 for all s ∈ D .

(1)

(2)

The cones Mn,d (D) and Pn,d (D) are related through duality; that is, Pn,d (D) = Mn,d (D)∗ .

(3)

Notice that we can rewrite (Ppw ) as zPpw =

sup

p X

Eµi (q i (s))

i=1

(Ppw )

s.t.

p X

Eµi (sα ) = σα , ∀ α ∈ Id

i=1

µi ∈ M(Di ),

i = 1, . . . , p.

Using (1) it follows that (Ppw ) is equivalent to zPCP = pw

sup

p X

hq i , y i i

i=1

(PCP pw )

s.t.

p X

yi = σ

(4)

i=1

y i ∈ Mn,d (Di ), i = 1, . . . , p, which is a conic program over the cone Mn,d (D1 ) × · · · × Mn,d (Dp ). Similarly, the dual (Dpw ) of (Ppw ) is equivalent to X zDpw = inf σα θα (Dpw ) α∈Id s.t. θ(s) ≥ q i (s) ∀ s ∈ Di , i = 1, . . . , p. Using (2) it follows that (Dpw ) is equivalent to (DCP pw )

zDCP = inf pw s.t.

hσ, θi θ − q i ∈ Pn,d (Di ), i = 1, . . . , p, 7

(5)

which is a conic program over the cone Pn,d (D1 ) × · · · × Pn,d (Dp ). As the notation suggests, from (3) it follows that (DCP pw ) is the conic programming dual of (PCP ). Thus, in general we have pw zPpw = zPCP ≤ zDCP = zDpw . pw pw

(6)

Propositions 1 and 2 yield the following sufficient conditions for strong duality. Proposition 4 If either (i) σ ∈ Int(Mn,d (D)) or (ii) there exists θ ∈ IRId such that ((θ − q 1 )T , . . . , (θ − q p )T )T ∈ Int(Pn,d (D1 ) × · · · × Pn,d (Dp )). Then zPpw = zPCP = zDCP = zDpw . pw pw In general, condition (i) in Proposition 4 is difficult to check numerically (see, e.g., Schm¨ udgen’s Theorem [25]). In contrast, it is easy to show that condition (ii), and hence strong duality, holds in the following important cases. In the statements below B(a, r) denotes the Euclidean ball of radius r centered at a. Proposition 5 If Di , i = 1, . . . , p are compact sets, then zPpw = zDpw . P Proof. Let θ(s) = c˜, where c˜ = max{˜ ci |i = 1, . . . , p} and c˜i = max{ α∈Id (qαi + ∆θα )sα | ||∆θ|| ≤ 1, s ∈ Di } < ∞ for i = 1, . . . , p. Then B(θ − q i , 1) ⊆ Pn,d (Di ) for i = 1, . . . , p. 2 Proposition 6 If d is even, and Di , i = 1, . . . , p − 1 are compact sets, then zPpw = zDpw . Proof. Fix θo ∈ Int(Pn,d (IRn )) (e.g., θo (s) = (s21 + · · · + s2n + 1)d/2 ). Let o p  > 0 be such that B(θo , ) ⊆ Pn,d (IRn ), and θ(s) = + c˜, where Pθ (s) +i q (s) i i c˜ = max{0, max{˜ c |i = 1, . . . , p − 1}} and c˜ = max{ α∈Id (qα − qαp )sα |s ∈ Di } for i = 1, . . . , p − 1. Then B(θ − q i , ) ⊆ Pn,d (Di ) for i = 1, . . . , p. 2 Proposition 7 If Di ⊆ IRn+ for i = 1, . . . , p, then zPpw = zDpw . Proof. Let θ = q ∗ +(1, . . . , 1)T , where qα∗ = max{qαi |i = 1, . . . , p} for all α ∈ Id . Then B(θ − q i , 1) ⊆ Pn,d (Di ) for i = 1, . . . , p. 2 It is worth mentioning that all the results presented so far extend in straightforward fashion if one changes supremum to infimum in the objectives of the problems (PCP ), (PGTI ) and (Ppw ). However, in the next section the supremum in the objective is crucial for the results presented therein. 8

5

Valid relaxations for special instances of (P ) pw

In this section, we consider two special instances of (Ppw ) that appear in many interesting applications. For these two cases, we show that solving a relaxation of the problem is equivalent to solving (Ppw ). This simplification is exploited when trying to solve these instances numerically; a subject that we discuss in Section 6.

