Geometry of polar cones and

superreplication prices in

incomplete financial markets

Frank Oertel Department of Mathematics

National University of Ireland, Cork (based on joint work with Mark Owen)

Positivity V

Queen’s University of Belfast, UK

July 20-27, 2007

Contents

- The market model

- The admissible case

- Separating probability measures

- Superreplication prices: the general approach - Pricing measures and utility functions I - Duality of polar cones

- Pricing measures and utility functions II - A very small extract of references

The market model

• We consider a security market with d + 1 assets - one risk free asset (a bond) and d risky assets.

• S ≡ (S^{1}, …, S^{d}) denotes the discounted price process of the d assets, where each price is modelled as an arbitrary - not necessarily locally bounded - càdlàg semimartingale including jumps (e. g., a Lévy process) in a filtered probability space (Ω, ℱ , 𝔽, ℙ) with finite time horizon T.

• A portfolio is a pair Π = (x, H), where

– x ∈ ℝ is the initial wealth;

– H ≡ (H^{1}, …, H^{d}) is a trading strategy, i. e., a predictable and S-integrable ℝ^{d}- valued stochastic process, representing the holdings of the risky d assets.

Example 1 Fix a scenario ω ∈ Ω. Assume you want to invest in one stock only, i. e., d = 1. The only trading dates are 0 = t_{1} < … < t_{n} = T. At any time point t_{i−1} you invest Z_{i−1}(ω) parts in the stock with price S_{t_{i−1}}(ω). The choice of Z_{i−1}(ω) is based on the information until t_{i−1} only! No trading is then possible until t_{i}. Thus, at t_{i}, your gain resp. loss is given by

Z_{i−1}(ω)(S_{t_{i}}(ω) − S_{t_{i−1}}(ω)).

In other words, if

H_{t}(ω) := 11_{{0}}(t)Z_{0}(ω)+ n ; ∑ ; i=1 11_{(t_{i−1},t_{i}]}(t)Z_{i−1}(ω),

denotes your trading strategy H, where each Z_{i} is a ℱ_{t_{i}}-measurable random variable, then your cumulative gain resp. loss process ∫ HdS is given by

( ∫ HdS)_{t}(ω) := n ; ∑ ; i=1 Z_{i−1}(ω)(S_{t_{i}∧t}(ω)−S_{t_{i−1}∧t}(ω)) .

The admissible case

A trading strategy H is said to be admissible if the gains process is uniformly bounded from below by a (non-random) constant, i. e., if there exists a constant c ≥ 0 such that for all t ∈ [0, T],

G_{t}(H) := ∫_{0}^{t} H_{u}dS_{u} := ( ∫ HdS)_{t} ≥ −c, ℙ−a. s. .

This disallows doubling strategies: investors have a fixed credit limit.

A road to positivity The set of all attainable terminal wealths G_{T} (H) from admissible trading strategies H with zero initial wealth is a convex cone (resp. wedge)

K^{adm} := {G_{T} (H) : H is admissible }

which is contained in the vector lattice L^{0} = L^{0}(Ω, ℱ , ℙ) of all random variables. Note that K^{adm} is not a subspace of L^{0}.

Let X ∈ L^{0} be an arbitrary contingent claim. We assume that X satisfies the Property (SR ^{adm} ) There exists an admissible portfolio Π = (x, H) which superreplicates X, i. e., we assume that

A_{X} := {x ∈ ℝ : ∃H adm. s.t. X ≤ x + G_{T} (H)}
= {x ∈ ℝ : X − x ∈ K^{adm} − L_{+}^{0} } ≠ ∅ .

Definition 1 Assume that the set A_{X} ⊆ ℝ is bounded from below. Then the superreplication price of X ∈ L^{0} is defined by the following real number:

\overline{π}(X) := inf (A_{X})
= inf {x ∈ ℝ : X − x ∈ K^{adm} − L_{+}^{0} }.

When is the set A_{X} bounded from below? How do we then calculate \overline{π}(X)?

Separating probability measures

Consider

M_{1}^{adm} (ℙ) := {ℚ ≪ ℙ : 𝔼_{ℚ} [G] ≤ 0 ∀ G ∈ K^{adm} }.

Remark 1 Recall:

(1) If S is bounded then M_{1}^{adm} (ℙ) is the set of all ℙ-absolutely continuous probability measures ℚ such that S is a ℚ-martingale.

