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Robust framework for quantifying the value of information

in pricing and hedging ∗

Anna Aksamit†, Zhaoxu Hou‡and Jan Obłój§

Mathematical Institute, University of Oxford

AWB, ROQ, Oxford OX2 6GG, UK

September 5, 2018

Abstract

We pursue the robust approach to pricing and hedging of financial derivatives.We investigate when the pricing–hedging duality for a regular agent, who only ob-serves the stock prices, extends to agents with some additional information. Weintroduce a general framework to express the superhedging and market model pricesfor an informed agent. Our key insight is that an informed agent can be seen as aregular agent who can restrict her attention to a certain subset of possible paths.We use results of Hou & Obłój [22] on robust approach with beliefs to establishthe pricing–hedging duality for an informed agent. Our results cover number ofscenarios, including information arriving before trading starts, arriving after staticposition in European options is formed but before dynamic trading starts or arrivingat some point before the maturity. For the latter we show that the superhedgingvalue satisfies a suitable dynamic programming principle, which is of independentinterest.

Keywords: robust superhedging, pricing–heding duality, informed investor, asym-metry of information, enlarged filtration, path restriction, dynamic programmingprinciple, modelling with beliefs

1 Introduction

Robust approach to pricing and hedging has been an active field of research in mathe-matical finance over the recent years. In this approach, instead of choosing one model,

∗The project has been generously supported by the European Research Council under the EuropeanUnion’s Seventh Framework Programme (FP7/2007-2013) / ERC grant agreement no. 335421. Theauthors also acknowledge the support of the Oxford-Man Institute of Quantitative Finance. ZhaoxuHou is further grateful to Balliol College in Oxford and Jan Obłój to St John’s College in Oxford fortheir financial support.

†E-mail: anna.aksamit@maths.ox.ac.uk‡E-mail: zhaoxu.hou@maths.ox.ac.uk§E-mail: jan.obloj@maths.ox.ac.uk

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one considers superhedging simultaneously under a family of models, or pathwise on a setof feasible trajectories. Typically dynamic trading strategy in stocks and static trading,i.e. at time zero, in some European options are allowed. This setup was pioneered in theseminal work of Hobson [21] who obtained robust pricing and hedging bounds for look-back options. Therein, all martingale models which calibrate to the given market optionprices, and superhedging for all canonical paths, were considered. Similar approach andsetup, both in continuous and discrete time, were used to study other derivatives andabstract pricing–hedging duality questions, see e.g. [11, 15, 14, 16, 1, 7, 8, 19, 20, 12]and the references therein. Other papers, often intertwined with the previous stream, fo-cused on the case when superhedging is only required under a given family of probabilitymeasures or, more recently, on a given set of feasible paths, see e.g. [27, 4, 29, 18, 10, 22]and the references therein.

The prevailing focus in the literature has been so far on the case when trading strate-gies are adapted to the natural filtration F of the price process S. In contrast, our maininterest in this paper is to understand what happens when a larger filtration G is con-sidered. One can think of F as the filtration of an unsophisticated or small agent and G

as the filtration of a sophisticated agent who invests in acquiring additional information.Equally, G may correspond to the filtration of an insider. In general, the two filtrationsmodel an asymmetry of information between the agents. We are primarily interested inthe case when the additional information does not lead to instant arbitrage opportunitiesbut does offer an advantage and we wish to quantify this advantage in the context ofrobust pricing and hedging of derivatives.

We develop pathwise approach. Our key insight is that the pricing and hedging prob-lem for the agent with information in G can often be reduced to that for the standardagent who only considers a subset of the pathspace1. This allows us to use the dual-ity results with beliefs obtained by Hou and Obłój [22]. Specifically, we consider theprice process as the canonical process on a restriction of the space of Rd-valued contin-uous functions on [0, T ]. Price process represents underlying stocks and possibly alsocontinuously traded options. We further allow static position in options from a givenset X with given prices P. These are less liquid options which are not assumed to betraded after time zero. The additional information could arrive both before and/or afterthe static trading is executed. To account for this we add to G an additional elementG−1, ∅,Ω ⊂ G−1 ⊂ G0 and require that the static position is G−1-measurable. Thesuperhedging cost of a derivative with payoff ξ, for an agent with filtration G, is then

