Duality formulas for robust pricing and hedging in discrete time

Duality formulas for robust pricing and hedging in discrete time∗ Patrick Cheridito

Michael Kupper

Ludovic Tangpi

February 2016

Abstract In this paper we derive robust super- and subhedging dualities for contingent claims that can depend on several underlying assets. In addition to strict super- and subhedging, we also consider relaxed versions which, instead of eliminating the shortfall risk completely, aim to reduce it to an acceptable level. This yields robust price bounds with tighter spreads. As applications we study strict super- and subhedging with general convex transaction costs and trading constraints as well as risk based hedging with respect to robust versions of the average value at risk and entropic risk measure. Our approach is based on representation results for increasing convex functionals and allows for general financial market structures. As a side result it yields a robust version of the fundamental theorem of asset pricing. 2010 Mathematics Subject Classification: 91G20, 46E05, 60G42, 60G48 Keywords: Robust price bounds, superhedging, subhedging, robust fundamental theorem of asset pricing, transaction costs, trading constraints, risk measures.

1. Introduction Super- and subhedging dualities lie at the heart of no-arbitrage arguments in quantitative finance. By relating prices to hedging, they provide bounds on arbitrage-free prices. But they also serve as a stepping stone to the application of duality methods to portfolio optimization problems. In traditional financial modeling, uncertainty is described by a single probability measure P, and the super- and subhedging prices of a contingent claim X are given by φ(X) = inf {m ∈ R : there exists a Y ∈ G such that m − X + Y ≥ 0 P-a.s.}

(1.1)

−φ(−X) = sup {m ∈ R : there exists a Y ∈ G such that X − m + Y ≥ 0 P-a.s.} ,

(1.2)

and

∗

We thank Daniel Bartl, Peter Carr, Samuel Drapeau, Marek Musiela, Jan Obl´ oj, Mete Soner and Nizar Touzi for fruitful discussions and helpful comments.

1

where G is the set of all realizable trading gains. Classical results assume that X depends on a set of underlying assets which can be traded dynamically without transaction costs or constraints. Then G is a linear space, and under a suitable no-arbitrage condition, one obtains dualities of the form φ(X) = sup EQ X and − φ(−X) = inf EQ X, (1.3) Q∈Me (P)

Q∈Me (P)

where Me (P) is the set of all (local) martingale measures equivalent to P, see e.g. [27] or [19] for an overview. In this paper we do not assume that the probabilities of all future events are known. So instead of starting with a predefined probability measure, we specify a collection of possible trajectories for a set of basic assets. This includes a wide range of setups, from a single binomial tree model to the model-free case, in which at any time, the prices of all assets can lie anywhere in R+ (or R). We replace the P-almost sure inequalities in (1.1) and (1.2) by a general set of acceptable positions A and consider super- and subhedging functionals of the form φ(X) = inf {m ∈ R : m − X ∈ A − G}

and

− φ(−X) = sup {m ∈ R : X − m ∈ A − G} . (1.4)

If A is the cone of non-negative positions, this describes strict super- and subhedging, which requires that the shortfall risk be eliminated completely. Alternatively, one can allow for a certain amount of risk by enlarging the set A. This reduces the spread between super- and subhedging prices. Moreover, the set of trading gains G does not have to be a linear space and can describe general market structures with transaction costs and trading constraints. Our main result, Theorem 2.1, yields dual expressions for the quantities in (1.4) in terms of expected values under the assumption that it is not possible to transform a debt into an acceptable position by investing in the market. As a byproduct, it contains a robust fundamental theorem of asset pricing (FTAP), which relates two different notions of no-arbitrage to the existence of generalized martingale measures. In the case where the underlying assets are bounded, it holds for general sets A and G such that G−A is convex. Otherwise, it needs that the set G−A is large enough. This can be guaranteed by either assuming that the financial market is sufficiently rich or, as shown in Proposition 2.2, that the acceptability condition is not too strict. In Sections 3 and 4 we study different specifications of A and G, for which the dual representations can be computed explicitly. Section 3 is devoted to the case where A consists of all non-negative positions, corresponding to strict super- and subhedging. We consider general semi-static trading strategies consisting of dynamic investments in the underlying assets and static derivative positions. Proposition 3.1 covers general convex transaction costs and constraints on the derivative positions. Proposition 3.2 deals with dynamic shortselling constraints. In Section 4 we relax the hedging requirement and control shortfall risk with a family of risk measures defined in different probabilistic models. This allows to introduce risk-tolerance in a setup of model-uncertainty and makes it possible to specify attitudes towards risk and ambiguity. Our price bounds then become robust good deal bounds. Proposition 4.2 gives an explicit duality formula in the case where risk is assessed with a robust average value at risk. Proposition 4.4 provides the same for a robust entropic risk measure. Our approach is based on representation results for increasing convex functionals and permits to combine robust methods with transaction costs, trading constraints, partial hedging and good deal bounds. Robust hedging methods go back to [31] and were further investigated in e.g. [16, 18, 37]. Various versions of robust FTAPs and superhedging dualities have been derived in [1, 2, 3, 4, 7, 8, 9, 12, 13, 21, 22, 23, 28, 38]. For FTAPs and superhedging dualities under transaction costs we

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refer to [20, 30, 33, 40] and the references therein. The literature on partial hedging started with the quantile-hedging approach of [25] and subsequently developed more general risk-based methods; see e.g. [10, 26, 39] and the closely related literature on good deals, such as e.g. [6, 32, 35, 41]. The rest of the paper is organized as follows: Section 2 introduces the notation and states the paper’s main results. In Section 3 we study strict super- and subhedging and give robust FTAPs with corresponding superhedging dualities under convex transaction costs and trading constraints. As special cases we obtain versions of Kantorovich’s transport duality [34] that include martingale or supermartingale constraints. In Section 4 we use robust risk measures to weaken the acceptability condition. This yields robust good deal bounds lying closer together than the strict super- and subhedging prices of Section 3. All proofs are given in the appendix.

2. Main results We consider a model with finitely many trading periods t = 0, 1, . . . , T and J + 1 financial assets S 0 , . . . , S J . As sample space we take a non-empty closed subset Ω of RJT . S 0 is used as num´eraire, and the prices of the assets S j , j ≥ 1, in units of S 0 are modeled as S0j = sj0 ∈ R

and Stj (ω) = ωtj ,

t ≥ 1, ω ∈ Ω.

We endow Ω with the Euclidean metric and denote the space of all Borel measurable functions X : Ω → R by B. We assume that the set of realizable trading gains is given by a subset G ⊆ B containing 0 and the set of acceptable positions by a subset A ⊆ B containing the positive cone B + := {X ∈ B : X ≥ 0} such that G − A is convex. The corresponding superhedging functional is φ(X) := inf {m ∈ R : m − X ∈ A − G} . It determines for every liability X ∈ B, the smallest amount of money m needed such that m−X can be transformed into an acceptable position by investing in the financial market. The case A = B + corresponds to strict superhedging, which requires that a contingent claim be superreplicated in every possible scenario. A larger acceptance set A relaxes the superhedging requirement and lowers the hedging costs. The subhedging functional induced by G and A is given by −φ(−X) = sup {m ∈ R : X − m ∈ A − G} . In the whole paper, Z : Ω → [1, +∞) is a continuous function with bounded sublevel sets {ω ∈ Ω : Z(ω) ≤ z}, z ∈ R+ . Let BZ ⊆ B be the subspace of positions X ∈ B such that X/Z is bounded, UZ the set of all upper semicontinuous X ∈ BZ and CZ the space of all continuous X ∈ BZ . By PZ we denote the set of all Borel probability measures on Ω satisfying the integrability condition EP Z < +∞. Define the convex conjugate φ∗ : PZ → R ∪ {±∞} by φ∗ (P) := sup (EP X − φ(X)). X∈CZ

Then the following holds: Theorem 2.1. Assume for every n ∈ N, there exists a z ∈ R+ such that n(Z − z)+ − Then the following three conditions are equivalent:

3

1 ∈ G − A. n

(2.1)

(i) R+ ∩ (G − A) = {0} (ii) there exists a probability measure P ∈ PZ such that EP X ≤ 0 for all X ∈ CZ ∩ (G − A) (iii) φ is real-valued on BZ with φ(0) = 0 and φ(X) = maxP∈PZ (EP X − φ∗ (P)) for all X ∈ CZ . If in addition to (2.1), one has φ(X) =

inf

Y ∈CZ ,Y ≥X

φ(Y )

for all X ∈ UZ ,

(2.2)

then (i)–(iii) are also equivalent to each of the following three: (iv) there exists no X ∈ G − A such that X(ω) > 0 for all ω ∈ Ω (v) there exists a probability measure P ∈ PZ such that EP X ≤ 0 for all X ∈ UZ ∩ (G − A) (vi) φ is real-valued on BZ with φ(0) = 0 and φ(X) = maxP∈PZ (EP X − φ∗ (P)) for all X ∈ UZ . (iii) and (vi) yield dual representations for the superhedging functional φ. They directly translate into dual representations for the subhedging functional −φ(−X). If condition (iii) holds, one has −φ(−X) = min (EP X + φ∗ (P))

for all X ∈ CZ ,

P∈PZ

and the representation extends to all X ∈ UZ if (vi) is satisfied. Moreover, note that as soon as φ is real-valued on BZ with φ(0) = 0, the same is true for the subhedging functional, and one obtains by convexity, φ(X) + φ(−X) ≥ 2φ(0) = 0, yielding the ordering φ(X) ≥ −φ(−X)

for all X ∈ BZ .

