Robust Pricing and Hedging around the Globe
Sebastian Herrmann Florian Stebegg
math.PR math.OC q-fin.MF 60G44, 49N05, 91G20





Robust Pricing and Hedging around the Globe
[The authors thank David Hobson, Sigrid Källblad, Marcel Nutz, and Yavor Stoev for stimulating discussions.]

    
  Sebastian Herrmann
  
  Department of Mathematics, University of Michigan, email
  mailto:sherrma@umich.edusherrma@umich.edu.
  
  Florian Stebegg
  
  Department of Statistics, Columbia University,
  email
  florian.stebegg@columbia.eduflorian.stebegg@columbia.edu.
  

    December 30, 2023
=====================================================================================================================================================================================================================================================================================




We consider the martingale optimal transport duality for càdlàg processes with given initial and terminal laws. Strong duality and existence of dual optimizers (robust semi-static superhedging strategies) are proved for a class of payoffs that includes American, Asian, Bermudan, and European options with intermediate maturity. We exhibit an optimal superhedging strategy for which the static part solves an auxiliary problem and the dynamic part is given explicitly in terms of the static part.





Keywords Robust superhedging; Semi-static strategies; Martingale optimal transport; Duality.



AMS MSC 2010
60G44, 
49N05, 
91G20. 



JEL Classification
G12, 
G23, 
C61. 





§ INTRODUCTION



This paper studies the robust pricing and superhedging of derivative securities with a payoff of the form

    F(X,A)
       = f(∫_[0,T] X_t  A_t).

Here, f is a nonnegative Borel function, X is a càdlàg price process (realized on the Skorokhod space), and A is chosen by the buyer from a given set  of exercise rights. More precisely,  is a set of so-called averaging processes, i.e., nonnegative and nondecreasing adapted càdlàg processes A with A_T ≡ 1. Setting = {_τ, T : τ a [0,T]-valued stopping time} or = { t ↦ t/T } reduces (<ref>) to the relevant special cases of American- or Asian-style derivatives, respectively:

    f(X_τ)
      or  
    f(1/T∫_0^T X_t  t).

Other relevant examples are Bermudan options and European options with intermediate maturity (cf. Examples <ref>–<ref>).



  
Robust pricing problem.
Let μ and ν be probability measures on . We denote by (μ,ν) the set of (continuous-time) martingale couplings between μ and ν, i.e., probability measures P under which X is a martingale with marginal distributions X_0P∼μ and X_TP∼ν. The value of the primal problem

    :=sup_P∈(μ,ν)sup_A∈[P]F(X,A)

can be interpreted as the maximal model-based price for F over all models which are consistent with the given marginals.

If  is a singleton, then (<ref>) is a so-called (continuous-time) martingale optimal transport problem. This problem was introduced (for general payoffs) by Beiglböck, Henry-Labordère, and Penkner <cit.> in a discrete-time setting and by Galichon, Henry-Labordère, and Touzi <cit.> in continuous time; cf. the survey <cit.>.



  
Semi-static superhedging problem.
The formal dual problem to (<ref>) has a natural interpretation as a superhedging problem.[The primal problem (<ref>) can be viewed as an optimization over finite measures P with three constraints: two marginal constraints and the martingale constraint. Its formal dual problem is the Lagrange dual problem where suitable functions φ and ψ and a suitable process H are used as Lagrange multipliers for the marginal and martingale constraints, respectively.] Loosely speaking, a semi-static superhedge is a triplet (φ,ψ,H) consisting of functions φ,ψ and a suitable process H such that for every A ∈, the superhedging inequality holds:

    φ(X_0) + ψ(X_T) + ∫_0^T H^A_t- X_t
    ≥ F(X,A)  pathwise.

Here, the strategy H = H^A may depend in an adapted way on A (cf. Section 3.2 for a precise formulation). For the example of an American-style payoff, this means that at the chosen exercise time τ, the buyer communicates her decision to exercise to the seller, who can then adjust the dynamic part of his hedging strategy (cf. <cit.>). The left-hand side in (<ref>) is the payoff of a static position in two European-style derivatives on X plus the final value of a self-financing dynamic trading strategy in X. The inequality (<ref>) says that the final value of this semi-static portfolio dominates the payoff F for every choice of exercise right and “all” price paths. The initial cost to set up a semi-static  superhedge (φ,ψ,H) equals the price μ(φ)+ν(ψ) of the static part.[We use the common notation μ(φ) for the integral of φ against μ.] The formal dual problem to (<ref>),

    := inf{μ(φ) + ν(ψ) : (φ,ψ,H)  is a semi-static superhedge},

amounts to finding the cheapest semi-static superhedge (if it exists) and its initial cost, the so-called robust superhedging price.



  
Main objectives and relaxation of the dual problem.
We are interested in strong duality, i.e., =, and dual attainment, i.e., the existence of a dual minimizer.

Dual attainment requires a suitable relaxation of the dual problem. Indeed, for the discrete-time martingale optimal transport problem, Beiglböck, Henry-Labordère, and Penkner <cit.> show strong duality for upper semicontinuous payoffs but provide a counterexample that shows that dual attainment can fail even if the payoff function is bounded and continuous. Beiglböck, Nutz, and Touzi <cit.> achieve strong duality and dual attainment for general payoffs and marginals in the one-step case by relaxing the dual problem in two aspects. First, they only require the superhedging inequality to hold in the quasi-sure sense, i.e., outside a set which is a nullset under every one-step martingale coupling between μ and ν. The reason is that the marginal constraints may introduce barriers on the real line which (almost surely) cannot be crossed by any martingale with these marginals; this was first observed by Hobson <cit.> (see also Cox <cit.> and Beiglböck and Juillet <cit.>). These barriers partition the real line into intervals and the marginal laws into so-called irreducible components. Then strong duality and dual attainment can be reduced to proving the same results for each irreducible component <cit.>. Second, Beiglböck, Nutz, and Touzi <cit.> extend the meaning of the expression μ(φ) + ν(ψ) to certain situations where both individual integrals are infinite. For example, it can happen that μ(φ) = -∞ and ν(ψ) = ∞, but the price [P]φ(X_0) + ψ(X_T) of the combined static part is well-defined, finite, and invariant under the choice of P∈(μ,ν). In this situation, this price is still denoted by μ(φ)+ν(ψ). We employ both relaxations for the precise definition of the dual problem in Section <ref>.

In continuous time, Dolinsky and Soner <cit.> show strong duality for uniformly continuous payoffs satisfying a certain growth condition. They use the integration by parts formula to define the stochastic integral ∫_0^T H_t- X_t pathwise for finite variation integrands H. However, dual attainment cannot be expected in this class in general. For our payoffs (<ref>), we need to allow integrands that are of finite variation whenever they are bounded but can become arbitrarily large or small on certain price paths. As the integrands are not of finite variation on these paths, the meaning of the pathwise integral needs to be extended appropriately.

For the purpose of the introduction, we discuss our results and methodology in a non-rigorous fashion, ignoring all aspects relating to the relaxation of the dual problem.



  
Main results.
We prove strong duality and dual attainment for payoffs of the form (<ref>) under mild conditions on f and  for irreducible marginals (Theorem <ref>); all results can be extended to general marginals along the lines of <cit.>. The key idea is the reduction of the primal and dual problems to simpler auxiliary problems, which do not depend on the set  of exercising rights. In particular, our results cover American-style derivatives f(X_τ) for Borel-measurable f and Asian-style derivatives f(1/T∫_0^T X_t  t) for lower semicontinuous f and show that both derivatives have (perhaps surprisingly) the same robust superhedging prices and structurally similar semi-static superhedges.



  
Methodology.
Our methodology relies on two crucial observations which allow us to bound the primal problem from below and the dual problem from above by simpler auxiliary primal and dual problems, respectively. To obtain a primal lower bound, we show that for any law θ which is in convex order between μ and ν, there is a sequence (P_n)_n≥1⊂(μ,ν) such that the law of ∫_[0,T]X_t A_t under P_n converges weakly to θ if A is a suitable averaging process. This allows us to bound  from below by the value of the auxiliary primal problem 

    := sup_μ≤_cθ≤_cνθ(f).

(The converse inequality also holds (cf. Lemma <ref>), so that in fact =.)

Regarding the dual upper bound, we prove (modulo technicalities) that if φ is concave and ψ is convex such that φ + ψ≥ f, then (φ,ψ,H) is a semi-static superhedge, where the dynamic part H is given explicitly in terms of φ and ψ by

    H_t
    := φ'(X_0) - ∫_[0,t]{φ'(X_0) + ψ'(X_s) } A_s.

This allows us to bound  from above by the value of the auxiliary dual problem

    := inf{μ(φ) + ν(ψ) : φ concave, ψ convex, φ + ψ≥ f}.


As a consequence, strong duality and dual attainment for  and  follow from the same assertions for the simpler problems  and , which we prove by adapting the techniques of <cit.>. Moreover, our reduction of the dual problem implies that if (φ,ψ) is optimal for , then it is also the static part of an optimal semi-static superhedge and the dynamic part H can be computed ex-post via (<ref>). This dramatically decreases the complexity of the superhedging problem: the optimization over two functions and a process satisfying an inequality constraint on the Skorokhod space is reduced to an optimization over two functions satisfying an inequality constraint on .

Our methodology reveals that many derivatives have the same robust superhedging prices and semi-static superhedges. Indeed,  and  do not depend on the set  of exercise rights granted to the buyer, and this independence transfers to  and  under mild conditions on f and . For example, if f is lower semicontinuous, then the Asian-style derivative f(1/T∫_0^T X_t  t), the American-style derivative f(X_τ), and the European-style derivative f(X_T') (for a fixed maturity T' ∈ (0,T)) all have the same robust superhedging price (Remark <ref>). This invariance breaks down when more than two marginals are given.



  
Related literature.
Much of the extant literature on robust superhedging in semi-static settings is concerned with strong duality and dual attainment. The results vary in their generality and explicitness as well as their precise formulation. The semi-static setting, where call options are available as additional hedging instruments, dates back to Hobson's seminal paper <cit.> on the lookback option.[We note that the superhedging strategies described in <cit.> are actually dynamic in the call options.] Many other specific exotic derivatives (mostly without special exercise rights) have been analyzed in this framework in the past two decades; cf. the survey <cit.>.

Securities with special exercise rights have been studied in the context of American-style derivatives in discrete-time settings. Bayraktar, Huang, and Zhou <cit.> obtain a duality result for a somewhat different primal problem (cf. <cit.>) and show that duality may fail in their setting if they formulate their primal problem in analogy to the present paper; see also <cit.> for related results with portfolio constraints. Hobson and Neuberger <cit.> (based on Neuberger's earlier manuscript <cit.>) resolve this issue by adopting a weak formulation for the primal problem: instead of optimizing only over martingale measures on a fixed filtered path space, the optimization there runs over filtered probability spaces supporting a martingale and thereby allows richer information structures and hence more stopping times. We also refer to <cit.> for recent developments in this regard. We note that all these papers permit significant restrictions on the set of possible price paths (e.g., binomial trees) while we allow all càdlàg paths. This difference may be the reason why strong duality holds in our setting without any relaxation of the primal problem.

The case of an Asian-style payoff as in (<ref>) has been studied in the case of a Dirac initial law μ. For convex or concave f, Stebegg <cit.> shows strong duality and dual attainment. For nonnegative Lipschitz f, Cox and Källblad <cit.> obtain a PDE characterization of the maximal model-based price for finitely supported ν. Bayraktar, Cox, and Stoev <cit.> provide a similar, but not identical PDE for the corresponding pricing problem for American-style payoffs as in (<ref>). A consequence of our main duality result is that the Asian and American pricing problems are the same, so that both these PDEs have the same (viscosity) solution.



  
Organization of the paper.
The remainder of the article is organized as follows. In Section 2, we recall basic results on the convex order and potential functions, introduce the generalized integral of <cit.> and its relevant properties, and present the extension of the pathwise definition of the stochastic integral for finite variation integrands. Section <ref> introduces the robust pricing and semi-static superhedging problems and presents our duality result. The duality between the auxiliary problems, the structure of their optimizers, and their relation to the robust pricing and superhedging problem are treated in Section <ref>. In Section <ref>, we provide simple geometric constructions of primal and dual optimizers for risk reversals and butterfly spreads. Finally, some counterexamples are collected in Section <ref>.