V GX ,P,Ω(ξ)(ω) := inf

P(X)(ω) : ∃ G-admissible (X, γ) s.t. X +

∫ T

0γudSu ≥ ξ on Ω

,

where X is a portfolio of cash and options in X . The pricing counterpart is obtained via

PGX ,P,Ω(ξ)(ω) := sup

P∈MG

X ,P,Ω

EP(ξ|G−1)(ω),

1When working on this paper we were made aware [2] of a related forthcoming paper which alsoconsiders informed agents using pathwise restrictions. However the technical setup therein is closelyrelated to the pathwise approach developed in [5], based on Vovk’s outer measure, and the paper developsa monotonicity principle in a similar sprit to [6].

2

where the supremum is taken over G-martingale measures calibrated to market prices Pof options in X and where we take a suitable version of the conditional expectations. Weshow that both V G

X ,P,Ω(ξ) and PGX ,P,Ω(ξ) are well defined and constant on atoms of G−1.

We first focus on the case of an initial enlargement: G0 = σ(Z), for some FT -measurable random variable Z. Initially, we treat the case when G−1 = G0 and showthat the duality for the informed agent is the same as duality for uninformed agentsin Hou & Obłój [22] with beliefs of the form ω : Z(ω) = c. We discuss two exam-ples: a very specific information Z = supt∈[0,T ] |St − 1| and a rather vague informationZ = 11St∈(a,b) ∀t∈[0,T ]. Subsequently, we show that the duality also extends to the caseof trivial G−1, under mild technical assumptions on Z.

Secondly, we focus on the case when the additional information is disclosed at sometime T1 ∈ (0, T ), i.e. the filtration G is of the form: Gt = Ft for t ∈ [0, T1) and Gt =Ft ∨ σ(Z) for t ∈ [T1, T ]. To prove the pricing–hedging duality we establish a dynamicprogramming principle for both the superhedging cost V (ξ) and the market model priceP (ξ). These are of independent interest even in the case of F = G.

We note that the pricing problem for an informed agent was also examined recentlyin [3]. However the focus therein is very different to ours. The authors do not study thepricing–hedging duality but instead focus on the pricing aspect PG

X ,P,Ω(ξ) with a trivialG−1. They show that it is enough to optimise over extreme measures and characterisethese, in analogy to the seminal work of Jacod-Yor [23], as ones under which semi-staticcompleteness holds, i.e. perfect hedging of all suitably integrable ξ using the underlyingassets and statically traded derivatives in X . Under some further assumptions on G,semi–static completeness under Q is equivalent to the filtrations F and G coincidingunder Q.

Our paper is organised as follows. In Section 2 we define our robust pricing andhedging setup. In Section 3 we define the relevant pricing and hedging notions for the in-formed agents and establish their characterisations via pathspace restrictions. In Section4 we present our main results on pricing–hedging duality under an initially enlarged fil-tration. Theorem 4.5 treats the case of instantly available information and Theorem 4.10the case when the information may not be used to construct static portfolios. In Section5 we study the dynamic situation when the additional information arrives at some timeT1 ∈ (0, T ). We establish relevant dynamic programming principles in Propositions 5.1and 5.2, and then we obtain pricing–hedging duality result in Theorem 5.5.

2 General set-up

2.1 Traded assets

We consider a financial market with d + 1 underlying assets: a numeraire and d ∈ N

risky underlying assets. The prices are denominated in the units of the numeraire whoseprice is thus normalised to 1. We suppose that the prices of the d risky underlying assetsbelong to the space Ωd := C1([0, T ],R

d+), i.e., the space of non-negative continuous

functions f from [0, T ] into Rd+ such that f0 = (1, ..., 1). We endow Ωd with the sup

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norm ||f ||d := supt≤T |ft| with |ft| := sup1≤n≤d |fnt | so that (Ωd, || · ||d) is a Polish space.