(i) and (iv) are no-arbitrage conditions, or in the case where the acceptance set A is larger than B + , no-good deal conditions. (i) means that there exists no trading strategy starting with zero initial capital generating an outcome that exceeds an acceptable position by a positive constant, or equivalently, that it is not possible to turn a debt into an acceptable position by investing in the market. The same condition was used by [29] and [36] in the standard setup. (iv) is stronger. In the case where A equals B + , it corresponds to absence of model-independent arbitrage as introduced by [17] and used e.g., in [1]. (ii) and (v) generalize the concept of a martingale measure. For instance, if A = B + and the underlying assets are liquidly traded, they consist of proper martingale measures. But in the presence of proportional transaction costs, they become ε-approximate martingale measures, and under short-selling constraints, supermartingale measures (see the examples in Section 3). A wide class of acceptance sets can be written as n o A = X ∈ BZ : EP X + α(P) ≥ 0 for all P ∈ PZ + B + (2.3) for a suitable mapping α : PZ → R+ ∪ {+∞}. In the extreme case α ≡ 0, A is the positive cone B + . On the other hand, it can be shown that if α grows fast enough, assumption (2.1) of Theorem 2.1 is automatically satisfied: Proposition 2.2. Condition (2.1) holds if A is given by (2.3) for a mapping α : PZ → R+ ∪ {+∞} satisfying

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(A1) inf P∈PZ α(P) = 0 and (A2) α(P) ≥ EP β(Z) for all P ∈ PZ , where β : [1, +∞) → R is an increasing1 function with the property limx→+∞ β(x)/x = +∞. The following result gives a dual condition for assumption (2.2) which will be useful in Sections 3 and 4. Proposition 2.3. Assume (2.1) holds. Then φ∗ (P) =

EP X ≤ sup (EP X − φ(X)) =

sup

X∈UZ

X∈CZ ∩(G−A)

sup

EP X

for all P ∈ PZ ,

X∈UZ ∩(G−A)

and the inequality is an equality if and only if φ satisfies (2.2).

3. Strict superhedging In this section we concentrate on the case where the acceptance set A is the positive cone B + . Ω 1 J is assumed to be a non-empty closed subset of RJT + , and the price processes S , . . . , S are given by S0j = sj0 ∈ R and Stj (ω) = ωtj , t ≥ 1, ω ∈ Ω. They generate the filtration Ft = σ(Ssj : j = P 1, . . . , J, s ≤ t), t = 0, . . . , T . As growth function we choose Z = 1 + j,t Stj . Then BZ is the space of all measurable functions X : Ω → R of linear growth, and for any set G ⊆ B of trading gains containing 0 such that G − B + is convex, condition (2.1) is equivalent to for every n ∈ N, j = 1, . . . , J, and t = 1, . . . , T , there exist X ∈ G and K ∈ R+ such that X ≥ n(Stj − K)+ − 1/n. Obviously, (3.1) holds automatically if Ω is compact. Let us define φ∗G (P) := sup EP X,

(3.1)

P ∈ PZ ,

X∈G

with the understanding that  P

E X :=

limm→+∞ EP (X ∧ m) −∞

if EP X − < +∞ otherwise,

and introduce the corresponding set of generalized martingale measures n o M := {P ∈ PZ : φ∗G (P) = 0} = P ∈ PZ : EP X ≤ 0 for all X ∈ G .

3.1. Semi-static hedging with convex transaction costs and constraints Let us first assume that the assets S 1 , . . . , S J can be traded dynamically according to any Jdimensional predictable strategy (ϑt )Tt=1 , and in addition, it is possible to form a static portfolio of derivatives with payoffs Hi ∈ CZ for i in an index set I. We denote by RI0 the set of vectors in RI with at most finitely many components different from 0 and suppose that the static part of the strategy θ is constrained to lie in a given convex subset Θ ⊆ RI0 containing 0. We model 1

We call a function f from a subset I ⊆ R to R increasing if f (x) ≥ f (y) for x ≥ y.

5

transactions costs in the derivatives market with a convex mapping h : Θ → CZ satisfying h(0) = 0. Transaction costs arising from dynamic trading are, as in [2, 11, 20], given by continuous functions gtj : Ω × R → R, j = 1, . . . , J, t = 0, . . . , T − 1, such that gtj (ω, x) is Ft -measurable in ω and convex in x with gtj (ω, 0) = 0. In the case where Ω ⊆ RJT is unbounded, we also assume that supx∈E |gtj (ω, x)|/Z(ω) is bounded in ω for every bounded subset E ⊆ R. As usual, we suppress the ω-dependence of gtj in the notation. Then the resulting set of trading gains G consists of outcomes of the form T X J   X X j j ϑjt ∆Stj − gt−1 (∆ϑjt St−1 ) + θi Hi − h(θ), t=1 j=1

i∈I

j where ∆Stj = Stj − St−1 and ∆ϑjt = ϑjt − ϑjt−1 with ϑj0 = 0. Under these circumstances we obtain the following version of Theorem 2.1:

Proposition 3.1. If (3.1) holds, the following four conditions are equivalent: (i) there exist no X ∈ G and ε ∈ R+ \ {0} such that X ≥ ε (ii) there exists no X ∈ G such that X(ω) > 0 for all ω ∈ Ω (iii) M = 6 ∅ (iv) φ is real-valued on BZ with φ(0) = 0 and φ(X) = maxP∈PZ (EP X − φ∗G (P)) for all X ∈ UZ . Moreover, φ∗G (P) =

  PT −1 PJ t=0



R j∗ j=1 {Stj >0} gt



EP [STj |Ft ]−Stj Stj



dP + h∗ (P)

if P ∈ R

(3.2)

if P ∈ / R,

+∞

for ! gtj∗ (y) and

:= sup(xy − x∈R

gtj (x)),

∗

P

h (P) := sup E θ∈Θ

X

θi Hi − h(θ) ,

i∈I

n o R := P ∈ PZ : P[Stj = 0 and STj > 0] = 0 for all j = 1, . . . , J and t ≤ T − 1 .

Proposition 3.1 extends results of [1, 2, 22]. [1] and [22] consider proportional transaction costs and make an assumption similar to condition (3.1). [2] treat the case of general convex transaction costs in a setup with a bounded sample space Ω. 3.1.1. Proportional transaction costs As a special case, let usPconsider proportional transaction costs and no trading constraints; that j + − − − + is, Θ = RI0 and h(θ) = i∈I h+ i θi − hi θi for bid and ask prices hi , hi ∈ R. Moreover, gt is of j j j the form gt (x) = εt |x| for Ft -measurable random coefficients εt ∈ CZ+ . The corresponding trading outcomes are of the form T X J  X

 X
j − − ϑjt ∆Stj − εjt−1 |∆ϑjt St−1 | + θi Hi − θi+ h+ i + θ i hi ,

t=1 j=1

i∈I

6

and gtj∗ (y)

 =

0 +∞

if |y| ≤ εjt otherwise,



∗

h (P) =

0 +∞

+ P if h− i ≤ E Hi ≤ hi for all i ∈ I otherwise.