(2) If S is locally bounded then M_{1}^{adm} (ℙ) = M^{a}(ℙ), where M^{a}(ℙ) denotes the set of all ℙ-absolutely continuous probability measures ℚ such that S is a ℚ-local martingale.

(3) In the general non-locally bounded case: if M_{σ}^{a}(ℙ) ≠ ∅ then M_{1}^{adm} (ℙ) is the closure, in the topology induced by the total variation norm, of the set M_{σ}^{a}(ℙ) of all ℙabsolutely continuous probability measures ℚ such that S is a ℚ-sigma-martingale.

Case 1: First, we reconsider the case of a contingent claim X ∈ L^{0} which is bounded from below (for example, X ≥ 0) and satisfies (SR^{adm} ).

If S is locally bounded and M^{e}(ℙ) ≠ ∅ then (cf. Delbaen-Schachermayer (1995))

\overline{π}(X) = sup_{ℚ∈M^{e}(ℙ)} 𝔼_{ℚ} [X].

If S is a general semimartingale and M_{σ}^{e}(ℙ) ≠ ∅ then (cf. Delbaen-Schachermayer (1998))

\overline{π}(X) = sup_{ℚ∈M_{σ}^{e}(ℙ)} 𝔼_{ℚ} [X].

Case 2: Assume now that M_{1}^{adm} (ℙ) ≠ ∅ and that X ∈ L^{0} is arbitrary, hence not necessarily bounded from below. However, since we are interested in pricing the contingent claim X, it is natural to assume that X ∈ L^{1}(ℚ) for any probability measure ℚ ∈ M_{1}^{adm} (ℙ), hence that

X ∈ L(M_{1}^{adm} (ℙ)) := _{ℚ∈M_{1}^{adm} (ℙ)}^{⋂} L^{1}(ℚ) ⊆ L^{0}.

In the following, let M_{1}^{adm} (ℙ) ≠ ∅.

Observation 1

K^{adm} ⊆ L(M_{1}^{adm} (ℙ)) .

Hence, if X ∈ L^{0} is bounded from below and satisfies (SR^{adm} ) then X ∈ L(M_{1}^{adm} (ℙ)). Proof. Let −r ≤ X. Then

|X| = |(X+r)−r| ≤ (X+r)+|r| ≤ (x+G_{T} (H))+2|r|.

□

Observation 2 If X ∈ L(M_{1}^{adm} (ℙ)) satisfies (SR^{adm} ), then the set A_{X} is bounded from below and {𝔼_{ℚ} [X] : ℚ ∈ M_{1}^{adm} (ℙ)} is bounded from above. Moreover, we have

−∞ < sup_{ℚ∈M_{1}^{adm} (ℙ)} 𝔼_{ℚ} [X] ≤ \overline{π}(X) < +∞.

Proof. Let x ∈ ℝ such that X − x ∈ K^{adm} − L_{+}^{0} and ℚ ∈ M_{1}^{adm} (ℙ) arbitrarily chosen. Then

−∞ < −𝔼_{ℚ} [|X|] ≤ 𝔼_{ℚ} [X] = 𝔼_{ℚ} [X − x]+x ≤ 0+x .

□ However, one can construct Y ∈ L(M_{1}^{adm} (ℙ)) (cf. Biagini-Frittelli (2004)) such that

sup_{ℚ∈M_{1}^{adm} (ℙ)} 𝔼_{ℚ} [Y]< \overline{π}(Y).

Therefore the (convex) cone K^{adm} of admissible trading strategies seems to be unsuitable for the superreplication of unbounded claims! Intuitively, the cone K^{adm} is too small.

To reveal the underlying – geometric – key ideas how to maintain equality even in the unbounded and (later) in the non-admissible case, recall that

K^{adm} ⊆ L(M_{1}^{adm} (ℙ)) = _{ℚ∈M_{1}^{adm} (ℙ)}^{⋂} L^{1}(ℚ).

Let X ∈ L(M_{1}^{adm} (ℙ)) such that X satisfies (SR^{adm} ). Let C be an arbitrary cone such that

K^{adm} ⊆ C ⊆ L(M_{1}^{adm} (ℙ))

and \overline{π}(X; C) := inf \{x ∈ ℝ : X ≤ x + G for some G ∈ C\}

exists. Then

sup_{ℚ∈M_{1}^{adm} (ℙ)} 𝔼_{ℚ} [X] ≤^{?} \overline{π}(X; C) ≤^{✓} \overline{π}(X; K^{adm} ) = \overline{π}(X) .