§ PRELIMINARIES



Fix a time horizon T and let Ω = D([0,T];) be the space of real-valued càdlàg paths on [0,T]. We endow Ω with the Skorokhod topology and denote by  the corresponding Borel σ-algebra, by X = (X_t)_t∈[0,T] the canonical process on Ω, and by = (_t)_t∈[0,T] the (raw) filtration generated by X. Unless otherwise stated, all probabilistic notions requiring a filtration pertain to .

For any process Y = (Y_t)_t∈[0,T] on Ω, we set Y_0- = 0, so that the jump of Y at time 0 is Δ Y_0 = Y_0.




 §.§ Martingale measures and convex order



Let μ and ν be finite[As in <cit.>, using finite measures as opposed to probability measures turns out to be useful.] measures on  with finite first moment. We denote by Π(μ,ν) the set of (continuous-time) couplings of μ and ν, i.e., finite measures P on Ω such that P ∘ X_0^-1 = μ and P ∘ X_T^-1 = ν. If, in addition, the canonical process X is a martingale under P (defined in the natural way if P is not a probability measure), then we write P∈(μ,ν) and say that P is a (continuous-time) martingale coupling between μ and ν.

We also consider discrete-time versions of these notions. To wit, we denote by Π^d(μ,ν) the set of finite measures Q on ^2 with marginal distributions μ and ν and by ^d(μ,ν) the subset of measures Q under which the canonical process on ^2 is a martingale (in its own filtration). The sets Π^d(μ,θ,ν) and (μ,θ,ν) of finite measures on ^3 with prescribed marginal distributions are defined analogously.

We write μ≤_c ν if μ and ν are in convex order in the sense that μ(φ) ≤ν(φ) holds for any convex function φ : →. In this case, μ and ν have the same mass and the same barycenter (μ) := 1/μ()∫ x  μ( x).

The potential function u_μ:→[0,∞] of μ is defined as 

    u_μ(x)
       := ∫ |x-y|  μ( y).

We refer to <cit.> for basic properties of potential functions. In particular, the following relationship between the convex order, potential functions, and martingale measures is well known.



Let μ and ν be finite measures with finite first moments and μ() = ν(). Then the following are equivalent: (i) μ≤_c ν, (ii) u_μ≤ u_ν, (iii) ^d(μ,ν) ≠∅, and (iv) (μ,ν)≠∅.


An analogous result holds for three marginals μ, θ, ν, the corresponding potential functions, and the set ^d(μ,θ,ν).

We recall the following definition from <cit.> (see also <cit.>).



A pair of finite measures μ≤_c ν is called irreducible if the set I = {u_μ < u_ν} is connected and μ(I) = μ(). In this situation, let J be the union of I and any endpoints of I that are atoms of ν; then (I,J) is the domain of μ≤_c ν.


We work with irreducible μ≤_c ν for the remainder of this article.




 §.§ Generalized integral




Let μ≤_c ν be irreducible with domain (I,J). Beiglböck and Juillet <cit.> and Beiglböck, Nutz, and Touzi <cit.> appropriately extend the meaning of the expression μ(φ) + ν(ψ) to the case where the individual integrals are not necessarily finite. We present here a slight extension of their work in order to deal with intermediate laws μ≤_cθ≤_cν for which the pairs μ≤_cθ and θ≤_cν may not be irreducible.

For the rest of this article, whenever we write μ≤_c ν for any two measures μ and ν, we implicitly assume that both measures are finite and have a finite first moment. Throughout this section, we fix μ≤_cθ_1≤_cθ_2≤_cν.



Let χ:J→ be concave. Denote by -χ” the second derivative measure of the convex function -χ and by Δχ the possible jumps of χ at the endpoints of I. We set

    (θ_1-θ_2)(χ)
       := 1/2∫_I (u_θ_1-u_θ_2) χ” + ∫_J∖ I|Δχ| (θ_2 - θ_1) ∈ [0,∞].

The right-hand side is well defined in [0,∞] because u_θ_1≤ u_θ_2 on I and θ_1({ b }) ≤θ_2({ b }) for b ∈ J∖ I.


If θ_1 = μ and θ_2 = ν, then (<ref>) coincides with Equation (4.2) in <cit.> because μ is concentrated on I. As in <cit.>, there is an alternative representation of (θ_1-θ_2)(χ) in terms of an iterated integral with respect to a disintegration of a (one-step) martingale coupling on ^2:



Let χ:J→ be concave and let Q∈^d(θ_1,θ_2). For any disintegration Q = θ_1 ⊗κ, we have

    (θ_1-θ_2)(χ)
       = ∫_J [ χ(x_1) - ∫_J χ(x_2) κ(x_1, x_2)] θ_1( x_1).




The proof of <cit.> does not use that μ≤_c ν is irreducible. Moreover, for χ̅:J → concave and continuous, the same arguments as in the proof of <cit.> yield

    1/2∫_I (u_θ_1-u_θ_2) χ̅”   = ∫_J [ χ̅(x_1) - ∫_J χ̅(x_2)  κ(x_1, x_2) ] θ_1( x_1).

(Note that ∫_Jχ̅(x_2)  κ(x_1, x_2) = χ̅(x_1) for boundary points x_1 ∈ J∖ I because κ is a martingale kernel concentrated on J.)

For a general concave χ:J→, we write χ = χ̅- |Δχ|_J∖ I with χ̅ continuous. Then we can replace χ̅ with χ on the left-hand side of (<ref>) and the integrand on the right-hand side reads as

    χ + |Δχ|_J∖ I - ∫_J χ(x_2) κ(·, x_2) - ∫_J∖ I|Δχ(x_2)| κ(·, x_2).

Integrating this against θ_1 and using Fubini's theorem yields

    ∫_J [ χ(x_1) - ∫_J χ(x_2) κ(x_1, x_2) ]θ( x_1) + ∫_J∖ I|Δχ|θ_1 - ∫_J∖ I|Δχ|θ_2.

Together with (<ref>), this proves the claim.


It can be shown as in <cit.> that (θ_1-θ_2)(χ) = θ_1(χ) - θ_2(χ) if at least one of the individual integrals is finite.

We can now define the integral θ_1(φ) + θ_2(ψ) as in <cit.>.



Let φ:J → and ψ:J→ be Borel functions. If there exists a concave function χ:J→ such that φ-χ∈ L^1(θ_1) and ψ + χ∈ L^1(θ_2), we say that χ is a concave moderator for (φ,ψ) with respect to θ_1≤_cθ_2 and set

    θ_1(φ) + θ_2(ψ)
       := θ_1(φ - χ) + θ_2(ψ + χ) + (θ_1-θ_2)(χ) ∈ (-∞,∞].



As in <cit.>, the expression θ_1(φ) + θ_2(ψ) defined in (<ref>) does not depend on the choice of the concave moderator.



We write L^c(θ_1,θ_2) for the space of pairs of Borel functions φ,ψ:J→ which admit a concave moderator χ with respect to θ_1≤_cθ_2 such that (θ_1-θ_2)(χ)<∞.


We next present additional properties of the notions introduced above.



Let (φ,ψ) ∈ L^c(θ_1,θ_2).


  * φ is finite on atoms of θ_1. If φ is concave, then φ < ∞ on J and φ > -∞ on the interior of the convex hull of the support of θ_1.

  * ψ is finite on atoms of θ_2. If ψ is convex, then ψ > -∞ on J and ψ < ∞ on the interior of the convex hull of the support of θ_2.

  * If a,b : → are affine, then (φ + a,ψ+b) ∈ L^c(θ_1,θ_2) and

    θ_1(φ+a) + θ_2(ψ+b)
       = {θ_1(φ)+θ_2(ψ) } + θ_1(a) + θ_2(b).





We only prove (iii). Let χ be a concave moderator for (φ,ψ) with respect to θ_1≤_cθ_2. Then φ - χ∈ L^1(θ_1), ψ + χ∈ L^1(θ_2), and (θ_1-θ_2)(χ) < ∞. Being affine, a and b are θ_1- and θ_2-integrable. It follows that χ is also a concave moderator for (φ + a, ψ + b) with respect to θ_1 ≤_c θ_2 and that (φ + a, ψ + b) ∈ L^c(θ_1,θ_2). The last assertion is a direct computation.




Recall that I is the interior of the convex hull of the support of ν and that J is the union of I and any endpoints of I that are atoms of ν. Hence, Lemma <ref> (ii) shows in particular, that if (φ,ψ)∈ L^c(θ_1,ν) with ψ convex, then ψ is finite on J.


We conclude this section with a number of calculation rules for the integrals defined above when φ is concave and ψ is convex.



Let μ≤_cθ_1≤_cθ_2≤_cθ_3≤_cν (where the pair μ≤_cν is irreducible) and let (φ,ψ)∈ L^c(θ_1,θ_3) be such that φ is concave and finite, ψ is convex and finite, and φ + ψ is bounded from below by a concave θ_3-integrable function.


  * φ and -ψ are concave moderators for (φ,ψ) with respect to θ_1≤_cθ_3.

  * (φ,ψ) ∈ L^c(θ_1,θ_2) ∩ L^c(θ_2,θ_3).

  * θ_1(φ) + θ_2(ψ) ≤θ_1(φ) + θ_3(ψ).

  * θ_2(φ) + θ_3(ψ) ≤θ_1(φ) + θ_3(ψ).




Denote by ξ a concave θ_3-integrable lower bound for φ+ψ. By the concavity of ξ, we have θ_1(ξ) ≥θ_2(ξ) ≥θ_3(ξ)>-∞, so that ξ is also θ_1- and θ_2-integrable.

(i): Regarding the concave moderator property of φ, it suffices to show that φ + ψ is θ_3-integrable. Denote by φ' the left-derivative of the concave function φ on I. Then for (x_1,x_3)∈ I× J,

    ξ(x_3)
       ≤φ(x_3) + ψ(x_3)
    ≤φ(x_1) + ψ(x_3) + φ'(x_1)(x_3-x_1).

Fix any Q ∈^d(θ_1,θ_3). Then (<ref>) also holds Q-a.e. on J× J (setting φ' = 0 on J∖ I for example); this uses that any mass in a point of J ∖ I stays put under a martingale transport plan. Since ξ is θ_3-integrable, the negative part of the right-hand side in (<ref>) is Q-integrable. Then it can be argued as in <cit.> that the right-hand side in (<ref>) is Q-integrable. It follows that φ+ψ is θ_3-integrable.

Regarding the assertion about -ψ, it suffices to show that φ + ψ is θ_1-integrable. We have 

    ξ(x_1)
       ≤φ(x_1) + ψ(x_1)
       = [φ(x_1) + ψ(x_3) + φ'(x_1)(x_3-x_1)] 
           + [ψ(x_1) - ψ(x_3) - φ'(x_1)(x_3-x_1)]   Q-a.e. on J× J.

By the above, the first term on the right-hand side is Q-integrable. Thus, the negative part of the second term is also Q-integrable. Hence, we may integrate the second term iteratively using Fubini's theorem as in <cit.>. The Q-integral equals -(θ_1-θ_3)(-ψ) ≤ 0. In particular, the right-hand side in (<ref>) is Q-integrable. It follows that φ+ψ is θ_1-integrable.

(ii)–(iv): We know from the above that φ+ψ is θ_3-integrable. It follows that φ is a concave moderator for (φ,ψ) with respect to θ_2≤_cθ_3. Because θ_1≤_cθ_2, we have that u_θ_1≤ u_θ_2 and θ_1({ b }) ≤θ_2({ b }) for b ∈ J ∖ I. Thus, (θ_2 - θ_3)(φ) ≤ (θ_1-θ_3)(φ) < ∞ (cf. Definition <ref>). Hence, (φ,ψ) ∈ L^c(θ_2,θ_3) and 

    θ_2(φ) + θ_3(ψ)
       = θ_2(φ-φ) + θ_3(φ+ψ) + (θ_2-θ_3)(φ)
       ≤θ_1(φ-φ) + θ_3(φ+ψ) + (θ_1-θ_3)(φ)
       = θ_1(φ) + θ_3(ψ).