By Proposition 3.1, one has φ∗G (P)

 =

if P ∈ M otherwise,

0 +∞

where M is the set of all probability measures P ∈ PZ satisfying the conditions a) (1 − εjt )Stj ≤ EP [STj | Ft ] ≤ (1 + εjt )Stj for all j = 1, . . . , J and t = 0, . . . , T − 1 + P b) h− i ≤ E Hi ≤ hi for all i ∈ I.

So if (3.1) and (i) of Proposition 3.1 hold, one obtains the duality φ(X) = max EP X

for all X ∈ UZ .

(3.3)

P∈M

If εjt ≡ 0, dynamic trading is frictionless, and a) reduces to the standard martingale condition. In this case, (3.3) is the superhedging duality derived in [1]. On the other hand, in the case where the coefficients εjt are constant and (Hi )i∈I consists of European options depending continuously on ST , (3.3) becomes the superhedging duality of [22]. 3.1.2. Superlinear transaction costs For Θ = RI0 and transaction costs corresponding to gtj (x)

εjt pj = |x| , pj

h(θ) =

X i∈I

hi θi +

δi qi |θi | qi

for positive Ft -measurable εjt ∈ CZ and constants δi > 0, pj , qj > 1, hi ∈ R, one obtains from Proposition 3.1, 0

φ∗G (P)

=

T X J X (εjt−1 )1−pj t=1 j=1

p0j



P j P E [ST

E

0 0 j pj X δ 1−qi | Ft−1 ] − St−1 i + j qi0 St−1

q 0 P i E H − h i i ,

i∈I

where p0j := pj /(pj − 1), qi0 := qi /(qi − 1), 0/0 := 0 and x/0 := +∞ for x > 0. Moreover, if (3.1) and condition (i) of Proposition 3.1 hold, one has φ(X) = max (EP X − φ∗G (P))

for all X ∈ UZ .

P∈PZ

3.1.3. European call options and constraints on the marginal distributions Let gtj : Ω × R → R be as in the beginning of Subsection 3.1 above and assume the family (Hi )i∈I consists of all European call options (Stj − K)+ ,

j = 1, . . . J, t = 1, . . . , T, K ∈ R+ .

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Moreover suppose that arbitrary quantities of (Stj − K)+ can be sold or bought at prices pj,− t,K and j,+ pj,+ t,K , respectively. If limK→+∞ pt,K = 0 for all j and t, condition (3.1) holds. So if in addition, (i) of Proposition 3.1 is satisfied, one obtains  !  T −1 X J Z j j P X E [ST | Ft ] − St dP for all X ∈ UZ , (3.4) φ(X) = max EP X − gtj∗ j j P S S >0 { } t t t=0 j=1

where the maximum is over all P ∈ PZ such that a) P[Stj and STj > 0] = 0 for all j = 1, . . . , J and t = 0, . . . , T − 1 j j,+ P + b) pj,− t,K ≤ E (St − K) ≤ pt,K for all j = 1, . . . , J, t = 1, . . . T and K ∈ R+ . R +∞ EP (Stj − K)+ can be written as K P[Stj > x]dx. So condition b) puts constraints on the distrij,+ butions of Stj under P, and in the limiting case pj,− t,K = pt,K , it fully determines the distributions j,+ j of Stj under P. In particular, if gtj ≡ 0 and pj,− t,K = pt,K = pt,K ∈ R+ for all j, t, K, it follows from

(3.1) (i) of Proposition 3.1 that there exist unique marginal distributions νtj on R+ specified by R +∞ and νtj (x, +∞)dx = pjt,K , K ∈ R+ , such that K φ(X) = max EP X

for all X ∈ UZ ,

(3.5)

P∈M

where M consists of all P ∈ PZ satisfying a) S 1 , . . . , S J are martingales under P b) the distribution of Stj under P is νtj for all j = 1, . . . , J and t = 1, . . . , T . (3.5) is a variant of Kantorovich’s transport duality [34] and has recently been studied in different setups under the name martingale transport duality; see e.g., [3, 4, 28].

3.2. Semi-static hedging with short-selling constraints Now assume that dynamic trading does not incur transaction costs, but only non-negative quantities of the assets S 1 , . . . , S J can be held. As above, one can invest statically in derivatives with payoffs Hi ∈ CZ , i ∈ I, according to a strategy θ lying in a convex subset Θ ⊆ RI0 containing 0. Let h : Θ → CZ be a convex mapping with h(0) = 0 and suppose that G consists of outcomes of the form T X J X X ϑjt ∆Stj + θi Hi − h(θ), t=1 j=1

i∈I

(ϑt )Tt=1

where is a non-negative J-dimensional predictable strategy and θ ∈ Θ. Then the following variant of Proposition 3.1 holds: Proposition 3.2. If (3.1) is satisfied, the conditions (i)–(iv) of Proposition 3.1 are equivalent, where  P supθ∈Θ EP ( i∈I θi Hi − h(θ)) if S 1 , . . . , S J are supermartingales under P ∗ φG (P) = +∞ otherwise, and M = {P ∈ PZ : φ∗G (P) = 0}.

8

3.2.1. Dynamic and static short-selling constraints If there are short-selling constraintsPon the dynamic as well as static part of the trading strategy, that is, Θ = RI0 ∩ RI+ , and h(θ) = i∈I hi θi for prices hi ∈ R, it follows by Proposition 3.2 from (3.1) and condition (i) of Proposition 3.1 that φ(X) = maxP∈M EP X for all X ∈ UZ , where M is the set of all measures P ∈ PZ such that a) S 1 , . . . , S J are supermartingales under P b) EP Hi ≤ hi for all i ∈ I. 3.2.2. Supermartingale transport duality Suppose now that only the dynamic part of the trading strategy is subject to short-selling constraints and (Hi )i∈I consists of all call options (Stj − K)+ , j = 1, . . . , J, t = 1, . . . , T , K ∈ R+ . If arbitrary j,+ quantities of (Stj − K)+ can be sold and bought at prices pj,− t,K ≤ pt,K , respectively, condition (3.1) is satisfied provided that limK→+∞ pj,+ t,K = 0 for all j and t. So, if in addition, (i) of Proposition 3.1 holds, one obtains from Proposition 3.2 that φ(X) = max EP X

for all X ∈ UZ

(3.6)

P∈M

where M consists of the measures P ∈ PZ satisfying a) S 1 , . . . , S J are supermartingales under P R +∞ j P + P[Stj > x]dx ≤ pj,+ b) pj,− t,K for all j = 1, . . . , J, t = 1, . . . , T and K ∈ R+ . t,K ≤ E (St − K) = K j j,+ In the special case pj,− t,K = pt,K = pt,K ∈ R+ , condition b) is satisfied if and only if for all j and R +∞ t, Stj under P is distributed according to a measure νtj satisfying K νtj (x, +∞)dx = pjt,K for all K ∈ R+ . In this case, (3.6) becomes a supermartingale version of Kantorovich’s transport duality [34].

4. Superhedging with respect to risk measures In this section we relax the strict superhedging requirement and consider acceptance sets of the form \ A= {X ∈ BZ : ρQ (X) ≤ 0} + B + , (4.1) Q∈Q

where Q ⊆ PZ is a non-empty set of probabilistic models, and for every Q ∈ Q, ρQ : BZ → R is a convex risk measure. More specifically, we concentrate on transformed loss risk measures:   ρQ (X) = min EQ lQ (s − X) − s s∈R

for loss functions lQ : R → R. Up to a minus sign they coincide with the optimized certainty equivalents of Ben-Tal and Teboulle [5]. For unbounded random variables, they were studied in Section 5 of [15]. We make the following assumptions:

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(l1) every lQ is increasing2 and convex with limx→±∞ (lQ (x) − x) = +∞ (l2) supQ EQ lQ (ϕ(Z)) < +∞ for an increasing function ϕ : [1, +∞) → R satisfying limx→+∞ ϕ(x)/x = +∞ ∗ (1) = 0 for all Q ∈ Q, where l∗ (y) = sup (l3) lQ x∈R (xy − lQ (x)), y ∈ R. Q

Then the following holds: Lemma 4.1. For every Q ∈ Q, ρQ is a real-valued convex risk measure on BZ with ρQ (0) = 0 and dual representation     dP P Q ∗ E [−X] − E lQ ρQ (X) = max . (4.2) P∈PZ , PQ dQ Moreover, the acceptance set A given in (4.1) is of the form (2.3) for a mapping α : PZ → R+ ∪ {+∞} satisfying (A1) and (A2).