Consequently, we have arrived at the following general problem which even will allow us to evade the too restrictive admissibility condition:

Problem: Fix an arbitrary cone K in L^{0} and a suitable (convex) set of pricing measures M_{1}(ℙ; K). Do there exist a non-empty set M ⊆ M_{1}(ℙ; K) and a sufficiently large cone C ⊆ _{ℚ∈M}^{⋂} L^{1}(ℚ) =:

L(M) such that K ⊆ C and

sup_{ℚ∈M} 𝔼_{ℚ} [X]= \overline{π}(X; C) ∀ X ∈ L(M) ?

To solve this problem, we now concentrate our attention on a suitable class of cones in vector lattices.

Definition 2 Let (E, ≤) be a vector lattice and C ⊆ E a cone in E. C is an umbrella (cone) in E if −E_{+} = {x ∈ E : x ≤ 0} ⊆ C.

Given an arbitrary cone K ⊆ E, we denote by s_{E}(K) the umbrella hull, i. e., the smallest umbrella in E which contains K. Note that

s_{E}(K) = {x ∈ E : ∃ g ∈ K s. t. x ≤ g} = K − E_{+} .

Consequently, a cone C ⊆ E is an umbrella in E if and only if C = s_{E}(C) = C − E_{+} .

Proposition 1 Let (E_{α})_{α∈A} be a family of vector sublattices of a vector lattice L.Then E := _{α∈A}^{⋂} E_{α} is a vector sublattice of L. Let K ⊆ E

be an arbitrary cone in E. Then

s_{E}(K) = _{α∈A}^{⋂} s_{E_{α}}(K) = _{α∈A}^{⋂} (K − (E_{α})_{+} ) .

Example 2 Assume that M_{1}^{adm} (ℙ) ≠ ∅. Set E := L(M_{1}^{adm} (ℙ)). Then K^{adm} ⊆ E, and

s_{E}(K^{adm} ) = _{ℚ∈M_{1}^{adm} (ℙ)}^{⋂} (K^{adm} − L_{+}^{1} (ℚ)) .

Superreplication prices: the general approach

We now extend the definition of the superreplication price to allow terminal wealths from an arbitrary cone K in L^{0}.

(i) Fix an arbitrary cone K ⊆ L^{0}.

(ii) Set M_{1}(ℙ; K) := {ℚ ≪ ℙ : K ⊆ L^{1}(ℚ) and 𝔼_{ℚ} [G] ≤ 0 for all G ∈ K}.

(iii) Consider an arbitrary non-empty subset M of M_{1}(ℙ; K) and set L(M) := ⋂_{ℚ∈M} L^{1}(ℚ) . Note that by construction

K ⊆ L(M_{1}(ℙ; K)) ⊆ L(M) .

(iv) Let X ∈ L(M) and C ⊆ L(M) be an arbitrary cone such that K ⊆ C. Assume that A_{X}(K) ≠ ∅, where

A_{X}(C) := {x ∈ ℝ : ∃ G ∈ C s.t. X ≤ x + G} . A_{X}(K) is bounded from below and we have A_{X}(K) ⊆ A_{X}(C) = A_{X}(s_{L(M)}(C)). If A_{X}(C) is bounded from below, we set

\overline{π}(X; C) := inf (A_{X}(C)) ≤ \overline{π}(X; K) .

Similarly to Observation 2, it follows that

−∞ < sup_{ℚ∈M} 𝔼_{ℚ} [X] ≤ \overline{π}(X; K) < +∞.

Moreover,

\overline{π}(X; C) = inf \{x ∈ ℝ:X − x ∈ s_{L(M)}(C) \}
= inf {x ∈ ℝ:X − x ∈ _{ℚ∈M}^{⋂} (C − L_{+}^{1} (ℚ)) } .

Thus, we have arrived at the following natural question: Given an arbitrary cone K ⊆ L^{0}, which cones C ⊇ K and which non-empty sets M of separating probability measures would then be suitable to solve our pricing problem? Would then s_{L(M)}(K) be a subset of C?