One can show similarly that (φ,ψ)∈ L^c(θ_1,θ_2) and that θ_1(φ) + θ_2(ψ) ≤θ_1(φ) + θ_3(ψ).





 §.§ Pathwise stochastic integration




For any -adapted càdlàg process H of finite variation, the integral H_-∙ X_T = ∫_(0,T] H_t- X_t can be defined pathwise, i.e., for each ω∈Ω individually, via integration by parts as follows:

    H_-∙ X_T
       := X_T H_T - X_0 H_0 - ∫_(0,T] X_t  H_t,

where the integral on the right-hand side is the pathwise Lebesgue–Stieltjes integral. Setting H_0- = 0, so that Δ H_0 = H_0, we can recast  (<ref>) as

    H_-∙ X_T
       = (X_T-X_0)H_0 + ∫_(0,T] (X_T - X_t)  H_t.

For any martingale measure P, if the (standard) stochastic integral of H_- with respect to X exists, then it is P-indistinguishable from the pathwise stochastic integral.

We need to give a sensible meaning to the integral H_-∙ X_T for certain integrands H which are not necessarily of finite variation, but may diverge in finite time.



The following example motivates our extension of the pathwise stochastic integral for finite variation integrands. Let μ = δ_0 and ν = 1/2δ_-1 + 1/2δ_1. Then μ≤_c ν are irreducible with domain (I,J) = ((-1,1),[-1,1]). Consider a payoff function f which is convex on [-1,1] and has infinite (one-sided) derivatives at -1 and 1, e.g., f(x) = 1-√(1-x^2)_[-1,1](x),  A semi-static superhedge for the Asian-style derivative f(1/T∫_0^T X_t  t) can be derived as follows. By Jensen's inequality and the convexity of f, for every path of X that evolves in [-1,1],

    f(1/T∫_0^T X_t  t)
       ≤∫_0^T f(X_t)   t/T≤∫_0^T (f(X_T) - f'(X_t)(X_T-X_t))   t/T
       = f(X_T) - ∫_0^T (X_T-X_t) f'(X_t)  t/T.

Comparing this with (<ref>), a semi-static superhedge for the Asian-style derivative is obtained from a European-style derivative with payoff f(X_T) maturing at T and a dynamic trading strategy H with H_0 = 0 and dynamics H_t = -f'(X_t)  t/T. Then H is of finite variation whenever X stays away from the boundaries of (-1,1). But, as X approaches -1 or 1, the derivative f'(X_t) becomes arbitrarily large (in absolute value), and H may fail to be of finite variation. It turns out, however, that the integral ∫_0^T (X_T-X_t)f'(X_t)   t/T is still well defined on these paths. The reason is that when paths of X come arbitrarily close to 1, say, then for any martingale coupling P∈(μ,ν), X_T=1 P-a.s. on these paths (because J = [-1,1]), so that X_T-X_t becomes small and counteracts the growth of f'(X_t).


We shall define a pathwise stochastic integral for -adapted càdlàg integrators X and integrands Ĥ_- of the form

    Ĥ_t
       = h_0 + ∫_(0,t] h_s  Y_s

for an -adapted càdlàg process Y = (Y_t)_t∈[0,T] of finite variation and an -adapted process h = (h_t)_t∈[0,T]—even in certain situations where the right-hand side of (<ref>) is not finite. The idea is to formally substitute (<ref>) into (<ref>), formally use the associativity of Lebesgue–Stieltjes integrals, and then employ the resulting expression as a definition for a pathwise stochastic integral. We first introduce a set of integrands for this integral.



Let Ω' ⊂ D([0,T];). We denote by L(Ω') the set of pairs (h,Y) consisting of an -adapted process h and an -adapted càdlàg process Y of finite variation such that the process ((X_T - X_t)h_t)_t∈[0,T] is Y-integrable on (0,T] for each path in Ω'.


If Y is an -adapted càdlàg process of finite variation, then (1,Y) ∈ L(Ω') for any Ω' ⊂ D([0,T];) (because any càdlàg function is bounded on compact intervals).

We fix a set Ω' ⊂ D([0,T];) for the rest of this section.



For H=(h,Y) ∈ L(Ω'), we set

    H ♢ X_T
       := (X_T-X_0)h_0 + ∫_(0,T] (X_T - X_t) h_t  Y_t  on  Ω'.



We note that the Lebesgue–Stieltjes integral on the right-hand side of (<ref>) is well defined and finite by the definition of L(Ω'). The following result shows that for pathwise bounded h, H♢ X_T coincides with Ĥ_-∙ X_T for Ĥ as in (<ref>). This motivates the interpretation of H♢ X_T as the gains from trading in X according to a self-financing  trading strategy Ĥ_-.



Let H = (h,Y) ∈ L(Ω') and ω∈Ω'. If the function t↦ h_t(ω) is bounded on [0,T], then 

    (H ♢ X_T)(ω)
       = (Ĥ_- ∙ X_T)(ω),

where Ĥ = h_0 + ∫_(0,·] h  Y.


If we set h_0 = Y_0 and h_t = 1 for t ∈ (0,T] for an -adapted càdlàg process Y of finite variation, then H=(h,Y)∈ L(Ω) and by Proposition <ref>,

    H ♢ X_T
       = Y_- ∙ X_T  on  Ω.

So the integral H♢ X_T embeds all pathwise stochastic integrals Y_-∙ X_T.


Since h(ω) is bounded on [0,T], Ĥ_t(ω) = h_0(ω) + ∫_(0,t] h_s(ω)  Y_s(ω) is a well-defined càdlàg finite variation function on [0,T]. Thus, by (<ref>),

    Ĥ_- ∙ X_T
       = (X_T - X_0)h_0 + ∫_(0,T] (X_T-X_s) h_s  Y_s
    = H ♢ X_T.






§ ROBUST PRICING AND SUPERHEDGING PROBLEMS




Throughout this section, we fix an irreducible pair μ≤_c ν with domain (I,J) and a Borel function f:→ which is bounded from below by a ν-integrable concave function.




 §.§ Pricing problem




Our pricing and hedging duality applies to a wide range of exotic derivatives including American options, fixed strike Asian options, Bermudan options, and European options with an intermediate maturity. We now describe this class of derivative securities. 


A nonnegative -adapted nondecreasing càdlàg process A=(A_t)_t∈[0,T] is called an averaging process if A_T(ω)=1 for every ω∈Ω. If in addition A_0(ω) = 0 and Δ A_T(ω) = 0 for each ω∈Ω, then A is called an interior averaging process. If in addition there is t∈(0,T) such that A_t(ω) = 0 for each ω∈Ω, then A is called a strictly interior averaging process.


Recall that we set A_0- = 0 and note that for each ω∈Ω, A(ω) can be identified with a Borel probability measure on [0,T]. If A is an interior averaging process, then this probability measure is supported on (0,T), and if A is a strictly interior averaging process then its support is (uniformly in ω) contained in [t,T) for some t ∈(0,T).

Given a nonempty set  of averaging processes, we consider a derivative security whose payoff at time T is

    f(∫_[0,T]X_t  A_t),

where A ∈ is chosen by the buyer and the seller observes (A_s)_s∈[0,t] at time t. Then the robust model-based price is defined as

    _μ,ν(f,)
       =sup_P ∈(μ,ν)sup_A ∈[P]f(∫_[0,T] X_t  A_t).

In other words, _μ,ν(f,) is the highest model-based price of the derivative security (<ref>) among all martingale models which are consistent with the given marginal distributions.


One can show that for each P ∈(μ,ν) and each averaging process A, the law of ∫_[0,T] X_t  A_t under P is in convex order between μ and ν; cf. Lemma <ref>. Because f is by assumption bounded from below by a ν-integrable concave function, the expectations in (<ref>) are well defined.



Important special cases are obtained for specific choices of .

[No special exercise rights]

Setting = { A } deprives the buyer of any special exercise rights and reduces (<ref>) to the more familiar robust pricing problem

    sup_P∈(μ,ν)[P]F

for the derivative security F = f(∫_[0,T]X_t A_t).



  * Asian options. Setting A_t = t/T recovers the Asian-style derivative f(1/T∫_0^T X_t  t); this includes fixed strike Asian puts and calls, but not floating strike Asian options. This robust pricing problem is analyzed in <cit.>.


  * European options. Setting A_t = _[T',T](t) yields a European-style payoff f(X_T') with an intermediate maturity T' ∈ (0,T).



[Special exercise rights] 

Fix a nonempty set  of [0,T]-valued -stopping times, and consider = {_τ, T :τ∈}. Then (<ref>) reduces to

    sup_P∈(μ,ν)sup_τ∈[P]f(X_τ).



  * American options. If  consists of all [0,T]-valued -stopping times, then (<ref>) is the robust American option pricing problem analyzed in <cit.>.


  * Bermudan options. Bermudan options with exercise dates 0≤ T_1 < ⋯ < T_n ≤ T are covered by choosing  to be the set of { T_1,…, T_n }-valued -stopping times. 






 §.§ Superhedging problem




In the case of robust semi-static superhedging of American options, it is well known that a pricing-hedging duality can in general only hold if the seller of the option can adjust the dynamic part of his trading strategy after the option has been exercised; cf. <cit.>. In other words, the buyer has to communicate her decision of exercising to the seller at the time of exercising. The analog in our setting is that the seller observes A_t at time t. That is, his trading strategy can be “adapted” to the averaging process chosen by the buyer.

To make this precise, let Ω̂ be the cartesian product of Ω and the set of nonnegative, nondecreasing, càdlàg functions a:[0,T]→[0,1] with a(T) = 1. As Ω̂ is a subspace of the Skorokhod space D([0,T];×[0,1]), we can equip it with the subspace Skorokhod topology and denote by  the corresponding Borel σ-algebra. We write = (_t)_t∈[0,T] for the (raw) filtration generated by the canonical process on Ω̂. For any process Z on Ω̂ and any averaging process A (on Ω), we define the process Z^A on Ω by

    Z^A_t(ω)
       = Z_t(ω,A(ω)),   ω∈Ω.

Note that if Z is -adapted, then Z^A is -adapted, and if Z is càdlàg or of finite variation, then so is Z^A.

Next, we define a suitable set of paths for the hedging problem. Let Ω_μ,ν⊂Ω denote the subset of paths which start in I, evolve in J, and are “captured” if they approach the boundary ∂ J:

    Ω_μ,ν
    := {ω∈Ω : 
         ω_0 ∈ I, ω_t ∈ J for all t ∈ (0,T],
         if ω_t-∈∂ J, then ω_u =ω_t- for all u∈[t,T], and
         if ω_t ∈∂ J, then ω_u = ω_t for all u∈[t,T]}.


One can show that every martingale coupling between μ and ν is concentrated on Ω_μ,ν:[The fact that μ and ν are concentrated on I and J, respectively, together with the martingale property implies that P-a.e. path has the first two properties in (<ref>). The other two properties can be shown similarly to the fact that nonnegative supermartingales are almost surely captured in zero (cf. <cit.>).]



Ω_μ,ν∈ and P[Ω_μ,ν] = P[Ω] for every P∈(μ,ν).


We are now ready to define the trading strategies for the robust superhedging problem. 



A semi-static trading strategy is a triplet (φ,ψ,H) consisting of a pair of functions (φ,ψ) ∈ L^c(μ,ν) and a pair H=(h_t,Y_t)_t∈[0,T] of -adapted processes on Ω̂ such that

    H^A
       := (h^A,Y^A) ∈ L(Ω_μ,ν)
      for every averaging process A.



The portfolio value at time T of a semi-static trading strategy is given by the sum of the static part with payoffs φ(X_0) and ψ(X_T) and the gains H^A♢ X_T of the dynamic part:

    φ(X_0) + ψ(X_T) + H^A♢ X_T.

The initial cost to set up this position is equal to the initial price of the static part:

    μ(φ) + ν(ψ).


We now turn our attention to semi-static trading strategies which dominate the payoff (<ref>) of our derivative security for  each path in Ω_μ,ν and every averaging process in .



A semi-static trading strategy (φ,ψ,H) is called a semi-static superhedge (for f and ) if for every A ∈,

    f(∫_[0,T] X_t  A_t)
       ≤φ(X_0) + ψ(X_T) + H^A ♢ X_T  on  Ω_μ,ν

and

    [P]φ(X_0) + ψ(X_T) + H^A ♢ X_T   ≤μ(φ) + ν(ψ),    P ∈(μ,ν).