4.1. Robust average value at risk j Now assume that Ω is a closed subset of RJT + and consider the filtration Ft := σ(Ss : j = 1, . . . , J, s ≤ P j j j t) generated by St (ω) = ωt . We also suppose Z ≥ 1 + j,t St . This ensures that Stj belongs to CZ for all j, t. For a fixed level 0 < λ ≤ 1 and Q from a given non-empty subset Q ⊆ PZ , consider the average value at risk Z 1 λ (X) := AVaRQ VaRQ X ∈ BZ . u (X)du, λ λ 0 It is well-known (see e.g. [27]) that it can be written as  Q  E (s − X)+ Q AVaRλ (X) = min − s , X ∈ BZ , s∈R λ

and has a dual representation of the form AVaRQ λ (X) =

max PQ, dP/dQ≤1/λ

EP [−X],

X ∈ BZ .

The corresponding robust acceptance set is o \ n (X) ≤ 0 + B+ A= X ∈ BZ : AVaRQ λ Q∈Q

Let us assume that the assets S 1 , . . . , S J can be traded dynamically with proportional transaction costs given by ε1 , . . . , εJ ≥ 0, and there exists a family of derivatives with payoffs (Hi )i∈I ⊆ CZ , + which can be traded statically with bid and ask prices h− i , hi ∈ R. The resulting set of trading gains G consists of outcomes of the form T X J X

j (ϑjt ∆Stj − εj St−1 |∆ϑjt |) +

t=1 j=1

X

− − (θi Hi − θi+ h+ i + θi hi ),

(4.3)

i∈I

where (ϑt ) is a J-dimensional (Ft )-predictable strategy and θ ∈ RI0 . Under these conditions, one has the following: 2

i.e., lQ (x) ≥ lQ (y) for x ≥ y

10

Proposition 4.2. If Q is convex, σ(PZ , CZ )-closed and satisfies sup EQ ϕ(Z) < +∞ for an increasing function ϕ : [1, +∞) → R such that Q∈Q

lim

x→+∞

ϕ(x) = +∞, x (4.4)

the following are equivalent: (i) R+ ∩ (G − A) = {0} (ii) there exists no X ∈ G − A such that X(ω) > 0 for all ω ∈ Ω (iii) M = 6 ∅ (iv) φ is real-valued on BZ and φ(X) = maxP∈M EP X for all X ∈ UZ , where M is the set of all probability measures P ∈ PZ satisfying j j a) (1 − εj )St−1 ≤ EP [STj | Ft−1 ] ≤ (1 + εj )St−1 for all j = 1, . . . , J and t = 1, . . . , T + P b) h− i ≤ E Hi ≤ hi for all i ∈ I

c) dP/dQ ≤ 1/λ for some Q ∈ Q. P Examples 4.3. If Z = 1 + t,j Stj , the integrability condition (4.4) is satisfied by the following four families of probability measures: 1. All Q ∈ PZ such that cjt ≤ EQ (Stj )2 ≤ Ctj for given constants 0 ≤ cjt ≤ Ctj . j − 1)2 | Ft−1 ] ≤ Ctj for given constants 0 ≤ cjt ≤ Ctj . 2. All Q ∈ PZ such that cjt ≤ EQ [(Stj /St−1 j ), j = 1, . . . , J, t = 1, . . . , T , forms a Gaussian family 3. All Q ∈ PZ under which Ytj = log(Stj /St−1 j Q with mean vector (E Yt ) in a bounded set M ⊆ RJT and covariance matrix CovQ (Ytj , Ysk ) in a bounded set Σ ⊆ RJT ×JT . 4. The σ(PZ , CZ )-closed convex hull of any set Q ⊆ PZ satisfying (4.4). It can easily be checked that the first two families are convex and σ(PZ , CZ )-closed. But the third one is in general not convex. So to satisfy the assumptions of Proposition 4.2, one has to pass to the σ(PZ , CZ )-closed convex hull.

4.2. Robust entropic risk measure j As above, suppose that Ω is a closed subset of RJT + , consider the filtration (Ft ) generated by (St ), P j = 1, . . . , J, and assume Z ≥ 1 + j,t Stj . For a fixed risk aversion parameter λ > 0 and Q in a given non-empty set Q ⊆ P, consider the entropic risk measure 1 Q EntQ X ∈ BZ . λ (X) = λ log E exp(−λX), It admits the alternative representations      exp(λs − 1 − λX) 1 Q dP dP P EntQ (X) = min − s = max E [−X] − E log , X ∈ BZ ; λ s∈R PQ λ λ dQ dQ

11

see e.g. [15]. The resulting robust acceptance set is o \ n + A= X ∈ BZ : EntQ λ (X) ≤ 0 + B . Q∈Q

If the set of trading gains G is the same as in (4.3), one obtains the following variant of Proposition 4.2: Proposition 4.4. If Q is convex, σ(PZ , CZ )-closed and satisfies sup EQ exp(ϕ(Z)) < +∞ for an increasing function ϕ : [1, +∞) → R with Q∈Q

ϕ(x) = +∞, x→+∞ x (4.5) lim

the following are equivalent: (i) R+ ∩ (G − A) = {0} (ii) there exists no X ∈ G − A such that X(ω) > 0 for all ω ∈ Ω (iii) there exists a P ∈ PZ such that EP X ≤ 0 for all X ∈ UZ ∩ (G − A) (iv) φ is real-valued on BZ with φ(0) = 0 and φ(X) = maxP∈PZ (EP X − η(P)) for all X ∈ UZ , where η : PZ → R+ ∪ {+∞} is given by   ( dP dP log dQ /λ inf Q∈Q, PQ EQ dQ η(P) = +∞

if P satisfies a)–b) and P  Q for some Q ∈ Q otherwise,

and a)–b) are the same conditions as in Proposition 4.2. P Examples 4.5. For Z = 1 + j,t Stj , the following are convex σ(PZ , CZ )-closed subsets of PZ satisfying (4.5): 1. All Q ∈ PZ such that cjt ≤ EQ (Stj )2 ≤ Ctj and EQ exp(εjt (STj )2 ) ≤ Dtj for given constants 0 ≤ cjt ≤ Ctj and εjt , Dtj > 0. j − 1)2 | Ft−1 ] ≤ Ctj and EQ exp(εjt (STj )2 ) ≤ Dtj for given 2. All Q ∈ PZ such that cjt ≤ EQ [(Stj /St−1 constants 0 ≤ cjt ≤ Ctj and εjt , Dtj > 0. 3. The σ(PZ , CZ )-closed convex hull of any set Q ⊆ PZ satisfying (4.5). Note that Example 4.3.3 also satisfies condition (4.5). Therefore, its σ(PZ , CZ )-closed convex hull fulfills the assumptions of Proposition 4.4.

A. Proofs A.1. Representation of increasing convex functionals In preparation for the proof of Theorem 2.1 we first derive representation results for general increasing convex functionals on CZ and UZ . For a discussion of related representation results based on the Daniell–Stone theorem we refer to [14] and the references therein. As in Section 2, Ω is a non-empty closed subset of RJT and Z : Ω → [1, +∞) a continuous function such that {ω ∈ Ω : Z(ω) ≤ z} is bounded for all z ∈ R+ . If (Xn ) is a sequence of functions Xn : Ω → R decreasing pointwise to a

12

function X : Ω → R, we write Xn ↓ X. The space CZ of continuous functions X : Ω → R such that X/Z is bounded is a Stone vector lattice; that is, it is a linear space with the property that for all X, Y ∈ CZ , the point-wise minima X R∧ Y and X ∧ 1 also belong to CZ . Let ca+ Z be the set of all Borel measures µ satisfying hZ, µi := Zdµ < +∞. We call a functional ψ : CZ → R increasing if ∗ : ca+ → R ∪ {+∞} by ψ(X) ≥ ψ(Y ) for X ≥ Y and define the convex conjugate ψC Z Z ∗ ψC (µ) := sup (hX, µi − ψ(X)). Z X∈CZ

Theorem A.1. Let ψ : CZ → R be an increasing convex functional with the property that for every X ∈ CZ there exists a constant ε > 0 such that lim ψ(X + ε(Z − z)+ ) = ψ(X).

(A.1)

z→+∞

Then ∗ ψ(X) = max (hX, µi − ψC (µ)) Z

for all X ∈ CZ .