5.1

Indicator functions

Consider the following problem studied by Lasserre [12], and Bertsimas and Popescu [2] : Given σ ∈ IRId , and Borel measurable sets S ⊆ D ⊆ IRn , compute: Eµ (1S )

zPind = sup (Pind )

Eµ (sα ) = σα , ∀ α ∈ Id µ ∈ M(D),

s.t.

where 1S : IRn → {0, 1} is the indicator function of the set S, that is, if s ∈ S and 1S (s) = 0 otherwise.

1S (s) = 1

Problem (Pind ) is a special instance of (Ppw ) with p = 2, q 1 (s) = 1, D1 = S, q (s) = 0, and D2 = D ∩ S c ; so by (4) its conic programming formulation is 2

zPCP = ind

1 y(0,...,0)

sup

y1 + y2 = σ y 1 ∈ Mn,d (S) y 2 ∈ Mn,d (D ∩ S c ).

s.t.

(PCP ind )

Similarly, by (5) the conic programming formulation of its dual is zDCP = ind

hσ, θi

inf

(DCP ind )

s.t.

θ − (1, 0, . . . , 0)T θ

∈ Pn,d (S) ∈ Pn,d (D ∩ S c ),

where the first component of θ is precisely θ(0,...,0) . Notice that from the simple observation: θ(s) ≥ 1S (s) ∀ s ∈ D ⇒ θ(s) ≥ 0 ∀ s ∈ D, it follows that (DCP ind ) is equivalent to zDg CP =

inf

hσ, θi

ind

g) (D CP ind

s.t.

θ − (1, 0, . . . , 0)T θ

9

∈ Pn,d (S) ∈ Pn,d (D).

Now consider the following relaxation of (PCP ind ): zPg CP =

sup

ind

s.t.

CP g (P ind )

1 y(0,...,0)

y1 + y2 = σ y 1 ∈ Mn,d (S) y 2 ∈ Mn,d (D).

CP CP g g Using (3) it readily follows that weak duality holds between (P ind ) and (Dind ); that is zPg g CP ≤ zD CP = zDCP . ind ind

ind

Furthermore, zPCP ≤ zPg CP ind

ind

CP CP g as (P ind ) is a relaxation of (Pind ). Using the last two equations and (6) we get

zPind = zPCP ≤ zPg g CP ≤ zD CP = zDCP = zDind . ind ind ind

ind

If any of the conditions for strong duality hold for (PCP ind ) (see Propositions 4 CP g through 7), then the relaxed problem (P ) solves (P ). Clearly, this is advanind ind CP CP g tageous when trying to solve (Pind ) numerically as (Pind ) is simpler than (PCP ind ) (see Section 6). A particular case in which the conditions for strong duality are satisfied by (PCP ind ) follows from Proposition 6. Corollary 1 If d is even and S is compact, then zPCP = zDCP . ind ind This means that in most of the cases considered in [2] and [12], strong duality holds without further assumptions. Both bounds in Example 1 are special instances of (Pind ), and many Tchebycheff Inequalities considered in probability theory fall into this category (see, e.g., Chapter XII in [9]).

5.2

Max-like functions

Consider the following problem: Given σ ∈ IRId , a set D ⊆ IRn , and n-variate polynomials q i (s), i = 1, . . . , p of degree d, compute: zPmax = sup (Pmax )

s.t.

Eµ (max{q i (s)|i = 1, . . . , p}) Eµ (sα ) = σα , ∀ α ∈ Id µ ∈ M(D).