The set of semi-static superhedges for f and  is denoted by _μ,ν(f,).


The requirement (<ref>) is an admissibility condition. It demands that for every P∈(μ,ν), the portfolio value, consisting of both the static and the dynamic part, is a one-step P-supermartingale between the time at which the static part is set up and time T. In other words, the expectation of the terminal portfolio value (<ref>) is less than or equal to the initial portfolio value (<ref>).


We define the robust superhedging price (for f and ) as the “minimal” initial capital required to set up a semi-static superhedge for f and :[We use the convention inf∅=∞.]

    _μ,ν(f,)
    = inf_(φ,ψ,H)∈_μ,ν(f,){μ(φ) + ν(ψ) }.





 §.§ Weak and strong duality




Weak duality between the robust pricing and hedging problems is an immediate consequence of their definitions:



Let f:→ be Borel and bounded from below by a ν-integrable concave function and let  be a nonempty set of averaging processes. Then

    _μ,ν(f,)
    ≤_μ,ν(f,).




Let P ∈(μ,ν), A∈, and (φ,ψ,H) ∈_μ,ν(f,) (there is nothing to show if this set is empty). Taking P-expectations in (<ref>) and using (<ref>) shows that [P]f(∫_[0,T] X_t  A_t)≤μ(φ) + ν(ψ).

This proves the claim as P, A, and (φ,ψ,H) were arbitrary.


With an additional mild assumption on either  or f, we obtain strong duality and the existence of dual minimizers:



Let μ≤_c ν be irreducible, let f:→[0,∞] be Borel, and let  be a set of averaging processes. Suppose that one of the following two conditions holds:


  * f is lower semicontinuous and  contains an interior averaging process;

  *  contains a strictly interior averaging process.

Then

    _μ,ν(f,)
       = _μ,ν(f,) ∈ [0,∞]

and this value is independent of  as long as one of the two conditions above holds. Moreover, if _μ,ν(f,) < ∞, then there exists an optimizer (φ,ψ,H)∈_μ,ν(f,) for _μ,ν(f,).


  


  * For fixed f, the robust model-based price _μ,ν(f,) is invariant under the choice of the set  (as long as the assumptions of Theorem <ref> hold). In particular, American, Bermudan, and European options with intermediate maturity (cf. Examples <ref>–<ref>) all have the same robust model-based price (because the corresponding sets  all contain a strictly interior averaging process). If f is lower semicontinuous, this extends to the Asian-style option of Example <ref> (i). If more than two marginals are given, then the robust model-based prices of these derivatives typically differ; see Example <ref>.


  * Derivatives of the form (<ref>) that depend distinctly on X_0 and/or X_T such as f(1/2(X_0 + X_T)) are not covered by Theorem <ref> ( does not contain an interior averaging process). In these cases, the robust model-based price is still bounded above by the corresponding robust model-based price of, say, the European-style derivative f(X_T/2). However, the inequality is typically strict; see Example <ref>.



  


  * Theorem <ref> can be extended to non-irreducible marginals along the lines of <cit.>.


  * Strong duality continues to hold if we restrict ourselves to finite variation strategies; cf. Remark <ref> for an outline of the argument. It is an open question whether there is (in general) a dual minimizer (φ,ψ,H) with a dynamic part H of finite variation.



We defer the proof of Theorem <ref> to the end of Section <ref>. The idea is as follows. We bound the pricing problem from below and the hedging problem from above by auxiliary maximization and minimization problems, respectively, and show that strong duality holds between those two auxiliary problems. Then all four problems have equal value and in particular strong duality for the pricing and hedging problems holds. Moreover, we show that the auxiliary dual problem admits a minimizer and that every element in the dual space of the auxiliary problem gives rise to a semi-static superhedge with the same cost. Then, in particular, the minimizer of the auxiliary dual problem yields an optimal semi-static superhedge for f and  (which is independent of ).




§ AUXILIARY PROBLEMS




Throughout this section, we fix an irreducible pair μ≤_c ν with domain (I,J) and a function f:→ which is bounded from below by a ν-integrable concave function.

The auxiliary primal and dual problems are formally derived in Section <ref>. They are rigorously introduced in Sections <ref>–<ref> and proved to be lower and upper bounds of the robust model-based price and the robust superhedging price, respectively. Their strong duality is proved in Section <ref>. Finally, structural properties of primal and dual optimizers of the auxiliary problems are studied in Section <ref>.




 §.§ Motivation




The key property of payoffs of the form (<ref>) is that the law of ∫_[0,T]X_t A_t under P∈(μ,ν) is in convex order between μ and ν. In this section, we explain this observation and how it can be used to estimate the robust pricing problem from below and the robust superhedging problem from above.

Let P ∈(μ,ν) and let τ be a [0,T]-valued -stopping time. An application of the optional stopping theorem and Jensen's inequality shows that for any convex function ψ,

    μ(ψ)
    = [P]ψ(X_0)
    = [P]ψ([P]X_τ_0)≤[P]ψ(X_τ)    and


    ν(ψ)
    = [P]ψ(X_T)
    ≥[P]ψ([P]X_T_τ)
    = [P]ψ(X_τ),

so that the law of X_τ under P is in convex order between μ and ν.

Using a time change argument and again Jensen's inequality and the optional stopping theorem, it can be shown that this property generalizes to the random variable ∫_[0,T]X_t  A_t for an averaging process A.



Let P∈(μ,ν) and let A be an averaging process. Then the law of ∫_[0,T]X_t A_t under P is in convex order between μ and ν.


In the sequel, we write = _μ,ν(f,) and = _μ,ν(f,) for brevity. Lemma <ref> implies that

    ≤sup_μ≤_cθ≤_cνθ(f)
    =: .

We show in Section <ref> that also the converse inequality holds under mild assumptions on f and . Thus, = and one is led to expect that = for a suitable dual problem  to .

Let us thus formally derive the Lagrange dual problem for . Dualizing the constraint μ≤_cθ≤_cν suggests to consider the Lagrangian

    L(θ,ψ_1,ψ_2)
       := θ(f) + (θ(ψ_1)-μ(ψ_1)) + (ν(ψ_2)-θ(ψ_2)),

where convex functions ψ_1,ψ_2 are taken as Lagrange multipliers.[Note that the last two terms in (<ref>) are nonnegative for all convex ψ_1,ψ_2 if and only if the primal constraint μ≤_cθ≤_cν holds.] Then the Lagrange dual problem is

    =inf_ψ_1,ψ_2sup_θ L(θ,ψ_1,ψ_2)
    = inf_ψ_1,ψ_2sup_θ{θ(f+ψ_1-ψ_2) -μ(ψ_1) + ν(ψ_2)}

where the infima are taken over convex functions and the suprema are taken over finite measures. Viewing the finite measure θ as a Lagrange multiplier for the constraint f ≤ -ψ_1 + ψ_2 and relabeling φ = -ψ_1 and ψ = ψ_2, we obtain

    = inf{μ(φ)+ν(ψ) : φ concave, ψ convex, and φ+ψ≥ f}.

In the precise definition of  in Section <ref>, μ(φ)+ν(ψ) is understood in the generalized sense of Definition <ref> and the inequality φ + ψ≥ f is required to hold on J. We then show that each feasible element (φ,ψ) for  entails an element (φ,ψ,H) ∈_μ,ν(f,) (Proposition <ref>), which implies that ≤.

Combining the above with the weak duality inequality (Lemma <ref>) yields

    = ≤≤.

Hence, strong duality and dual attainment for the robust pricing and superhedging problems reduce to the same assertions for the simpler auxiliary problems, which are proved in Section <ref>.




 §.§ Auxiliary primal problem




Consider the auxiliary primal problem

    _μ,ν(f)
    = sup_μ≤_c θ≤_c νθ(f),

where θ(f) is understood as the outer integral if f is not Borel-measurable. Under suitable conditions on f and , the primal value _μ,ν(f) is a lower bound for the robust model-based price (<ref>):



Let  be a set of averaging processes. Suppose that one of the following two sets of conditions holds:


  *  contains an interior averaging process and f is lower semicontinuous and bounded from below by a ν-integrable concave function φ:J→;

  *  contains a strictly interior averaging process and f is Borel.

Then

    _μ,ν(f)
    ≤_μ,ν(f,).



The proof of Proposition <ref> is given at the end of this section. It is based on the following construction of measures in (μ,ν) under which the law of ∫_[0,T] X_t  A_t equals (approximately or exactly) a given θ. This construction also highlights the importance of  containing an interior averaging process, which does not put any mass on the times 0 and T at which the marginal distributions of X are given; see Example <ref> for a counterexample.



Let μ≤_c θ≤_c ν.


  * There is a sequence (P_n)_n≥1⊂(μ,ν) such that

    ^P_n(∫_[0,T]X_t  A_t) θ  weakly

for every interior averaging process A.

  * If A is a strictly interior averaging process, then there is P ∈(μ,ν) (depending on A) such that ^P(∫_[0,T]X_t  A_t) = θ.




(i): By the two-step adaptation of Proposition <ref>, there exists a measure Q ∈^d(μ,θ,ν). For all n large enough, let ι^n:^3 →Ω be the embedding of ^3 in Ω which maps (y_1,y_2,y_3) to the piecewise constant path

    [0,T]∋ t ↦ y_1_[0,1/n)(t) + y_2_[1/n,T)(t) + y_3_{T}(t)

(which jumps (at most) at times 1/n and T), and denote by P_n := Q ∘ (ι^n)^-1 the associated pushforward measure. Then P_n ∈(μ,ν) by the corresponding properties of Q. Moreover, denoting the canonical process on ^3 by (Y_1,Y_2,Y_3) and setting A^n = A ∘ι^n for an interior averaging process A, we have

    ∫_[0,T] (ι^n)_t  A^n_t - Y_2
       = Y_1 A^n_1/n- + Y_2 (A^n_T--A^n_1/n-) + Y_3 Δ A^n_T -Y_2A_T 
       = (Y_1 - Y_2)A^n_1/n- +(Y_3 - Y_2)Δ A^n_T 
       = (Y_1 - Y_2)A^n_1/n-  on  ^3,

where we use the properties A_T = 1 and Δ A_T=0 of an interior averaging process.

By construction, the law of ∫_[0,T] (ι^n)_t  A^n_t under Q coincides with the law of ∫_[0,T] X_t  A_t under P_n and the law of Y_2 under Q is θ. It thus suffices to prove that the right-hand side in (<ref>) converges to zero in L^1(Q) as n→∞. To this end, note that |Y_1 - Y_2| ≤ |Y_1| + |Y_2| is Q-integrable because μ and θ have finite first moments. Thus, by dominated convergence, it is enough to show that A^n_1/n-→ 0 pointwise as n→0. So fix (y_1,y_2,y_3) ∈^3. Since A is -adapted, A_1/n-(ω) only depends on the values of the path ω on the interval [0,1/n). In view of the embedding (<ref>), this means that 

    A^n_1/n-(y_1,y_2,y_3)
       = A_1/n-(ι^n(y_1,y_2,y_3))
    = A_1/n-(y_1 _[0,T]),

where y_1 _[0,T] denotes the constant path at y_1. Hence, the asserted pointwise convergence follows from the fact that A_0 = 0 and A is right-continuous.

(ii): If A is a strictly interior averaging process, then the last expression in (<ref>) is identically zero for n large enough and setting P=P_n gives the desired result.




Part (i) of Lemma <ref> remains true if we restrict ourselves to martingale measures with almost surely continuous paths. The analog of part (ii) for continuous martingales requires the additional assumption that there exists t < T such that A_t ≡ 1.

The main ingredient for this assertion is <cit.>: for every discrete time-martingale {Y_n}_n ≥ 0, there is a continuous-time martingale {Z_t}_t ≥ 0 with continuous sample paths such that the processes {Y_n}_n ≥ 0 and {Z_n}_n ≥ 0 have the same (joint) distribution.



Let μ≤_cθ≤_cν. Assume first that condition (ii) holds and let A be a strictly interior averaging process. Then by Lemma <ref> (ii), there is P∈(μ,ν) such that ^P(∫_[0,T]X_t A_t) = θ. Hence,

    θ(f)
    = [P]f(∫_[0,T]X A)≤_μ,ν(f,).