µ∈ca+ Z

(A.2)

∗ that Proof. Fix X ∈ CZ . It is immediate from the definition of ψC Z ∗ ψ(X) ≥ sup (hX, µi − ψC (µ)). Z

(A.3)

µ∈ca+ Z

On the other hand, it follows from the Hahn–Banach extension theorem that there exists a positive linear functional ζX on CZ such that ζX (Y ) ≤ ψX (Y ) := ψ(X + Y ) − ψ(X)

for all Y ∈ CZ .

Let (Xn ) be a sequence in CZ satisfying Xn ↓ 0. If we can show that there exists a constant η > 0 such that ψX (ηXn ) ↓ 0, then ζX (Xn ) ↓ 0, and we obtain from the Daniell–Stone theorem that ζX ∗ is of the form ζX (Y ) = hY, µX i for a measure µX ∈ ca+ Z . As a result, one obtains ψCZ (µX ) = hX, µX i − ψ(X) and the representation (A.2) follows from (A.3). Choose ε > 0 such that (A.1) holds and m > 0 so that X1 ≤ mZ. Set η = ε/(4m) and fix δ > 0. Let z ∈ R+ such that ψX (ε(Z − z)+ ) ≤ δ. By assumption, the set Λ = {Z ≤ 2z} is compact. Therefore, one obtains from Dini’s lemma that xn := max Xn (ω) ↓ 0. ω∈Λ

Since x 7→ ψX (x) is a convex function from R to R, it is continuous. In particular, there exists an n0 such that ψX (2ηxn ) ≤ δ for all n ≥ n0 . Moreover, it follows from Xn ≤ Xn 1{Z≤2z} + X1 1{Z>2z} ≤ xn 1{Z≤2z} + mZ1{Z>2z} ≤ xn + 2m(Z − z)+ , that

Xn − xn ≤ (Z − z)+ , 2m

and therefore,  ψX (2η(Xn − xn )) = ψX

Xn − xn ε 2m

 ≤δ

for all n.

This gives ψX (2ηxn ) + ψX (2η(Xn − xn )) ≤δ 2 Hence, ψX (ηXn ) ↓ 0, and the proof is complete. ψX (ηXn ) ≤

13

for all n ≥ n0 .

To extend the representation (A.2) beyond CZ , we need a slightly stronger condition than (A.1). Lemma A.2. An increasing convex functional ψ : CZ → R satisfies condition (A.1) if lim ψ(n(Z − z)+ ) = ψ(0)

z→+∞

for every n ∈ N.

(A.4)

Proof. If (A.4) holds, one has for any X ∈ CZ , ε ∈ R+ and λ ∈ (0, 1),     (Z − z)+ X + (1 − λ)ψ ε ψ(X + ε(Z − z)+ ) ≤ λψ λ 1−λ as well as

  (Z − z)+ ψ ε → ψ(0) 1−λ

for z → +∞.

Moreover, since z 7→ ψ(zX) is a real-valued convex function on R, it is continuous. In particular,   X → ψ(X) and (1 − λ)ψ(0) → 0 for λ → 1. λψ λ This shows that ψ(X + ε(Z − z)+ ) → ψ(X) for z → +∞. We also need the following Lemma A.3. For every X ∈ UZ there exists a sequence (Xn ) in CZ such that Xn ↓ X. Proof. For X ∈ UZ , define 

 0) X j X(ω ˜ n (ω) := sup  X −n |ωt − ωt0j | . Z(ω 0 ) ω 0 ∈Ω j,t

˜ n is a sequence in CZ such that Xn ↓ X. Then Xn = Z X The next results gives conditions under which a representation of the form (A.2) holds on the set UZ of all upper semicontinuous functions X : Ω → R such that X/Z is bounded. For µ ∈ ca+ Z , we define ψU∗ Z (µ) := sup (hX, µi − φ(X)). X∈UZ

Theorem A.4. Let ψ : UZ → R be an increasing convex functional satisfying condition (A.4). Then the following are equivalent: ∗ (µ)) for all X ∈ U (i) ψ(X) = maxµ∈ca+ (hX, µi − ψC Z Z Z

(ii) ψ(Xn ) ↓ ψ(X) for all X ∈ UZ and every sequence (Xn ) in CZ such that Xn ↓ X (iii) ψ(X) = inf Y ∈CZ , Y ≥X ψ(Y ) for all X ∈ UZ ∗ (µ) = ψ ∗ (µ) for all µ ∈ ca+ . (iv) ψC UZ Z Z

14

Proof. Since, by Lemma A.2, (A.4) implies (A.1), we obtain from Theorem A.1 that
∗ ψ(X) = max hX, µi − ψC (µ) for all X ∈ CZ . Z µ∈ca+ Z

+ ∗ Moreover, by Lemma A.5 below, the sublevel sets {µ ∈ ca+ Z : ψCZ (µ) ≤ a}, a ∈ R, are σ(caZ , CZ )compact, and there exists a function ϕ : R+ → R ∪ {+∞} such that limx→+∞ ϕ(x)/x = +∞ and ∗ (µ) ≥ ϕ(hZ, µi) for all µ ∈ ca+ . ψC Z Z (i) ⇒ (ii) is now a consequence of part (iii) of Lemma A.6 below, and (ii) ⇒ (iii) follows since by Lemma A.3, there exists for every X ∈ UZ a sequence (Xn ) in CZ such that Xn ↓ X. ∗ . On the other hand, if for every X ∈ U , there is a (iii) ⇒ (iv): One obviously has ψU∗ Z ≥ ψC Z Z sequence (Xn ) in CZ such that Xn ≥ X and ψ(Xn ) ↓ ψ(X), then

sup(hXn , µi − ψ(Xn )) ≥ hX, µi − ψ(X), n

∗ ≥ ψ∗ . from which one obtains ψC UZ Z (iv) ⇒ (i): For given X ∈ UZ , one obtains from the definition of ψU∗ Z that ∗ ψ(X) ≥ sup (hX, µi − ψU∗ Z (µ)) = sup (hX, µi − ψC (µ)). Z µ∈ca+ Z

µ∈ca+ Z

Conversely, we know from Lemma A.3 that there exists a sequence (Xn ) in CZ such that Xn ↓ X. So we can conclude by (iii) of Lemma A.6 below that ∗ ψ(X) ≤ inf ψ(Xn ) = max (hX, µi − ψC (µ)). Z n

µ∈ca+ Z

Lemma A.5. For every increasing convex functional ψ : CZ → R the following hold: (i) There exists an increasing convex function ϕ : R+ → R ∪ {+∞} satisfying limx→+∞ ϕ(x)/x = ∗ (µ) ≥ ϕ(hZ, µi) for all µ ∈ ca+ . +∞ such that ψC Z Z + ∗ (ii) If ψ satisfies (A.4), the sublevel sets {µ ∈ ca+ Z : ψCZ (µ) ≤ a}, a ∈ R, are σ(caZ , CZ )-compact.

Proof. For every µ ∈ ca+ Z , one has ∗ ψC (µ) ≥ sup (hyZ, µi − ψ(yZ)) = ϕ(hZ, µi) Z y∈R+

for the increasing convex function ϕ : R+ → R ∪ {+∞} given by ϕ(x) := sup (xy − ψ(yZ)). y∈R+

It follows from the fact that ψ is real-valued that limx→+∞ ϕ(x)/x = +∞. This shows (i). + ∗ ∗ Since ψC is σ(ca+ Z , CZ )-lower semicontinuous, the sets Λa := {µ ∈ caZ : ψCZ (µ) ≤ a} are Z + σ(caZ , CZ )-closed. If (A.4) holds, every µ ∈ Λa satisfies

 ∗ m (Z − z)+ , µ − ψ(m(Z − z)+ ) ≤ ψC (µ) ≤ a for all m, z ∈ R+ . Z

15

So for given m ∈ R+ , there exists a z ∈ R+ such that

 a + ψ(0) + 1 (Z − z)+ , µ ≤ . m

In particular,

 lim sup (Z − z)+ , µ = 0,

z→+∞ µ∈Λa

and, as a result,



 lim sup Z1{Z>2z} , µ ≤ lim sup 2(Z − z)+ , µ = 0.

z→+∞ µ∈Λa

z→+∞ µ∈Λa

(A.5)

From (i) we know that hZ, µi ≤ ϕ−1 (a) < +∞ for all µ ∈ Λa ,

(A.6)

where ϕ−1 : R → R+ is the right-continuous inverse of ϕ given by ϕ−1 (y) := sup {x ∈ R+ : ϕ(x) ≤ y}

with sup ∅ := 0.