Pricing measures and utility functions I

First answers were provided by Biagini and Frittelli (for K^{adm} only!) and Oe.-Owen (already for arbitrary cones K ⊆ L^{0}), where preferences of an investor were incorporated in the construction of an enlarged cone in the space L(M_{Φ} ) by means of the Fenchel-Legendre conjugate Φ of the investor’s utility function U. Given an arbitrary cone K ⊆ L^{0}, this cone was defined as

C_{Φ} := _{ℚ∈M_{Φ}}^{⋂} \overline{K − L_{+}^{1} (ℚ)}^{L^{1}(ℚ)},

where

M_{Φ} ≡ M_{Φ} (K) := \{ℚ ∈ M_{1}(ℙ; K) | Φ ∘ \frac{dℚ}{dℙ} ∈ L^{1}(ℙ)\}

(which is the set of all pricing measures with finite entropy if K := K^{adm} ). If M_{Φ} ≠ ∅, then

s_{L(M_{Φ} )}(K) = _{ℚ∈M_{Φ}}^{⋂} (K − L_{+}^{1} (ℚ)) ⊆ C_{Φ} ⊆ L(M_{Φ} ) .

If M_{Φ} ≡ M_{Φ} (K^{adm} ), the following result holds:

Theorem 1 (Biagini-Frittelli (2004)) Let U : ℝ ⟶ ℝ be a strictly concave, strictly increasing C^{1}-utility function, satisfying the Inada conditions and Reasonable Asymptotic Elasticity at −∞ and +∞. Assume that M_{Φ}^{adm} ≠ ∅ and Φ(0) < +∞, where Φ denotes the Fenchel-Legendre conjugate of U and let X ∈ L(M_{Φ}^{adm} ). Then

\overline{π}(X; C_{Φ}^{adm} ) = sup_{ℚ∈M_{Φ}^{adm}} 𝔼_{ℚ} [X].

By slightly enlarging the set M_{Φ} , we will recognise that we only need Reasonable Asymptotic Elasticity of U at −∞ (but not at +∞) and that we can drop the boundedness assumption Φ(0) < +∞.

The proof of this generalisation is built on a duality theory for (convex) cones which does not only allow a transfer of Farkas’ Lemma (i. e., dual LPs) to infinite-dimensional normed lattices. It also gives us a very helpful bipolar relation where none was possible before.

Duality of polar cones

Let (E, F) be an arbitrary bilinear system of vector spaces over ℝ. For a non-empty set A ⊆ E the polar of A is often defined in two different ways in the literature:

Polar I:

A∘ := {w ∈ F : ⟨z, w⟩ ≤ 1 ∀z ∈ A};

Polar II:

A• := {w ∈ F : |⟨z, w⟩| ≤ 1 ∀z ∈ A}.

Note that A• ⊆ A∘. Both definitions coincide if A is circled (i. e., if [−1, 1] · A ⊆ A).

In addition, we introduce the polar cone of A:

A^{◁} := {w ∈ F : ⟨z, w⟩ ≤ 0 ∀z ∈ A}.

Clearly, A^{◁} is a cone in F and A^{◁} ⊆ A∘.

Similarly, for a non-empty set B ⊆ F, we define its polar cone B^{◁} ⊆ E.

Let ∅ ≠ A ⊆ E. Let cone (A) be the smallest cone in E which contains A.

Proposition 2 Let (E, F) be an arbitrary bilinear system of real vector spaces and A be an arbitrary non-empty subset of E. Then:

A^{◁} = (cone (A))∘.

In particular, A^{◁} is σ(F, E)-closed.

Corollary 1 Let (E, F) be an arbitrary bilinear system of real vector spaces and A be an arbitrary non-empty subset of E. Then:

A^{◁◁} = \overline{cone (A)}^{σ(E,F)}.

In other words, A^{◁◁} is the smallest σ(E, F)- closed cone in E which contains A. If in addition A is a cone, then A^{◁◁} = \overline{A}^{σ(E,F)}.

Let K be an arbitrary cone in L^{0}. Recall the set

M_{1}(ℙ; K) := {ℚ ≪ ℙ : K ⊆ L^{1}(ℚ) and
𝔼_{ℚ} [G] ≤ 0 for all G ∈ K}.

Let ∅ ≠ M ⊆ M_{1}(ℙ; K). By identifying probability measures ℚ ≪ ℙ with their Radon-Nikodym derivative \frac{dℚ}{dℙ} ∈ L_{+}^{1} (ℙ) of norm 1, we set

E := L(M) := _{ℚ∈M}^{⋂} L^{1}(ℚ) ⊇ K

and

F := Lin(M) ⊆^{✓} L^{1}(ℙ).