As θ was arbitrary, the claim follows.

Next, assume instead that condition (i) holds and let A be an interior averaging process and φ as in condition (i). By Lemma <ref> (i), there is a sequence (P_n)_n∈⊂(μ,ν) such that  θ_n:=^P_n(∫_[0,T]X_t A_t) →θ weakly. Define f_k = f ∨ (-k), k ≥ 1. Then f_k is bounded from below and lower semicontinuous, so lim inf_n→∞θ_n(f_k) ≥θ(f_k) by the Portmanteau theorem.

Fix ε > 0. Choose first k large enough such that ν((φ+k)^-) ≤ε/2 and then N large enough such that θ_n(f_k)-θ(f_k) ≥ -ε/2 for all n ≥ N. Using that 0 ≤ f_k-f ≤ (φ + k)^- and that (φ + k)^- is convex, we obtain for n ≥ N,

    θ_n(f) - θ(f)
       = θ_n(f-f_k) + (θ_n(f_k) - θ(f_k)) + θ(f_k - f)
       ≥ -θ_n((φ+k)^-)-ε/2≥ -ν((φ+k)^-) - ε/2≥ -ε.

Thus, lim inf_n→∞θ_n(f) ≥θ(f). Now the claim follows from

    θ(f)
       ≤lim inf_n→∞θ_n(f)
    = lim inf_n→∞[P_n]f(∫_[0,T]X_t A_t)≤_μ,ν(f,).






 §.§ Auxiliary dual problem




Consider the auxiliary dual problem

    _μ,ν(f)
    = inf_(φ,ψ)∈_μ,ν(f){μ(φ) + ν(ψ) },

where _μ,ν(f) denotes the set of (φ,ψ) ∈ L^c(μ,ν) with concave φ:J→ and convex ψ:J→ such that φ + ψ≥ f on J.

The dual value _μ,ν(f) is an upper bound for the robust superhedging price (<ref>):



Let f:→[0,∞] be Borel. Then _μ,ν(f,) ≤_μ,ν(f).


Proposition <ref> follows immediately from the next result (Proposition <ref>) which shows that every (φ,ψ)∈_μ,ν(f) gives rise to a semi-static superhedge for f and . More precisely, the semi-static superhedge is of the form (φ,ψ,H) and the dynamic part H can be explicitly written in terms of the “derivatives” of φ and ψ.

Given a convex function ψ: J →, a Borel function ψ':I → is called a subderivative of ψ if for every x_0 ∈ I, ψ'(x_0) belongs to the subdifferential of ψ at x_0, i.e.,

    ψ(x) - ψ(x_0)
    ≥ψ'(x_0)(x-x_0),   x ∈ J.

Symmetrically, for a concave function φ: J→, a Borel function φ':I → is called a superderivative of φ if -φ' is a subderivative of -φ.



If (φ,ψ)∈_μ,ν(f) and f > -∞ on J, then φ and ψ are both finite (so that sub- and superderivatives are well defined). Indeed, we already know from Remark <ref> that ψ is finite on J. Moreover, φ < ∞ on J by Lemma <ref> (i) and if f > -∞ on J, then φ≥ f-ψ > -∞, so that also φ is finite on J.




Let f:→[0,∞] be Borel and let (φ,ψ)∈_μ,ν(f). Denoting the canonical process on Ω̂ by (X,A), define the -adapted process h = (h_t)_t∈[0,T] (on Ω̂) by

    h_0
       = φ'(X_0)(1-A_0) - ψ'(X_0)A_0,
    
    h_t
       = -φ'(X_0) - ψ'(X_t),   t ∈ (0,T],

where φ' is any superderivative of φ, ψ' is any subderivative of ψ, and we set φ'=ψ' = 0 on ∖ I.
Set H = (h,A). Then (φ,ψ, H) ∈_μ,ν(f,) for any nonempty set  of averaging processes.


The proof of Proposition <ref>, given at the end of this section, relies on the following two technical lemmas. The definition of Ω_μ,ν in (<ref>) is crucial for the first one. We recall that (real-valued) càdlàg functions are bounded on compact intervals.



Let ψ and ψ' be as in Proposition <ref>. For each ω∈Ω_μ,ν, the function [0,T] ∋ t ↦ (ω_T-ω_t)ψ'(ω_t) is bounded.



Fix ω∈Ω_μ,ν and write I= (l,r) with l,r ∈. We consider three cases: (i) J = I, (ii) J=[l,r), and (iii) J = [l,r]. The case (l,r] is symmetric to (ii).

(i): Suppose that J = I = (l,r). We claim that ω evolves in a compact (and hence strict) subset of I. Suppose for the sake of contradiction hat inf_t∈[0,T]ω_t = l ∈ [-∞,∞). Then there is a sequence (t_n)_n∈⊂ [0,T] such that lim_n→∞ω_t_n = l. Passing to a subsequence if necessary, this sequence may be chosen to be either (strictly) increasing or nonincreasing to a limit t^⋆∈ [0,T]. Then, as ω is càdlàg, ω_t^⋆- = l or ω_t^⋆ = l. But then ω_t^⋆  =l in any case by the definition of Ω_μ,ν, a contradiction to ω_t^⋆∈ J = I. Thus, inf_t∈[0,T]ω_t > l and symmetrically sup_t∈[0,T]ω_t < r. This proves the claim. It follows that (ω_T-ω_t)ψ'(ω_t) is bounded over t∈[0,T] because the subderivative ψ' is bounded on compact subsets of I.

(ii): Suppose that J = [l,r), i.e., ν has an atom in l > -∞. If ω evolves in I, then we can argue as in (i). We may thus assume that t^⋆ := inf{ t∈ [0,T] : ω_t = l }∈ (0,T]. Then, as ω is càdlàg and by the definition of Ω_μ,ν, we have ω_u = l for all u ∈ [t^⋆,T]). In particular, ω_T = l and it is enough to show that [0,t^⋆)∋ t ↦ (ω_T - ω_t)ψ'(ω_t) is bounded.

We can argue similarly as in (i) that r':=sup_t∈[0,T]ω_t < r, so that the path ω evolves in the compact interval [l,r']. Because ψ is convex and finite on (l,r), ψ' is bounded from above on [l,r']. It follows that t↦(ω_T-ω_t)ψ'(ω_t) is bounded from below on [0,t^⋆). To show that this function is also bounded from above, we observe that by the convexity of ψ,

    (ω_T-ω_t)ψ'(ω_t)
    ≤ψ(ω_T) - ψ(ω_t)
    =ψ(l) - ψ(ω_t).

Now ψ(l) is finite because ν has an atom at l, and ψ is bounded from below on [l,r'] because it is finite and convex on the compact interval [l,r']. Using this in (<ref>) shows the assertion.

(iii): Suppose that J = [l,r], i.e., ν has atoms at l > -∞ and r<∞. As in (ii), we may assume that ω hits one of the endpoints of J before T. By symmetry, we may assume that ω hits l. By definition of Ω_μ,ν, the path ω is then bounded away from the right endpoint r (otherwise it would be captured in r), i.e., sup_t∈[0,T]ω_t < r. Now the same argument as in (ii) proves the assertion.


The second technical lemma is an adaptation of <cit.> to our setting. It is used to show the admissibility condition (<ref>) of the semi-static trading strategy in Proposition <ref>.



Let (φ,ψ) ∈ L^c(μ,ν) and let g_0,g_1:J → be Borel. Let τ be a [0,T]-valued -stopping time such that

    φ(X_0) + ψ(X_T) + g_0(X_0)(X_τ - X_0) + g_1(X_τ)(X_T - X_τ)

is bounded from below on Ω_μ,ν. Then for all P ∈(μ,ν),

    [P]φ(X_0) + ψ(X_T) + g_0(X_0)(X_τ - X_0) + g_1(X_τ)(X_T - X_τ)
    = μ(φ)+ν(ψ).




Let χ be a concave moderator for (φ,ψ) with respect to μ≤_c ν and let θ be the law of X_τ. By optional stopping, μ≤_c θ≤_c ν. We expand (<ref>) to

    (φ - χ)(X_0)    + (ψ + χ)(X_T)
    + [χ(X_0) - χ(X_T) 
       + g_0(X_0)(X_τ - X_0) + g_1(X_τ)(X_T - X_τ)],

and observe that the first two terms are P-integrable. Then the assumed lower bound yields that the last term has a P-integrable negative part. We can therefore apply Fubini's theorem and evaluate its integral iteratively. To this end, let Q be the law of (X_0,X_τ,X_T) on the canonical space ^3 with a disintegration 

    Q
       = μ( x_0) ⊗κ_0(x_0, x_1) ⊗κ_1(x_0,x_1, x_2)

for martingale kernels κ_0 and κ_1. In view of the definition of μ(φ) + ν(ψ) in (<ref>), we have to show that the P-expectation of the last term in (<ref>) is (μ-ν)(χ).

To this end, we observe that for μ⊗κ_0-a.e. (x_0,x_1) ∈ J^2,

    ∫_J    [ χ(x_0) - χ(x_2) + g_0(x_0)(x_1-x_0) + g_1(x_1)(x_2-x_1) ] κ_1(x_0,x_1, x_2) 
        = ∫_J [χ(x_0) - χ(x_2) + g_0(x_0)(x_1-x_0) ] κ_1(x_0,x_1, x_2)
        = χ(x_0) - ∫_J χ(x_2) κ_1(x_0,x_1, x_2) + g_0(x_0)(x_1-x_0).

Integrating the left-hand side of (<ref>) against μ⊗κ_0 gives the P-expectation of the last term in (<ref>). It thus remains to show that the corresponding integral of the right-hand side equals (μ-ν)(χ). Integrating the right-hand side of (<ref>) first against κ_0(x_0, x_1) yields for μ-a.e. x_0∈ J,

    χ(x_0) - ∫_Jχ(x_2)  κ(x_0, x_2),

where κ(x_0,·) = ∫_J κ_1(x_1,·) κ_0(x_0, x_1) is again a martingale kernel. Finally, the integral of (<ref>) against μ is

    ∫_J [ χ(x_0) - ∫_Jχ(x_2)  κ(x_0, x_2) ] μ( x_0).

Noting that μ⊗κ is a disintegration of a one-step martingale measure on ^2 with marginals μ and ν, the last term equals (μ-ν)(χ) by Lemma <ref>.



First, we show that (φ,ψ,H) is a semi-static trading strategy. As h and A are clearly -adapted and (φ,ψ)∈ L^c(μ,ν) by assumption, it remains to check condition (<ref>) (with Y^A replaced by A). So fix an averaging process A and note that H^A = (h^A,A). The only nontrivial part in proving H^A ∈ L(Ω_μ,ν) is to show that (X_T-X_t)h^A_t is A-integrable on (0,T] for each path in Ω_μ,ν. To this end, note that φ'(X_0) and ψ'(X_0) are finite because X_0 ∈ I. It thus suffices to show that (X_T-X_t)ψ'(X_t) is bounded on [0,T] for each path in Ω_μ,ν; this is the content of Lemma <ref>.

Second, we show the superhedging property (<ref>). Fix an averaging process A and a path in Ω_μ,ν. To ease the notation, we write h instead of h^A in the following. Note, however, that h^A has the same formal expression as h in (<ref>), but with A being the fixed averaging process (and not the second component of the canonical process on Ω̂).

Using the definitions of H♢ X_T and h as well as the fact that A_0 = Δ A_0, we obtain

    H ♢ X_T
       = (X_T-X_0)h_0 + ∫_(0,T](X_T-X_t) h_t  A_t
       = (X_T-X_0)φ'(X_0) - ∫_[0,T] (X_T-X_t) (φ'(X_0) + ψ'(X_t))  A_t.

Then, using that A is a probability measure on [0,T], the concavity of φ and the convexity of ψ, and Jensen's inequality, we can estimate

    H ♢ X_T
       = ∫_[0,T]φ'(X_0)(X_t - X_0)  A_t - ∫_[0,T]ψ'(X_t)(X_T - X_t)  A_t 
       ≥φ'(X_0) (∫_[0,T] X_t  A_t - X_0) - ∫_[0,T]( ψ(X_T) - ψ(X_t))  A_t 
       ≥φ(∫_[0,T] X_t  A_t) - φ(X_0) - ψ(X_T) + ∫_[0,T]ψ(X_t)  A_t 
       ≥φ(∫_[0,T] X_t  A_t) - φ(X_0) - ψ(X_T) + ψ(∫_[0,T] X_t  A_t).