The mapping f : X 7→ X/Z identifies CZ with the space of bounded continuous functions Cb , and + g : µ 7→ Zdµ identifies ca+ Z with the set of all finite Borel measures ca . It follows from (A.5) and (A.6) that g(Λa ) is tight. So one obtains from Prokhorov’s theorem that g(Λa ) is σ(ca+ , Cb )compact, which is equivalent to Λa being σ(ca+ Z , CZ )-compact. Lemma A.6. Let α : ca+ Z → R ∪ {+∞} be a mapping such that inf µ∈ca+ α(µ) ∈ R and α(µ) ≥ Z

ϕ(hZ, µi) for all µ ∈ ca+ Z and a function ϕ : R+ → R ∪ {+∞} satisfying limx→+∞ ϕ(x)/x = +∞. Then the following hold: (i) ψ(X) := supµ∈ca+ (hX, µi − α(µ)) defines an increasing convex functional ψ : BZ → R. Z

 + (ii) If all sublevel sets µ ∈ ca+ Z : α(µ) ≤ a , a ∈ R, are relatively σ(caZ , CZ )-compact, then ψ(Xn ) ↓ ψ(X) for every sequence (Xn ) in CZ such that Xn ↓ X for an X ∈ CZ .  + (iii) If all sublevel sets µ ∈ ca+ Z : α(µ) ≤ a , a ∈ R, are σ(caZ , CZ )-compact, then ψ(Xn ) ↓ ψ(X) for every sequence (Xn ) in CZ such that Xn ↓ X for an X ∈ UZ , and ψ(X) = max (hX, µi − α(µ)) µ∈ca+ Z

for all X ∈ UZ .

Proof. For every X ∈ BZ , there exists an m ∈ R such that |X| ≤ mZ. Therefore, ψ(X) ≥ sup (−m hZ, µi − α(µ)) > −∞ µ∈ca+ Z

as well as ψ(X) ≤ sup (m hZ, µi − α(µ)) ≤ sup (m hZ, µi − ϕ(hZ, µi)) < +∞. µ∈ca+ Z

µ∈ca+ Z

That ψ is increasing and convex is clear. So (i) holds.

16

Now, let (Xn ) be a sequence in CZ such that Xn ↓ X for some X ∈ UZ . By replacing ϕ with ϕ(x) ˜ = inf ϕ(y) ∨ inf α(µ), µ∈ca+ Z

y≥x

one can assume that ϕ is increasing. Then the right-continuous inverse ϕ−1 : R → R+ given by ϕ−1 (y) := sup {x ∈ R+ : ϕ(x) ≤ y}

with sup ∅ := 0,

satisfies limy→+∞ ϕ−1 (y)/y = 0 and hZ, µi ≤ ϕ−1 (α(µ)) for all  µ ∈ dom α := ν ∈ ca+ Z : α(ν) < +∞ . Choose m ∈ R+ such that X1 ≤ mZ. Then hXn , µi − α(µ) ≤ m hZ, µi − α(µ) ≤ mϕ−1 (α(µ)) − α(µ) for all n and µ ∈ dom α.  If the sets µ ∈ ca+ Z : α(µ) ≤ a , a ∈ R, are σ(PZ , CZ )-compact, α is σ(PZ , CZ )-lower secicontinu ous, and it follows that for a ∈ R large enough, there exists a sequence (µn ) in µ ∈ ca+ Z : α(µ) ≤ a such that ψ(Xn ) = hXn , µn i − α(µn ) for all n. Since σ(ca+ , Cb ) is metrizable, the same is true for σ(ca+ Z , CZ ). Therefore,  after possibly passing to a subsequence, one can assume that (µn ) converges to a measure µ ∈ µ ∈ ca+ Z : α(µ) ≤ a in σ(ca+ Z , CZ ). Then α(µ) ≤ lim inf α(µn ). n

Moreover, for every ε > 0, there is an that hXn0 , µn i ≤ hXn0 , µi + ε. Then

n0

such that hXn0 , µi ≤ hX, µi + ε. Now choose n ≥ n0 such

hXn , µn i ≤ hXn0 , µn i ≤ hXn0 , µi + ε ≤ hX, µi + 2ε showing that, lim supn hXn , µn i ≤ hX, µi, and therefore, inf ψ(Xn ) ≤ lim sup (hXn , µn i − α(µn )) ≤ hX, µi − α(µ) ≤ ψ(X). n

n

In particular, ψ(Xn ) ↓ ψ(X) = max (hX, µi − α(µ)) . µ∈ca+ Z

Since by Lemma A.3, every function X ∈ UZ can be appromixated from above by a decreasing sequence (Xn ) in CZ , this shows (iii). It remains to prove (ii). To do that we note that if α satisfies the assumption of (ii), the + σ(ca+ Z , CZ )-lower semicontinuous hull α∗ has σ(caZ , CZ )-compact sublevel sets and α∗ (µ) ≥ ϕ∗ (hZ, µi) ∨ inf α(µ) µ∈ca+ Z

for all µ ∈ ca+ Z,

where ϕ∗ is the lower semicontinuous hull of ϕ. Since limx→+∞ ϕ∗ (x)/x = +∞ and ψ(X) = sup (hX, µi − α∗ (µ)) ,

for all X ∈ CZ ,

µ∈ca+ Z

it follows from (iii) that ψ(Xn ) ↓ ψ(X) for every sequence (Xn ) in CZ such that Xn ↓ X for an X ∈ CZ . This shows (ii).

17

A.2. Proof of Theorem 2.1 Having Theorems A.1 and A.4 in hand, we now are ready to prove Theorem 2.1. That (iv) implies (i) is clear, and the implications (iii) ⇒ (ii) ⇒ (i) as well as (vi) ⇒ (v) ⇒ (iv) ⇒ (i) can be shown without assumptions: (iii) ⇒ (ii): It follows from (iii) that min φ∗ (P) = − max −φ∗ (P) = −φ(0) = 0. P∈PZ

P∈PZ

In particular, since φ(X) ≤ 0 for all X ∈ G − A, there exists a P ∈ PZ such that 0 = φ∗ (P) = sup (EP X − φ(X)) ≥ X∈CZ

sup

(EP X − φ(X)) ≥

X∈CZ ∩(G−A)

sup

EP X,

X∈CZ ∩(G−A)

and therefore, EP X ≤ 0 for all X ∈ CZ ∩ (G − A). (ii) ⇒ (i): By our assumptions on A and G, G − A contains 0. On the other hand, if (ii) holds, there exists a P ∈ PZ such that EP m ≤ 0 for all m ∈ R ∩ (G − A), implying that R+ ∩ (G − A) = {0}. (vi) ⇒ (v): If (vi) holds, then φ∗ (P) ≥ supX∈UZ (EP X − φ(X)) for all P ∈ PZ . So (v) follows from (vi) like (ii) from (iii). (v) ⇒ (iv): Assume there exists an X ∈ G − A such that X(ω) > 0 for all ω ∈ Ω and fix a P ∈ PZ . Since P is regular, there exist an ε > 0 and a closed set E ⊆ {X ≥ ε} such that P[E] > 0, or equivalently, EP 1E > 0. But since ε1E belongs to UZ ∩ (G − A), this contradicts (v). Now we assume (2.1) and show (i) ⇒ (iii). Since A − G is convex and contains B + , the mapping φ : BZ → R ∪ {±∞} is increasing and convex. Moreover, it follows from (i) that φ(m) = m for all m ∈ R, from which one obtains φ(X) ∈ R for every bounded X ∈ B. By condition (2.1), there exists for every n ∈ N a z ∈ R+ such that φ(n(Z − z)+ ) ≤ 1/n, and therefore, φ

 n  φ(nz) + φ(n(Z − z)+ ) nz 1 Z ≤ ≤ + . 2 2 2 2n

So one obtains from monotonicity and convexity that φ is real-valued on BZ . In addition, it follows from Lemma A.2 that φ satisfies condition (A.1). Therefore, Theorem A.1 yields φ(X) = maxµ∈ca+ (hX, µi − φ∗ (µ)) for all X ∈ CZ . Since φ(m) = m for all m ∈ R, φ∗ (µ) is +∞ for all Z

P ∗ µ ∈ ca+ Z \ PZ , and one obtains φ(X) = maxP∈PZ (E X − φ (P)) for all X ∈ CZ . Finally, if both conditions, (2.1) and (2.2), hold, one obtains from Theorem A.4 that (iii) implies (vi), and the proof of Theorem 2.1 is complete.