Hence, for any pair (Z, W) ∈ E × F, the following dual pairing is well-defined:

⟨Z, W ⟩ := 𝔼_{ℙ} [ZW].

In particular,

⟨Z, ℚ⟩ ≡ ⟨Z, \frac{dℚ}{dℙ}⟩ = 𝔼_{ℚ} [Z] ∀ ℚ ∈ M.

A standard exercise in measure theory shows that (E, F) is a well-defined left-dual (bilinear) system. Consequently,

F = (E, τ)′ ,

where τ := σ(E, F). To understand the following crucial result, recall that

s_{E}(K) = _{ℚ∈M}^{⋂}(K − L_{+}^{1} (ℚ)) = K − E_{+} ⊆ E

describes the umbrella hull of the cone K in the vector sublattice E of L^{0}.

Proposition 3 Let K be an arbitrary cone in L^{0}, and let ∅ ≠ M ⊆ M_{1}(ℙ; K). Then

L_{+}^{1} (ℙ)∩K^{◁} = F ∩ cone (M_{1}(ℙ; K)) = (s_{E}(K))^{◁}.

In particular, the set F ∩ cone (M_{1}(ℙ; K)) is σ(F, E)-closed, and it contains \overline{cone (M)}^{σ(F,E)}.

Corollary 2 Let K be an arbitrary cone in L^{0} and let ∅ ≠ M ⊆ M_{1}(ℙ; K). Set C_{E}(K) : = \overline{s_{E}(K)}^{σ(E,F)}. TFAE:

(i) F ∩ cone (M_{1}(ℙ; K)) = \overline{cone (M)}^{σ(F,E)};

(ii) M^{◁} = C_{E}(K) ;

(iii) \overline{cone (M)}^{σ(F,E)} = (C_{E}(K))^{◁} = (s_{E}(K))^{◁}.

Example 3 Set E_{1} := L(M_{1}(ℙ; K)) and F_{1} := Lin(M_{1}(ℙ; K)). If M_{1}(ℙ; K) ≠ ∅, Proposition 3 implies that

cone (M_{1}(ℙ; K)) = (C_{E_{1}}(K))^{◁}

is σ(F_{1}, E_{1})-closed. Moreover,

M_{1}(ℙ; K)^{◁} = C_{E_{1}}(K) .

Theorem 2 Let K be an arbitrary cone in L^{0}, and let ∅ ≠ M ⊆ M_{1}(ℙ, K) such that

\overline{cone (M)}^{σ(F,E)} = F ∩ cone (M_{1}(ℙ, K)) .

Then K ⊆ C_{E}(K) = M^{◁} ⊆ E. Let X ∈ E. Then the set A_{X}(C_{E}(K)) is bounded from below and

\overline{π}(X; C_{E}(K)) = sup_{ℚ∈M} 𝔼_{ℚ} [X] .

Proof. Since the cone C_{E}(K) obviously is an umbrella in E, it follows that

\overline{π}(X; C_{E}(K)) = inf \{x ∈ ℝ : X − x ∈ C_{E}(K)\}
= inf {x ∈ ℝ : X − x ∈ M^{◁}}
= inf {x ∈ ℝ : 𝔼_{ℚ} [X − x] ≤ 0 ∀ ℚ ∈ M}
= inf {x ∈ ℝ : 𝔼_{ℚ} [X] ≤ x ∀ ℚ ∈ M}
= sup_{ℚ∈M} 𝔼_{ℚ} [X].
□

Corollary 3 Let K be an arbitrary cone in L^{0}. Set E_{1} := _{ℚ∈M_{1}(ℙ;K)}^{⋂} L^{1}(ℚ) = L(M_{1}(ℙ; K)) and

F_{1} := Lin(M_{1}(ℙ; K)). Then

\overline{π}(X; C_{E_{1}}(K)) = sup_{ℚ∈M_{1}(ℙ,K)} 𝔼_{ℚ} [X] ∀ X ∈ E_{1} .

Consequently, due to Biagini-Frittelli’s counterexample, the (convex) cone s_{E_{1}^{adm}} (K^{adm} ) is not σ(E_{1}^{adm} , F_{1}^{adm} )-closed.

Pricing measures and utility functions II

Let K be an arbitrary cone in L^{0}.