Rearranging terms and using that φ + ψ≥ f on J, we find

    φ(X_0) + ψ(X_T) + H ♢ X_T ≥ f(∫_[0,T] X_t  A_t).


Third, we show the admissibility condition (<ref>). Fix an averaging process A and P ∈(μ,ν). Define the family of -stopping times C_s, s∈(0,1), by

    C_s
       = inf{ t ∈ [0,T] : A_t > s }

and note that 0 ≤ C_s ≤ T for s∈(0,1) because A_T =1. Then using the family C_s as a time change (cf. <cit.>) for the integral in (<ref>) yields

    φ(X_0) + ψ(X_T) + H♢ X_T 
        =∫_0^1 {φ(X_0) + ψ(X_T) +φ'(X_0) (X_C_s - X_0) - ψ'(X_C_s)(X_T-X_C_s)} s.

Now, suppose that the integrand in (<ref>) is bounded from below, uniformly over s∈(0,1) and ω∈Ω_μ,ν. Then by Lemma <ref>, the P-expectation of the integrand equals μ(φ) + ν(ψ) for each s∈(0,1). Using this together with Tonelli's theorem and (<ref>) gives

    [P]φ(X_0) + ψ(X_T) + H ♢ X_T   = μ(φ) + ν(ψ),

so that (<ref>) holds.

It remains to show that the integrand in (<ref>) is uniformly bounded from below. This follows from concavity of φ and convexity of ψ together with the fact that φ + ψ≥ f ≥ 0 on J:

    φ(X_0)    + ψ(X_T) +φ'(X_0) (X_t - X_0) - ψ'(X_t)(X_T-X_t)
        ≥φ(X_t) + ψ(X_t)
    ≥ f(X_t)
    ≥ 0,    t∈[0,T].

This completes the proof.






 §.§ Duality




We now turn to the duality between the auxiliary problems _μ,ν(f) and _μ,ν(f).



Let μ≤_c ν be irreducible with domain (I,J) and let f: → [0,∞].


  * If f is upper semianalytic, then _μ,ν(f) = _μ,ν(f) ∈ [0,∞].

  * If _μ,ν(f) < ∞, then there exists a dual minimizer (φ,ψ) ∈_μ,ν(f).



A couple of remarks are in order.


We only state the duality for one irreducible component. One can formulate and prove the full duality for arbitrary marginals μ≤_c ν in analogy to <cit.>. We omit the details in the interest of brevity.




The lower bound on f in Theorem <ref> can be relaxed. Indeed, suppose that f:→ is upper semianalytic and bounded from below by an affine function g.

We first consider the primal problem. Because g is affine and any μ≤_cθ≤_cν has the same mass and barycenter as μ,

    θ(f-g)
       = θ(f) - θ(g)
    = θ(f) - μ(g).

Thus,

    _μ,ν(f-g)
       = _μ,ν(f) - μ(g).


Regarding the dual problem, we note that (φ,ψ)∈_μ,ν(f-g) if and only if (φ+g,ψ)∈_μ,ν(f) and that by Lemma <ref> (iii),

    μ(φ) + ν(ψ)
       = {μ(φ+g) + ν(ψ) } - μ(g).

Hence,

    _μ,ν(f-g)
       = _μ,ν(f) - μ(g).

Because f-g is nonnegative, the left-hand sides of (<ref>)–(<ref>) coincide by Theorem <ref> (i). Therefore, _μ,ν(f) = _μ,ν(f) ∈ (-∞,∞].

Moreover, if _μ,ν(f) < ∞, then also _μ,ν(f-g) < ∞ and a dual minimizer (φ,ψ)∈_μ,ν(f-g) for _μ,ν(f-g) exists by Theorem <ref> (ii). Now the above shows that (φ+g,ψ) ∈_μ,ν(f) is a dual minimizer for _μ,ν(f).


The proof of Theorem <ref> is based on several preparatory results. We start with the crucial closedness property of the dual space in the spirit of <cit.>.



Let μ≤_c ν be irreducible with domain (I,J), let f,f_n: J → [0,∞] be such that f_n → f pointwise, and let (φ_n,ψ_n) ∈_μ,ν(f_n) with sup_n {μ(φ_n) + ν(ψ_n)} < ∞. Then there is (φ,ψ) ∈_μ,ν(f) such that μ(φ) + ν(ψ) ≤lim inf_n →∞{μ(φ_n) + ν(ψ_n) }.



Let h_n = φ_n':I→ be a superderivative of the concave function φ_n. As

    φ_n(x) + ψ_n(y) + h_n(x)(y-x)
       ≥φ_n(y) + ψ_n(y)
    ≥ f_n(y) ≥ 0,    (x,y) ∈ I× J,

(φ_n,ψ_n,h_n) is in the dual space ^c_μ,ν(0) of <cit.>. Hence, following the line of reasoning in the proof of <cit.> (which is based on Komlos' lemma; we recall that convex combinations of convex (concave) functions are again convex (concave)), we may assume without loss of generality that 

    φ_n →φ̅  μ-a.e.  and  ψ_n →ψ̅  ν-a.e.

for some (φ̅,ψ̅)∈ L^c(μ,ν). Moreover, the arguments in <cit.> also show that μ(φ̅) + ν(ψ̅) ≤lim inf_n →∞{μ(φ_n) + ν(ψ_n) }.

Now, define the functions φ,ψ:J→ by φ := lim inf_n→∞φ_n and ψ := lim sup_n→∞ψ_n. Then φ is convex, ψ is concave, φ = φ̅ μ-a.e., and ψ = ψ̅ ν-a.e. In particular, (φ,ψ) ∈ L^c(μ,ν) and μ(φ) + ν(ψ) ≤lim inf_n →∞{μ(φ_n) + ν(ψ_n) }. Furthermore, as φ_k + ψ_k ≥ f_k on J, we have for each n that

    inf_k ≥ nφ_k + sup_k ≥ nψ_k
       ≥inf_k ≥ n (φ_k + ψ_k)
    ≥inf_k ≥ n f_k  on  J.

Sending n→∞ gives φ + ψ≥ f. In summary, (φ,ψ) ∈_μ,ν(f).


We proceed to show strong duality for bounded upper semicontinuous functions. 



Let f: → [0,∞] be bounded and upper semicontinuous. Then _μ,ν(f) = _μ,ν(f).


The proof is based on a Hahn–Banach separation argument similar to <cit.>.


We first show the weak duality inequality. Let μ≤_cθ≤_cν and (φ,ψ)∈_μ,ν(f). In particular, φ+ψ is bounded from below. Then by Lemma <ref> (iii)–(iv), 

    θ(f)
       ≤θ(φ+ψ)
    = θ(φ) + θ(ψ)
    ≤θ(φ) + ν(ψ)
    ≤μ(φ) + ν(ψ),

and the inequality _μ,ν(f) ≤_μ,ν(f) follows.

The converse inequality is based on a Hahn–Banach argument, so let us introduce a suitable space. By the de la Vallée–Poussin theorem, there is an increasing convex function ζ_ν:_+→_+ of superlinear growth such that x ↦ζ_ν(|x|) is ν-integrable. Now, set ζ(x) = 1+ζ_ν(|x|), x ∈, and denote by C_ζ the space of all continuous functions f:→ such that f/ζ vanishes at infinity. We endow C_ζ with the norm f_ζ := f/ζ_∞. With this notation, the same arguments as in the proof of <cit.> show that the dual space C^*_ζ of continuous linear functionals on C_ζ can be represented by finite signed measures.

Fix f ∈ C_ζ. Then

    -ζ(x) f _ζ   ≤ f(x)
    ≤ζ(x) f _ζ,   x ∈ J.

Because ζ_ν is convex and x ↦ζ_ν(|x|) is ν-integrable, we have θ(ζ) ≤ν(ζ) < ∞ for all μ≤_cθ≤_cν. This together with (<ref>) shows that _μ,ν(f) is finite. Thus, adding a suitable constant to f, we may assume that _μ,ν(f) = 0. For the following Hahn–Banach argument, we consider the convex cone

    K
       := {g ∈ C_ζ : _μ,ν(g) ≤ 0}.

Proposition <ref> implies that K is closed.

Suppose for the sake of contradiction that _μ,ν(f) > 0. Then, by the Hahn–Banach theorem, K and f can be strictly separated by a continuous linear functional on C_ζ. That is, there is a finite signed measure ρ such that ρ(f) > 0 and ρ(g) ≤ 0 for all g∈ K. For any compactly supported nonnegative continuous function g∈ C_ζ, we have _μ,ν(-g) ≤ 0. That is, -g ∈ K and hence ρ(-g) ≤ 0. This shows that ρ is a (nonnegative) finite measure. Multiplying ρ by a positive constant if necessary, we may assume that ρ has the same mass as μ and ν. Next, let ψ be convex and of linear growth. Then ψν() - ν(ψ)∈ K and -ψμ() + μ(ψ) ∈ K. Using that ρ≤0 for these two functions yields μ(ψ) ≤ρ(ψ) ≤ν(ψ). We conclude that μ≤_cρ≤_cν. But now ρ(f) > 0 contradicts _μ,ν(f) = 0. Thus, _μ,ν(f) ≤_μ,ν(f).

Finally, let f be bounded and upper semicontinuous and choose f_n ∈ C_b() ⊆ C_ζ such that f_n ↘ f. By the above, we have _μ,ν(f_n) = _μ,ν(f_n) for all n. We show below that lim_n→∞_μ,ν(f_n) = _μ,ν(f). Using this and the monotonicity of _μ,ν, we obtain

    _μ,ν(f)
       ≤lim_n→∞_μ,ν(f_n)
    = lim_n→∞_μ,ν(f_n)
    = _μ,ν(f)
    ≤_μ,ν(f).

So strong duality holds for bounded upper semicontinuous functions.

It remains to argue that lim_n→∞_μ,ν(f_n) = _μ,ν(f). We show more generally that _μ,ν is continuous along decreasing sequences of bounded upper semicontinuous functions. So let f_n ↘ f be a convergent sequence of bounded upper semicontinuous functions. Fix ε > 0 and set ℓ := lim_n→∞_μ,ν(f_n). Then for each n, ℓ≤_μ,ν(f_n) < ∞ and thus the set

    A_n
       := {μ≤_c θ≤_c ν : θ(f_n) ≥ℓ - ε}

is nonempty. Moreover, each A_n is a closed subset of the weakly compact set {θ : μ≤_c θ≤_c ν} and A_n+1⊆ A_n. Therefore, there exists a θ' in the intersection ∩_n ≥ 1 A_n. We then obtain by monotone convergence that

    _μ,ν(f)
       ≥θ'(f)
    = lim_n→∞θ'(f_n) ≥ℓ - ε.

This implies that _μ,ν(f) ≥ℓ as ε was arbitrary. The converse inequality follows from the monotonicity of _μ,ν. This completes the proof.



(i): This is a consequence of Lemma <ref> and a capacitability argument that is almost verbatim to <cit.>. The same arguments can be found in <cit.>. We therefore omit these elaborations.

(ii): Applying Proposition <ref> to the constant sequence f_n = f and a minimizing sequence (φ_n,ψ_n) ∈_μ,ν(f) of _μ,ν(f) yields a dual minimizer.


We are now in a position to prove the duality between the robust pricing and superhedging problems.


By Proposition <ref>, Lemma <ref>, and Proposition <ref>,

    _μ,ν(f)
       ≤_μ,ν(f,)
    ≤_μ,ν(f,)
    ≤_μ,ν(f),

and Theorem <ref> shows that _μ,ν(f) = _μ,ν(f). Hence,

    _μ,ν(f)
       = _μ,ν(f,)
    = _μ,ν(f,)
    = _μ,ν(f).

In particular, the quantities in (<ref>) are all independent of the choice of  (as long as one of the two conditions of Theorem <ref> holds).