A.3. Proofs of Propositions 2.2 and 2.3 Proof of Proposition 2.2 If the acceptance set A is of the form (2.3) for a mapping α : PZ → R+ ∪{+∞} satisfying (A1)–(A2), it can be written as   A = {X ∈ BZ : ρ(X) ≤ 0} + B + for ρ(X) := sup EP (−X) − α(P) . P∈PZ

By passing to the lower convex hull, one can assume that β is convex. Then, Jensen’s inequality yields α(P) ≥ EP β(Z) ≥ β(EP Z) for all P ∈ PZ .

18

It follows from Prokhorov’s theorem that the sets {P ∈ PZ : EP β(Z) ≤ a}, a ∈ R, are σ(PZ , CZ )compact. As a consequence, the sets {P ∈ PZ : α(P) ≤ a}, a ∈ R, are relatively σ(PZ , CZ )-compact, and one obtains from part (ii) of Lemma A.6 that for every n ∈ N,  
1 1 1 + ρ − n(Z − z) = − + ρ −n(Z − z)+ ↓ − as z → +∞. n n n In particular, 1/n − n(Z − z)+ ∈ A ⊆ A − G for z large enough, showing that condition (2.1) holds. Proof of Proposition 2.3 By our assumptions on G and A, one has φ(0) ≤ 0. If φ(0) = −∞, G−A contains R, which by (2.1), implies G − A = BZ . Then φ ≡ −∞, φ∗ ≡ +∞ and all the statements of Proposition 2.3 become obvious. On the other hand, if φ(0) > −∞, it follows from (2.1) like in the proof of Theorem 2.1, that φ is real-valued on BZ . Then φ∗ (P) ≤ sup (EP X − φ(X))

for all P ∈ PZ ,

X∈UZ

and by Theorem A.4, the inequality is an equality if and only if φ satisfies condition (2.2). Next, note that since X − φ(X) − ε ∈ CZ ∩ (G − A) for all X ∈ CZ and ε > 0, one has φ∗ (P) = sup EP (X − φ(X)) = X∈CZ

EP X,

sup X∈CZ ∩(G−A)

and analogously, sup (EP X − φ(X)) = X∈UZ

EP X.

sup X∈UZ ∩(G−A)

This completes the proof of Proposition 2.3.

A.4. Proofs of Propositions 3.1 and 3.2 Proof of Proposition 3.1 Let us first show (iv) ⇒ (iii) ⇒ (ii) ⇒ (i). If (iv) holds, one has 0 = φ(0) = maxP∈PZ −φ∗G (P), yielding (iii). Moreover, since EP X ≤ 0 for every X ∈ G and P ∈ M, (iii) implies (ii). That (ii) implies (i) is clear. To prove (i) ⇒ (iv), we first note that (3.1) implies (2.1) and (i) is a reformulation of condition (i) of Theorem 2.1 in the case A = B + . So, by Theorem 2.1, it follows from (i) that φ is real-valued on BZ with φ(0) = 0. We know from Proposition 2.3 that φ∗ (P) ≤

sup X∈UZ

∩(G−B + )

EP ≤ sup EP X = φ∗G (P),

P ∈ PZ .

X∈G

So if we can show φ∗ (P) ≥ φ∗G (P),

P ∈ PZ ,

(A.7)

we obtain φ∗ = φ∗G , and by Proposition 2.3, condition (2.2) holds. Then it follows from Theorem 2.1 that (i) implies (iv). To prove (A.7), we observe that hX i X  j j sup EP X = sup EP ϑjt ∆Stj − gt−1 (∆ϑjt St−1 ) + sup θi EP Hi − h(θ) , X∈G

ϑ

θ

t,j

19

i

j and since gt−1 (0) = 0, the first supremum can be taken over strategies ϑ such that hX i j j EP ϑjt ∆Stj − gt−1 (∆ϑjt St−1 ) ≥ 0. t,j

P P j j j j Then EP [ t,j ϑjt ∆Stj − gt−1 (∆ϑjt St−1 )] can be approximated by EP [ t,j ϑ˜jt ∆Stj − gt−1 (∆ϑ˜jt St−1 )] j for continuous Ft−1 -measurable functions ϑ˜t : Ω → R with compact support. Since X j X j j (ϑ˜t ∆Stj ) − gt−1 (∆ϑ˜jt St−1 )) + θi Hi − h(θ) ∈ CZ ∩ G, t,j

i

it follows that φ∗ (P) ≥ supX∈CZ ∩G EP X ≥ φ∗G (P) for all P ∈ PZ . It remains to show that φ∗G is of the form (3.2). To do this we note that T X

ϑjt ∆Stj

=

t=1

T X t X

∆ϑjs ∆Stj

=

T X T X

∆ϑjs ∆Stj

=

s=1 t=s

t=1 s=1

T X

j ∆ϑjt (STj − St−1 ).

t=1

Hence, sup EP X = sup EP X∈G

ϑ

hX

i X  j j j ) + sup (∆ϑjt St−1 θi EP Hi − h(θ) , ) − gt−1 ∆ϑjt (STj − St−1 θ

t,j

i

where the first supremum can be taken over strategies ϑ such that hX i j j j EP ∆ϑjt (STj − St−1 ) − gt−1 (∆ϑjt St−1 ) ≥ 0. t,j

Now EP EP

i j j j j j j ) can be approximated by (∆ϑ S ) − g − S ∆ϑ (S t t−1 t t−1 t−1 t,j T

hP

hX

i i X h j j j j j j ) (∆ϑ˜jt St−1 ) − gt−1 ) = (∆ϑ˜jt St−1 EP ∆ϑ˜jt (EP [STj | Ft−1 ] − St−1 ) − gt−1 ∆ϑ˜jt (STj − St−1 t,j

t,j

j for bounded Ft−1 -measurable mappings ∆ϑ˜jt with compact support. On {St−1 > 0}, one has   j j j (∆ϑ˜jt St−1 ) ) − gt−1 sup ∆ϑ˜jt (EP [STj | Ft−1 ] − St−1 ∆ϑ˜jt

= sup ∆ϑ˜jt j∗ = gt−1

P j j E [ST ∆ϑ˜jt St−1

j | Ft−1 ] − St−1 j St−1

j EP [STj | Ft−1 ] − St−1 j St−1

! −

j j gt−1 (∆ϑ˜jt St−1 )

! ,

j and on {St−1 = 0},   j j j sup ∆ϑ˜jt (EP [STj | Ft−1 ] − St−1 ) − gt−1 (∆ϑ˜jt St−1 ) = sup ∆ϑ˜jt EP [STj | Ft−1 ] = +∞1{EP [S j |Ft−1 ]>0} . ∆ϑ˜jt

∆ϑ˜jt

20

T

j j Since P[St−1 = 0 and EP [STj > 0 | Ft−1 ] > 0] > 0 if and only if P[St−1 = 0 and STj > 0] > 0, this proves (3.2).

Proof of Proposition 3.2 It follows as in the proof of Proposition 3.1 that   φ∗ (P) = sup EP X − φ(X) = φ∗G (P)

for all P ∈ PZ ,

X∈UZ

and (i)–(iv) are equivalent. Moreover, φ∗G (P) = sup EP ϑ≥0

hX

!

i

ϑjt ∆Stj + sup EP θ∈Θ

t,j

X

θi Hi − hi

,

i∈I

and since the first supremum can be taken over non-negative predictable strategies (ϑt ) such that hX i EP ϑjt ∆Stj ≥ 0, t,j

it can equivalently be taken over bounded non-negative predictable strategies. Now, it is easy to see that  P supθ EP ( i∈I θi Hi − h(θ)) if S 1 , . . . , S J are supermartingales under P ∗ φG (P) = +∞ otherwise.