Assumption 1 We assume that the investor has a critical wealth a ∈ {−∞} ∪ ℝ and a utility function U : (a, ∞) → ℝ which is increasing, strictly concave, continuously differentiable, and satisfies the Inada conditions

lim_{x↓a} U′(x) = ∞,  lim_{x↑∞} U′(x) = 0.

Furthermore, if the domain of U is the whole real line (i. e., if a = −∞) then we assume that U has Reasonable Asymptotic Elasticity at −∞, in the sense that

AE_{−∞}(U) := lim inf_{x→−∞} \frac{xU′(x)}{U(x)} > 1.

The Fenchel-Legendre conjugate Φ of the utility function U is defined for y > 0 by

Φ(y) := sup_{x∈(a,∞)}(U(x) − xy) .

In the following, we set M_{1} := M_{1}(ℙ, K). Note that

M_{Φ} = \{ℚ ∈ M_{1} | 𝔼_{ℙ} [Φ^{+} (\frac{dℚ}{dℙ})] < ∞\} .

Definition 3 A measure ℚ ≪ ℙ is said to have finite loss-entropy if there exists a constant b > 0 such that

𝔼_{ℙ} [Φ^{+} (\frac{dℚ}{dℙ}) 11_{\{\frac{dℚ}{dℙ}≥b\}}] < ∞.

The set of separating measures with finite lossentropy is therefore given by

M̂_{Φ} := \{ℚ ∈ M_{1} | 𝔼_{ℙ} [Φ^{+} (\frac{dℚ}{dℙ}) 11_{\{\frac{dℚ}{dℙ}≥b\}}] < ∞\} .

Clearly, the definition of the set M̂_{Φ} does not depend on the choice of the constant b > 0.

Since our machinery is set up, we only have to adapt the previous definitions. Assume throughout that M̂_{Φ} ≠ ∅. Our left-dual system is now given by

E_{Φ} := L(M̂_{Φ} ) := _{ℚ∈M̂_{Φ}}^{⋂} L^{1}(ℚ)

and

F_{Φ} := Lin(M̂_{Φ} ) .

Set C_{Φ} := C_{E_{Φ}} (K). It turns out that cone (M_{Φ} ) might not be σ(F_{Φ} , E_{Φ} )-closed. However,

Theorem 3

cone (M̂_{Φ} ) = F_{Φ} ∩ cone (M_{1}(ℙ; K)) .

Consequently, cone (M̂_{Φ} ) is σ(F_{Φ} , E_{Φ} )-closed and the pair (C_{Φ} , cone (M̂_{Φ} )) satisfies a “symmetric cone duality”.

Corollary 4

\overline{π}(X; C_{Φ} ) = sup_{ℚ∈M̂_{Φ}} 𝔼_{ℚ} [X] ∀ X ∈ E_{Φ} .

The next result links our approach with the very useful theory of Orlicz spaces:

Observation 3 Let a = −∞. Assume that in addition to Assumption 1 Φ is of “De La Vallée Poussin” type, i. e., assume that

lim_{x→∞} \frac{Φ(x)}{x} = ∞ ,

then Φ satisfies the Δ_{2} condition:

∃b > 0 ∀α > 1 ∃k_{α} > 0 : Φ(αy) ≤ k_{α}Φ(y) ∀y > b .

There is a very nice economic interpretation of

C_{Φ} : Since C_{Φ} =^{(!)} (M̂_{Φ} )^{◁} depends only on the finite loss-entropy measures, it depends only on the values of the Fenchel-Legendre conjugate Φ(y) for arbitrarily large values of y, and hence on values of the utility function U(x) for arbitrarily large negative values of x. So the only important information is contained in the preferences of the investor to asymptotically large losses.

Note also that the utility preferences of the investor to asymptotically large losses are encoded in the spaces E_{Φ} and F_{Φ} .

A very small extract of references

• S. Biagini and M. Frittelli (2004). On the super replication price of unbounded claims. Ann. Appl. Prob. 14, 1970–1991.

• F. Delbaen and W. Schachermayer (2001). Applications to mathematical finance. Handbook of the geometry of Banach spaces, Vol. 1, North-Holland, 369–391.

• H. H. Schaefer (1974). Banach Lattices and Positive Operators. Springer

• F. Oertel and M. Owen (2006). On utility-based super-replication prices of contingent claims with unbounded payoffs. Submitted.