If _μ,ν(f,) < ∞, then _μ,ν(f) < ∞ and hence there is an optimizer (φ,ψ) ∈_μ,ν(f) for _μ,ν(f). Then Proposition <ref> provides an H = (h,A) such that (φ,ψ,H) ∈_μ,ν(f,). By (<ref>) and the definition of _μ,ν(f,), (φ,ψ,H) is an optimizer for _μ,ν(f,).




Strong duality (without dual attainment) for the robust pricing and superhedging problems continues to hold if we restrict ourselves to trading strategies whose dynamic part is of finite variation.

First, observe that the process Ĥ defined by (<ref>) is of finite variation when h is bounded. Recalling the definition (<ref>) of h in Proposition <ref>, we see that h is bounded on {ω : ω_T ∈ J^∘} as these paths are bounded in a compact subset of J, on which ψ' is bounded. This will more generally hold for almost all paths if ψ' is uniformly bounded on J. Therefore, strong duality (and dual attainment in strategies of finite variation) holds if J is open.

Second, consider the case J = [a,b) for some -∞ < a < b ≤∞. Suppose that the assumptions of Theorem <ref> hold and that _μ,ν(f,) < ∞, and let (φ,ψ) ∈_μ,ν(f) be a dual auxiliary optimizer. Then ψ(a) < ∞ as ν has an atom at a (cf. Lemma <ref>). If ψ'(a) > -∞, then the same argument as above shows that the dynamic trading strategy constructed in Proposition <ref> is of finite variation. If ψ'(a) = -∞, then we construct a sequence of functions

    ψ_k(x)
    := ψ(x)    for  x ≥ a+1k, 
    ψ(a) + k(x-a) (ψ(a+1k) - ψ(a))    for  x < a+1k.

that approximates ψ by linear interpolation on the interval [a,a+1/k]. We then have ψ_k ↘ψ and μ(φ) + ν(ψ_k) ↘μ(φ) + ν(ψ) = _μ,ν(f) as k→∞. Since ψ'_k(a) > -∞, the associated process H^(k) is of finite variation almost surely. The cases J = (a,b] and J = [a,b] are analogous.





 §.§ Structure of primal and dual optimizers




If a primal optimizer to the auxiliary problem exists, we can derive some necessary properties for the dual optimizer.



Let μ≤_c ν be irreducible with domain (I,J) and let f: → [0,∞] be Borel. Suppose that _μ,ν(f) = _μ,ν(f), that μ≤_cθ≤_cν is an optimizer for _μ,ν(f), and that (φ,ψ)∈_μ,ν(f) is an optimizer for _μ,ν(f). Then


  * φ+ψ = f θ-a.e.,

  * φ is affine on the connected components of {u_μ < u_θ},

  * ψ is affine on the connected components of {u_θ < u_ν},

  * φ does not have a jump at a finite endpoint b of J if θ({b}) > 0, and

  * ψ does not have a jump at a finite endpoint b of J if θ({b}) < ν({b}).




As in the proof of Lemma <ref>, we obtain (cf. (<ref>)) that

    θ(f)
       ≤θ(φ+ψ)
    ≤μ(φ) + ν(ψ).

By the absence of a duality gap as well as the optimality of θ and (φ,ψ), all inequalities are equalities:

    θ(f)
       = θ(φ+ψ)
    = μ(φ) + ν(ψ).


Now (i) follows from the first equality in (<ref>) and the fact that φ+ψ≥ f on J. Rearranging the second equality, we can write

    0
       = {μ(φ) + ν(ψ) } - θ(φ+ψ) 
       = {μ(φ) + ν(ψ) } - {θ(φ) + ν(ψ)} + {θ(φ) + ν(ψ)}- θ(φ+ψ).

Using the definition (<ref>) of the first three expressions (using φ as a concave moderator for the first two terms and -ψ for the third; cf. Lemma <ref> (i)), we obtain

    0
       = (μ - ν)(φ) - (θ - ν)(φ) + (θ - ν)(-ψ) 
       = (μ - θ)(φ)+(θ-ν)(-ψ),

where the last equality is a direct consequence of the definitions of (μ - ν)(φ) and (θ - ν)(φ) (cf. (<ref>)). Both terms on the right-hand side of (<ref>) are nonnegative by definition and hence must vanish:

    0
       = (μ - θ)(φ)
    = ∫_I (u_μ - u_θ) φ” + ∫_J ∖ I |Δφ| θ

and similarly for (θ-ν)(-ψ). This implies that φ” = 0 on {u_μ < u_θ} (which is assertion (ii)) and that |Δφ| = 0 for every endpoint of J on which θ has an atom (which is assertion (iv)). The proofs of (iii) and (v) are similar.


The next result shows that for upper semicontinuous f, there is a maximizer for _μ,ν(f) which is maximal with respect to the convex order. We omit the proof in the interest of brevity.



Let μ≤_c ν be irreducible and let f:→ [0,∞] be upper semicontinuous and bounded from above by a convex, continuous, and ν-integrable function. Furthermore, fix a strictly convex function g:→ with linear growth, and consider the “secondary” optimization problem

    sup_θ∈Θ(f)θ(g),

where Θ(f) := {θ : μ≤_c θ≤_c ν  and  θ(f) ≥_μ,ν(f) } is the set of optimizers of the auxiliary primal problem.


  * Θ(f) is non-empty, convex, and weakly compact and (<ref>) admits an optimizer.

  * Any optimizer θ of (<ref>) has the following properties:


  * θ is maximal in Θ(f) with respect to the convex order.

  * If O is an open interval such that O ⊆{ u_θ < u_ν} and f|_O is convex, then θ(O) = 0.

  * If K is an interval such that K^∘⊆{ u_μ < u_θ}, f|_K is strictly concave, and θ(K)>0, then θ|_K is concentrated in a single atom.




The following example shows that the set optimizers for _μ,ν(f) can have multiple maximal or minimal elements with respect to the convex order; there is in general no greatest or least element for this partially ordered set.


Let μ = δ_0 and ν = 1/3(δ_-1 + δ_0 + δ_1) and let f be piecewise linear with f(-1) = f(1) = 3, f(-1/2) = f(1/2) = 2, and f(0) = 0. We claim that there is no greatest or least primal optimizer.

We construct candidate primal and dual optimizers as follows. On the primal side, set θ_1 = 23δ_-1/2 + 13δ_1 and θ_2 = 13δ_-1 + 23δ_1/2. On the dual side, set φ≡ 0 and let ψ be the convex function that interpolates linearly between ψ(-1) = ψ(1) = 3 and ψ(0) = 1. Direct computations yield θ_1(f) = θ_2(f) = 7/3 = ν(ψ) which shows that θ_1 and θ_2 are primal optimizers and that (φ,ψ) is a dual optimizer.

First, we show that there is no primal optimizer which dominates both θ_1 and θ_2 in convex order. Indeed, one can check that u_ν = max(u_θ_1,u_θ_2), so that ν is the only feasible primal element which dominates both θ_1 and θ_2 in convex order. But ν(f) = 2 < 7/3 and therefore ν is not optimal.

Second, we show that there is no primal optimizer which is dominated by both θ_1 and θ_2. Indeed, one can check that { u_μ < min(u_θ_1,u_θ_2)} = (-1/2,1/2), so that every feasible primal element that is dominated by both θ_1 and θ_2 must be concentrated on [-1/2,1/2]. But f ≤ 2 on [-1/2,1/2], so that no primal optimizer can be concentrated on this interval.


We conclude this section with an example that shows that primal attainment does not hold in general if f is not upper semicontinuous.



Let μ = δ_0, ν = 1/2(δ_-1+δ_1), and set f(x) := |x|_(-1,1)(x). Then μ≤_c ν is irreducible with domain ((-1,1),[-1,1]). Considering the sequence θ_n := 1/2(δ_-1+1/n + δ_1-1/n), one can see that _μ,ν(f) ≥ 1. But there is no μ≤_c θ≤_c ν such that θ(f) ≥ 1 because f < 1 on [-1,1].





§ EXAMPLES




Two common payoff functions are risk reversals and butterfly spreads. In this section, we provide solutions to the auxiliary primal and dual problems for these payoffs. Throughout this section, we fix irreducible marginals μ≤_c ν and denote their common total mass and first moment by m_0 and m_1, respectively.




 §.§ Risk reversals



The payoff function of a risk reversal is of the form

    f(x)
       = -(a-x)_+ + (x-b)_+,

for fixed a<b. The following result provides a simple geometric construction of the primal and dual optimizers in terms of the potential functions of μ and ν.[The authors thank David Hobson for the idea of this construction.] We recall that any convex function u lying between the potential functions u_μ and u_ν is the potential function of a measure θ which is in convex order between μ and ν (cf., e.g., <cit.>).





Consider the line through the point (a,u_μ(a)) of maximal slope lying below (or on) the graph of u_ν; cf. Figure <ref>. This line is either (i) a tangent line to the graph of u_ν with a tangent point (z,u_ν(z)) for some z ∈ (a,∞) or (ii) the asymptote line for the graph of u_ν near +∞.[Note that case (ii) can only happen when (a,μ(a)) lies on the increasing part of the dashed potential function in Figure <ref>. In particular, in this case, μ is concentrated on the left of a.]

In case (i), define the concave function φ and the convex function ψ by

    φ(x)
       =
     - α (x-a)_+,
    ψ(x)
       = x-a + α(x-(z∨ b))_+,

where α = (b-a)/((z∨ b)-a). Moreover, let u be the unique convex function that coincides with u_μ on (-∞,a] and with u_ν on [z,∞) and is affine on [a,z] (i.e., u coincides on [a,z] with the tangent line considered above). Denote by θ the unique measure with potential function u_θ = u. In case (ii), set φ(x) = 0, ψ(x) = x-a, and θ = μ.

Then, θ is an optimizer for the auxiliary primal problem _μ,ν(f), (φ,ψ) is an optimizer for the auxiliary dual problem _μ,ν(f), and the common optimal optimal value is given in terms of the potential functions of μ and ν by

    _μ,ν(f)
    = _μ,ν(f)
    = 
    
    m_1 - a+b/2m_0 + b-a/2u_ν(z∨ b) - u_μ(a)/(z∨ b)-a   in case (i),
    
    m_1 - a m_0    in case (ii).




We first note that θ and (φ, ψ) are admissible elements for the auxiliary primal and dual problems, respectively. Indeed, by construction, u_θ is convex and lies between u_μ and u_θ. Thus, the associated measure θ satisfies μ≤_c θ≤_c ν. Moreover, a straightforward computation shows that φ + ψ≥ f, and (φ,ψ) ∈ L^c(μ,ν) by Lemma <ref>.

By the weak duality inequality (<ref>) (this also holds if f is bounded from below by an affine function; cf. Remark <ref>), θ(f) ≤μ(φ) + ν(ψ) holds for any admissible primal and dual elements. It thus suffices to show that θ(f) = μ(φ) + ν(ψ) for our particular choices for θ and (φ,ψ). 

Case (i): Using the identity (t-s)_+ = 1/2(| t-s | + t-s), the integrals θ(f), μ(φ), and ν(ψ) can be expressed in terms of the potential functions of μ,θ, and ν as follows:

    θ(f)
       = 1/2(u_θ(b) - u_θ(a)) + m_1 - a+b/2m_0,
    μ(φ)
       = -α/2(u_μ(a) + m_1 - am_0),
    ν(ψ)
       = m_1 - am_0 + α/2(u_ν(z ∨ b) + m_1 - (z∨ b)m_0).

Substituting α = (b-a)/((z∨ b)-a) and simplifying gives

    μ(φ) + ν(ψ)
       = m_1 + 1/2((a - (z∨ b)) α - 2a) m_0 - α/2(u_μ(a) - u_ν(z∨ b))
       = m_1 - a+b/2m_0 + b-a/2u_ν(z∨ b) - u_μ(a)/(z∨ b)-a.

Hence,

    θ(f) - (μ(φ) + ν(ψ))
       = b-a/2(u_θ(b) - u_θ(a)/b-a - u_ν(z∨ b) - u_μ(a)/(z∨ b)-a),

and it suffices to show that the two quotients inside the brackets are equal. To this end, we distinguish two cases. On the one hand, if z ≤ b, it is enough to observe that u_θ(a) = u_μ(a) and u_θ(b) = u_ν(b) by the construction of u = u_θ. On the other hand, if z ≥ b, then the two quotients are the same because u = u_θ is affine on [a, z] ⊃ [a,b] and coincides with u_μ at a and with u_ν at z.