By choosing disjoint S Di , i = 1, . . . , p such that max{q j (s)|j = 1, . . . , p} = p q (s) if s ∈ Di and D = i=1 Di , it follows that (Pmax ) is a special instance of i

10

CP (Ppw ), and we can identify (4) with (PCP max ) and (5) with (Dmax ). Furthermore, i notice that from the simple observation: θ(s) ≥ max{q (s)|i = 1, . . . , p} ∀ s ∈ D ⇒ θ(s) ≥ q i (s) ∀ s ∈ D for i = 1, . . . , p, it follows that (DCP max ) is equivalent to

CP ] (D max )

zD] CP =

hσ, θi

inf

max

θ − qi

s.t.

∈ Pn,d (D), i = 1, . . . , p.

Now consider the following relaxation of (PCP max ) zPg CP =

sup

max

CP (Pg max )

p X

hq i , y i i

i=1

s.t.

p X

yi = σ

i=1

y i ∈ Mn,d (D), i = 1, . . . , p. CP CP ] Using (3) it readily follows that weak duality holds between (Pg max ) and (Dmax ); that is CP zPg CP ≤ zD ] CP = zDmax . max

max

Furthermore, zPCP ≤ zPg CP max

max

CP CP as (Pg max ) is a relaxation of (Pmax ). Using the last two equations and (6) we get CP zPmax = zPCP ≤ zPg CP ≤ zD ] CP = zDmax = zDmax . max max

max

As in Section 5.1, if any of the conditions for strong duality hold for (PCP max ) CP (see Propositions 4 through 7), then the relaxed problem (Pg ) solves (P ). max max Again, this is advantageous, and key in obtaining numerical solutions for (Pmax ) (see Section 6). The bound in Example 2 is a special instance of (Pmax ). This bound is used by Lo [14] to obtain bounds on the prices of European call options, and by Grundy [6] to study the distribution of the underlying stock price. In [1], Bertsimas and Popescu generalize Lo’s bound by studying the special instance of (Pmax ) with f (s) = max{s − K, 0}. As the following three examples illustrate, by solving special instances of (Pmax ), this type of financial analysis can be extended to European rainbow options; that is, European options whose payoff depends on the value of several stock prices. Example 3 The tight upper bound on the payoff of a European exchange option on a pair of stocks whose prices at maturity have known means τ1 , τ2 , variances

11

ρ21 , ρ22 and covariance ρ12 is given by zexch =

max s.t.

Eµ (max{0, s1 − s2 }) Eµ (1) = 1 Eµ (s1 ) = τ1 , Eµ (s2 ) = τ2 Eµ (s1 s2 ) = ρ12 + τ1 τ2 Eµ (s21 ) = ρ21 + τ12 , Eµ (s22 ) = ρ22 + τ22 µ ∈ M(IR2+ ),

(7)

where s1 , s2 represent the random underlying stock prices at maturity. From Proposition 7 and by letting p = 2, q 1 (s) = 0, q 2 (s) = s1 − s2 , and D = IR2+ CP in (Pg max ), it follows that the upper bound (7) can be obtained by solving the following conic program: zexch =

2 2 − y(0,1) y(1,0)

sup s.t.

2 X

i i i i i i (y(0,0) , y(1,0) , y(0,1) , y(1,1) , y(2,0) , y(0,2) )T = σ 0

i=1

y i ∈ M2,2 (IR2+ ), i = 1, 2, where σ 0 = (1, τ1 , τ2 , ρ12 + τ1 τ2 , ρ21 + τ12 , ρ22 + τ22 )T . Example 4 The tight upper bound on the payoff of a European call option with strike K on the maximum of three stocks whose prices at maturity have known means τj , variances ρ2j , j = 1, 2, 3 and covariances ρjk , 1 ≤ j < k ≤ 3 is given by zcall = max Eµ (max{max{s1 , s2 , s3 } − K, 0}) s.t. Eµ (1) = 1 Eµ (sj ) = τj Eµ (sj sk ) = ρjk + τj τk Eµ (s2j ) = ρ2j + τj2 µ ∈ M(IR3+ ),

j = 1, 2, 3 1≤j