A.5. Proofs of Lemma 4.1 and Propositions 4.2 and 4.4 Proof of Lemma 4.1 It follows from (l1)–(l2) that for all X ∈ BZ , s 7→ EQ lQ (s − X) is a real-valued increasing convex function on R such that lim EQ lQ (s − X) − s = +∞. s→±∞

In particular, there exists a minimizing s, and it is easy to see that ρQ is a real-valued decreasing convex functional on BZ with the translation property ρQ (X + m) = ρQ (X) − m, m ∈ R. Moreover, EQ lQ (cZ) < +∞ for all c ∈ R+ . So BZ is contained in the Orlicz heart M lQ corresponding to Q and the Young function lQ (.) − lQ (0). By Theorem 4.6 and the computation in Section 5.4 of [15],     dP P Q ∗ ρQ (X) = max E [−X] − E lQ , X ∈ BZ , P dQ where the maximum is over all P  Q such that dP/dQ is in the norm-dual of M lQ . For all these P, one has EP Z = EQ [ZdP/dQ] < +∞, yielding that P ∈ PZ . On the other hand, since ∗ (y) for all x, y ∈ R, one has for every X ∈ B , s ∈ R and P  Q, lQ (x) ≥ xy − lQ Z  Q

Q

E lQ (s − X) − s ≥ E

       dP dP dP Q ∗ P Q ∗ − E lQ − s = E [−X] − E lQ . (s − X) dQ dQ dQ

21

This proves the duality (4.2). By Jensen’s inequality and (l3), ∗ ∗ Q EQ lQ (dP/dQ) ≥ lQ E [dP/dQ] = 0, ∗ (dP/dQ) = EQ l∗ (dQ/dQ) = 0, showing that ρ (0) = 0. Moreover, and therefore, minP∈PZ EQ lQ Q Q since \ {X ∈ BZ : ρQ (X) ≤ 0} = {X ∈ BZ : sup ρQ (X) ≤ 0}, Q∈Q

Q∈Q

the acceptance set A can be written as n o A = X ∈ BZ : EP X + α(P) ≥ 0 for all P ∈ PZ + B + , where α(P) := inf Q∈Q αQ (P) with  αQ (P) :=

∗ (dP/dQ) EQ lQ +∞

if P  Q otherwise.

It follows from minP∈PZ αQ (P) = 0 for all Q ∈ Q that minP∈PZ α(P) = 0. Hence (A1) holds. ∗ (y) ≥ xy − l (x) for all x, y ∈ R, one has for every P ∈ P for which there exists Moreover, since lQ Z Q a Q ∈ Q such that P  Q,     dP dP Q ∗ Q ≥ inf E ϕ(Z) − lQ (ϕ(Z)) ≥ EP ϕ(Z) − sup EQ lQ (ϕ(Z)), α(P) = inf E lQ Q∈Q, PQ Q∈Q, PQ dQ dQ Q∈Q which implies (A2) for β = ϕ − supQ EQ lQ (ϕ(Z)). Proof of Proposition 4.2 + Since AVaRQ λ is a transformed loss risk measure with loss function lQ (x) = x /λ, it follows from the integrability condition (4.4) that the assumptions of Lemma 4.1 are satisfied. So one obtains from Proposition 2.2 that condition (2.1) holds. Moreover, G − A is a convex cone. Therefore, by Proposition 2.3,  0 if EP X ≤ for all X ∈ CZ ∩ (G − A) ∗ P φ (P) = sup E X= (A.8) +∞ otherwise. X∈CZ ∩(G−A) ˆ := {P ∈ PZ : φ∗ (P) = 0} and write M = MG ∩ MA , where MG is the set of all Let us denote M P ∈ PZ satisfying a)–b) and MA the set of all P ∈ PZ satisfying c). Since for X ∈ A, the negative part X − belongs to BZ , one obtains from (A.8), φ∗ (P) ≤

EP X ≤

sup X∈UZ ∩(G−A)

EP X − inf EP X

sup X∈G, EP X>−∞

X∈A

for all P ∈ PZ .

(A.9)

It follows as in the proof of Proposition 3.1 that the second supremum in (A.9) is zero for all P ∈ MG , while it can be seen from the dual representation AVaRQ λ (X) =

sup

EP [−X]

P∈PZ , dP/dQ≤1/λ

ˆ ⊇ M = MG ∩ MA . that the infimum is zero for all P ∈ MA . In particular, M

22

ˆ ⊆ MG . Moreover, On the other hand, it can be shown as in the proof of Proposition 3.1 that M it follows from our assumptions on Q that MA is convex and σ(PZ , CZ )-closed. Indeed, for P1 , P2 ∈ MA and 0 ≤ µ ≤ 1, there exist Y1 , Y2 ∈ B + bounded by 1/λ such that P1 = Y1 ·Q1 and P2 = Y2 ·Q2 . Therefore, 1 (µP1 + (1 − µ)P2 )[E] ≤ (µQ1 + (1 − µ)Q2 )[E] λ for all measurable sets E, and it follows that d(µP1 + (1 − µ)P2 ) 1 ≤ , d(µQ1 + (1 − µ)Q2 ) λ showing that MA is convex. Furthermore, if (Pn ) is a sequence in MA converging to a P ∈ PZ in σ(PZ , CZ ), there exist Qn ∈ Q and Yn ∈ B + bounded by 1/λ such that Pn = Yn · Qn . It follows from condition (4.4) that Q is σ(PZ , CZ )-compact. So by passing to a subsequence, one can assume that Qn converges to a Q in Q with respect to σ(PZ , CZ ). Then EP X = lim EPn X ≤ n

1 1 lim EQn X = EQ X n λ λ

for all X ∈ CZ+ .

Since P and Q are regular, this implies dP/dQ ≤ 1/λ and hence, shows that MA is σ(PZ , CZ )ˆ ∈ PZ \ MA , an X ∈ CZ closed. By a separating hyperplane argument, one obtains for every P ˆ P P ∗ ˆ = supX∈C ∩(G−A) EP X = +∞. So such that E X < inf P∈MA E X = 0, implying X ∈ A and φ (P) Z ˆ ⊆ MA , and as a consequence M ˆ = M. This shows that M  0 if P ∈ M ∗ P φ (P) = sup E X= +∞ otherwise, X∈UZ ∩(G−A) and it follows from Proposition 2.3 that (2.2) holds. As a result, all conditions (i)–(vi) of Theorem 2.1 are equivalent, which implies that the conditions (i)–(iv) of Proposition 4.2 are equivalent. Proof of Proposition 4.4 EntQ λ is a transformed loss risk measure corresponding to the loss function lQ (x) = exp(λx − 1)/λ. Therefore, it follows from condition (4.5) that Lemma 4.1 applies. So we know that condition (2.1) holds. As in the proof of Proposition 4.2, one has φ∗ (P) ≤

EP X ≤

sup X∈UZ ∩(G−A)

EP X − inf EP X

sup

X∈A

X∈G, EP X>−∞

for all P ∈ PZ ,

(A.10)

and supX∈G, EP X>−∞ EP X = 0 for P in the set MG of all measures in PZ satisfying a)–b). Furthermore, since P EntQ λ (X) = sup (E [−X] − ηQ (P)) for all X ∈ BZ , P∈PZ

where

( ηQ (P) =

EQ



dP dQ

log

dP dQ



+∞

/λ

if P  Q otherwise,

one obtains ! inf ηQ (P) ≥ sup

Q∈Q

X∈BZ

P

E [−X] − sup

EntQ λ (X)

Q∈Q

23

≥ sup EP [−X] ≥ φ∗ (P) X∈A

for P ∈ MG .

It follows from the assumptions that there exists a continuous function ϕ˜ : [1, +∞) → R such that ϕ(x) ˜ = +∞ x→+∞ x lim

and

ϕ(x) = +∞. x→+∞ ϕ(x) ˜ lim

By (4.5), Q is σ(PZ˜ , CZ˜ )-compact for Z˜ = exp(ϕ(Z)). ˜ Moreover, exp(X) ∈ CZ˜ for all X ∈ CZ , and EntQ λ (X) =

1 log EQ exp(−λX) λ

is concave and σ(PZ˜ , CZ˜ )-continuous in Q. Therefore, one obtains from a minimax result, such as e.g. the one of Ky Fan [24], that for all P ∈ PZ ,   inf ηQ (P) = inf sup EP [−X] − EntQ (X) λ Q∈Q Q∈Q X∈CZ ! =

sup X∈CZ

EP [−X] − sup EntQ λ (X) Q∈Q

=

sup X∈CZ ∩A

EP [−X] ≤ φ∗ (P).

Since φ∗ (P) = +∞ for P ∈ PZ \ MG , this shows that   inf Q∈Q ηQ (P) if P ∈ MG = φ∗ (P), η(P) = +∞ otherwise which, by (A.10), implies φ∗ (P) = supX∈UZ ∩(G−A) EP X. So Proposition 2.3 gives us that condition (2.2) holds. Now Proposition 4.4 is a consequence of Theorem 2.1.

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