Case (ii): On the one hand, since θ = μ and μ is concentrated on the left of a, we have θ(f) = μ(f)
= m_1-am_0. On the other hand, μ(φ) + ν(ψ) = ∫ (x-a)  ν( x) = m_1 - am_0.






 §.§ Butterfly spreads



The payoff function of a butterfly spread is of the form

    f(x)
       = (x-(a-h))_+ - 2(x-a)_+ + (x-(a+h))_+,

for fixed a and h > 0. We have the following analog to Proposition <ref>; we omit the proof.





Consider the two lines l_+,l_- through the point (a,u_μ(a)) of maximal and minimal slope, respectively, lying below (or on) the graph of u_ν. We distinguish the cases (i+) l_+ is a tangent line with tangent point (z_+,u_ν(z_+)),
(ii+) l_+ is an asymptote, (i-) l_- is a tangent line with tangent point (z_-,u_ν(z_-)), (ii-) l_- is 
an asymptote. In case (ii±), we set z_± = ±∞.

Let u be the convex function that coincides with u_ν on (-∞,z_-] ∪ [z_+,∞) and is affine on [z_-,a] and on [a,z_+], and define the concave function φ and the convex function ψ by

    φ(x)
       = -(α+β)(x-a)_+,
    ψ(x)
       = α(x-(z_- ∧ (a-h)))_+ + β(x-(z_+ ∨ (a+h)))_+,

where α = h/a-(z_- ∧ (a-h)) and β = h/(z_+ ∨ (a+h)) - a. Here, in the asymptote cases (ii±), φ,ψ need to be interpreted as the limiting functions that arise as z_±→±∞.[For instance, if z_- = -∞ and z_+ < ∞, then φ(x) = -β(x-a)_+ and ψ(x) = h + β(x-(z_+ ∨ (a+h)))_+.]

Then, the intermediate law θ with potential function u_θ = u is an optimizer for the auxiliary primal problem _μ,ν(f), (φ,ψ) is an optimizer for the auxiliary dual problem _μ,ν(f), and the common optimal optimal value is given in terms of the potential functions of μ and ν by

    _μ,ν(f)
    = _μ,ν(f)
    = h2 (s_+ + s_-),

where

    s_+
       =
    u_ν(z_+ ∨ (a+h)) - u_μ(a)/(z_+ ∨ (a+h)) - a   in case (i+),
    
    m_0    in case (ii+),
    
    
    s_-
       =
    u_ν(z_- ∧ (a-h))-u_μ(a)/a-(z_- ∧ (a-h))   in case (i-),
    
    -m_0    in case (ii-).







§ COUNTEREXAMPLES




In this section, we give four counterexamples. Example <ref> shows that strong duality for the auxiliary problems may fail for general (not necessarily irreducible) marginals if the dual elements φ,ψ are required to be globally concave and convex, respectively. Example <ref> shows that strong duality may fail if the dual elements φ and ψ are required to be μ- and ν-integrable, respectively.
Example <ref> shows that the robust model-based prices of Asian- and American-style derivatives are typically not equivalent when more than two marginals are given. Example <ref> shows that the equality _μ,ν(f,)= _μ,ν(f) may fail when the assumptions of Proposition <ref> are violated.

[Duality gap with globally convex/concave dual elements]

Let μ = 1/2δ_-1 + 1/2δ_1, let ν be the uniform distribution on (-2,2), and set f(x) := | x |^-1/2, x ∈ (with f(0) = ∞).

First, we show that _μ,ν(f) is finite. Fix any μ≤_cθ≤_cν. Computing the potential functions u_μ and u_ν shows that μ≤_c ν and that { u_μ < u_ν} = I_1 ∪ I_2 with I_1 = (-2,0) and I_2 = (0,2). Because ν does not have an atom at the common boundary 0 of I_1 and I_2, also θ cannot have an atom at 0. Thus, we can write θ = θ_1 + θ_2 with

    1/2δ_-1≤_c θ_1
    ≤_c ν|_I_1  and  1/2δ_1≤_c θ_2
    ≤_c ν|_I_2.

Since f is convex when restricted to I_1 or I_2, we have

    θ(f)
    = θ_1(f) + θ_2(f)
    ≤ν|_I_1(f) + ν|_I_2(f)
    = ν(f)
    
    < ∞.

It follows _μ,ν(f) = ν(f) < ∞.

Second, let φ be concave and ψ be convex such that φ + ψ≥ f. We show that then necessarily μ(φ) + ν(ψ) = ∞. To this end, we may assume that φ < ∞ on (μ) = { -1, 1}. Then φ < ∞ everywhere by concavity. Thus, evaluating φ + ψ≥ f at 0 implies that ψ(0) = ∞. Therefore, ψ = ∞ on (-∞,0] or on [0,∞) by the convexity of ψ. In both cases, we have μ(φ) + ν(ψ) = ∞.


[Duality gap with individually integrable dual elements]

We consider the marginals

    μ
    := C ∑_n≥ 1 n^-3μ_n
      and  ν
    := C∑_n≥ 1 n^-3ν_n,

where C := (∑_n≥ 1 n^-3)^-1, μ_n
:= δ_n and ν_n
:= 1/3(δ_n-1+δ_n + δ_n+1)
 for n≥ 1.
These are the same marginals as in <cit.> where it is shown that μ≤_cν is irreducible with domain ((0,∞),[0,∞)). We now let f:_+→[0,1] be the piecewise linear function through the points given by f(n) = 0 and f(2n + 12) = 14 for n ≥ 0; cf. Figure <ref>.

We proceed to construct candidates for optimizers for _μ,ν(f) and _μ,ν(f). For the primal problem, define the sequence (θ̅_n)_n ≥ 1 by

    θ̅_n
       =
    1/3(δ_n-1 + 2δ_n+1/2)    for  n  even, 
    1/3(2δ_n-1/2 + δ_n+1)    for  n  odd,

and set θ̅:= C∑_n≥ 1 n^-3θ̅_n. One can check that μ_n ≤_c θ̅_n ≤_c ν_n and compute θ̅_n(f) = 1/6. Hence, μ≤_c θ̅≤_c ν (by linearity of potential functions in the measure) and θ̅(f) = 1/6.



We now turn to the dual problem. Let φ̅ and ψ̅ be the unique concave and convex functions, respectively, with second derivative measures

    -φ̅”
    = ∑_n ≥ 0δ_2n+1/2  and  ψ̅”
    = 1/2∑_n ≥ 1δ_n

and φ̅(0) = ψ̅(0) = 0, φ̅'(0) = f'(0) = 1/2, and ψ̅'(0) = 0. The “initial conditions” are chosen such that f(0) = φ̅(0) + ψ̅(0) and f'(0) = φ̅'(0) + ψ̅'(0) and the choice of the second derivative measures ensures that φ and ψ pick up the negative and positive curvature of f, respectively. Thus, φ̅+ ψ̅= f on _+ by construction. We proceed to compute μ(φ̅) + ν(ψ̅) in the sense of Definition <ref>. (The individual integrals are infinite because φ̅ and ψ̅ have quadratic growth while μ and ν have no second moments.) To this end, we note that φ̅+ ψ̅= f vanishes on the support of ν. This implies that φ̅ is a concave moderator for (φ̅,ψ̅) with respect to μ≤_cν. We can then compute

    μ(φ̅) + ν(ψ̅)
       = μ(φ̅-φ̅) + ν(ψ̅+ φ̅) +  (μ-ν)(φ̅)
    = (μ-ν)(φ̅) 
       = C ∑_n ≥ 1 n^-3 (μ_n - ν_n)(φ̅).

Fix n ≥ 1. Because φ̅ is continuous, we have

    (μ_n - ν_n)(φ̅)
    = 1/2∫_I (u_μ_n - u_ν_n) φ̅”.

The difference u_μ_n - u_ν_n vanishes outside (n-1,n+1) and on this interval, φ̅” is concentrated on either n-1/2 (if n is odd) or on n+1/2 (if n is even) with mass 1. Therefore, the right-hand side of (<ref>) collapses to 1/2(u_μ_n - u_ν_n)(n±1/2) = 1/6. It follows that μ(φ̅) + ν(ψ̅) = 1/6 = θ̅(f). Hence, by (weak) duality, θ̅ and (φ̅,ψ̅) are primal and dual optimizers, respectively.

We are now in a position to argue that no dual optimizer lies in L^1(μ) × L^1(ν). Suppose for the sake of contradiction that (φ,ψ) ∈ L^1(μ)× L^1(ν) is a dual optimizer and note that (θ̅) = { 0.5, 1, 2, 2.5, 3,…}. We have φ + ψ = f θ̅-a.e. by Proposition <ref> (i). One can show that the following modifications of (φ,ψ) do not affect its optimality nor the individual integrability of φ and ψ; we omit the tedious details. First, ψ is replaced by its piecewise linear interpolation at the atoms of ν. Second, φ is replaced by its piecewise linear interpolation at the kinks of f. Third, a suitable convex function is added to φ and subtracted from ψ (preserving their concavity and convexity, respectively) such that the second derivative measures -φ” and ψ” become singular.

Because φ + ψ = f on (θ̅) and both sides are piecewise linear, we conclude that φ + ψ = f holds on [1/2,∞). As -φ” and ψ” are singular, φ and ψ must then account for the negative and positive curvature of f, respectively. It follows that both φ and ψ have quadratic growth. Since μ and ν do not have a second moment, we conclude that μ(φ) = -∞ and ν(ψ) = ∞, a contradiction.


[Different robust model-based prices for Asian- and American-style derivatives for multiple marginals]

For n ≥ 2 given marginals μ_0≤_c μ_1 ≤_c ⋯≤_cμ_n corresponding to the time points 0,1,…,n (say), the robust model-based price _μ_0,…,μ_n(f,) is defined analogously. But this robust model-based price now depends non-trivially on , as the following example shows. Fix a strictly convex function f. On the one hand, if  corresponds to American-style derivatives, then one can check that _μ_0,…,μ_n(f,) = μ_n(f). On the other hand, for Asian-style derivatives, i.e., ' = {t ↦ t/n}, Jensen's inequality yields

    f(1/n∫_0^n X_t  t)
    ≤1/n∑_i=0^n-1 f(∫_i^i+1 X_t  t ),

so that

    _μ_0,…,μ_n(f,')
    ≤1/n∑_i=1^n μ_i(f) ≤μ_n(f).

For a generic choice of marginals, both inequalities are strict. Hence, the robust model-based price of an Asian-style derivative with a strictly convex payoff function is typically smaller than that of the corresponding American-style derivative.



[Necessity of the assumptions of Proposition <ref>]  


  * We show that _μ,ν(f,) = _μ,ν(f) may fail if  does not contain an interior averaging process. Set = { A } = {t ↦12 + 12_{ t = T }}, so that ∫_0^T X_t dA_t = (X_0 + X_T)/2, and consider f(x) = x^2. Then, using the martingale property of X under any P ∈(μ,ν), one can check that _μ,ν(f,) = (3μ(f) + ν(f))/4, whereas
_μ,ν(f) = ν(f) since f is convex. Now, choose μ and ν such that μ(f) < ν(f) (f is strictly convex). Then, _μ,ν(f,) < _μ,ν(f).


  * We show that _μ,ν(f,) = _μ,ν(f)  may fail if  contains an interior averaging process but f is not lower semicontinuous. Set = {t ↦ t/T} and f(x) = _{| x |≥ 1 }, and choose μ = δ_0 and ν = (δ_1 + δ_-1)/2. On the one hand, since ν(f) = 1 and f ≤ 1, we have _μ,ν(f) = 1. On the other hand, we claim that _μ,ν(f,) = 0. To this end, fix P ∈(μ,ν). Since P-a.e. path of X starts in 0, evolves in [-1,1], and is right-continuous, |1/T∫_0^T X_t  t | < 1 P-a.s. Thus, E^P[f(1/T∫_0^T X_t  t)] = 0. Since P ∈(μ,ν) was arbitrary, _μ,ν(f,) = 0.





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