We will denote by FdT the Borel σ-field of subsets of (Ωd, || · ||d) which is the σ-field

generated by canonical process on Ωd.Apart from the risky underlying assets, there may be derivative products traded on

the market. Some of these could be particularly liquid – we will assume they trade dy-namically – and others may be less liquid and only available for trading at the initialtime. We only consider European derivatives which can be seen as Fd

T –measurable func-tions X : Ωd → R. In line with [22], we further assume all the payoffs X are boundedand uniformly continuous. The options traded at time zero are assumed to have a welldefined price, denoted P(X). The set of market options available for static trading isdenoted by X . We consider here a frictionless setting with no transaction costs so thatin particular P is a linear operator on X . There are further K options, K ≥ 0, with non-

negative payoffs X(c)1 , ...,X

(c)K , which are traded dynamically. We normalise their payoffs

so that their initial prices are equal to 1 and we model this situation by augmenting theset of risky assets. We simple consider d+K assets that may be traded at any time, dunderlyings and K options, with paths in an extended path space, Ωd+K , with the norm|| · || := || · ||d+K . Specifically, as the prices at maturity T must be consistent with payoffs,only the following subset of Ωd+K is considered:

Ω := ω ∈ Ωd+K : ω(d+i)T = X

(c)i (ω(1), ..., ω(d))/P(X

(c)i ) ∀i ≤ K.

The space Ω was called the information space in [22] since it encodes the informationabout the initial prices and the payoffs of continuously traded options. We note that(Ω, || · ||) is a Polish space since it is a closed subset of the Polish space (Ωd+K , || · ||).

2.2 Information

Let S be the canonical process on Ω, i.e., St(ω) := ωt. We introduce the filtrationF := (Ft)t≤T generated by S, i.e., Ft := σ(Ss : s ≤ t) for each t ∈ [0, T ]. It is a rightcontinuous filtration since S is a continuous process. Each element Ft, or more generallyFτ for a stopping time τ , is countably generated as the Borel σ-field on a Polish space.

We also consider an enlarged filtration G := (Gt)t≤T defined as Gt := Ft∨Ht, for eacht ∈ [0, T ], where H := (Ht)t≤T is another filtration. In the literature, e.g. [24, 25, 28],typically two special cases of the filtration H are studied. First, the filtration H can betaken constant, Ht = σ(Z) where Z is a random variable, and G is then called the initialenlargement of F with Z. Second, the filtration H can be taken as Ht = σ(11τ≤s : s ≤ t)where τ is a random time (non-negative random variable) and G is then called theprogressive enlargement of F with a random time τ .

In our considerations it is important to specify when the additional information arrives– some decisions may have to be taken before and some after, the additional informationis acquired. To model such situations, we add to each filtration G an additional elementG−1 with ∅,Ω ⊂ G−1 ⊂ G0. For a given arbitrary filtration G we denote by G+ thefiltration such that G+

−1 = G0 and G+t = Gt for t ∈ [0, T ]. Similarly, we denote by G−

the filtration such that G−−1 = ∅,Ω and G−

t = Gt for t ∈ [0, T ]. We note that for thenatural filtration of the price process F the only choice is F−1 = F0 = ∅,Ω.

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2.3 Trading strategies

We now discuss the notion of atoms of a σ-field and introduce the right class of tradingstrategies with respect to a general filtration G. We refer to Dellacherie & Meyer [17]Chapter 1 § 9-12 pages 14-16 for useful details.

For a measurable space (Ω,FT ) and a sub σ-field G ⊂ FT we introduce the followingequivalence relation.

Definition 2.1. Let ω and ω be two elements of Ω, and G ⊂ FT be a σ-field. Thenwe say that ω and ω are G-equivalent, and write ω ∼G ω, if for each G ∈ G we have11G(ω) = 11G(ω).

We call G-atoms the equivalence classes in Ω with respect to this relation. We denoteby Aω the atom which contains ω: Aω =

⋂A : A ∈ G, ω ∈ A.

Note that if G is countably generated, G = σ(Bn : n ≥ 1), then each atom is anelement of G as Aω =

⋂nCn is a countable intersection, where Cn = Bn if ω ∈ Bn and

Cn = Bcn if ω ∈ Bc

n. Also, it is then enough to check the relation from Definition 2.1 onthe generators of G. Finally, we note that in our setting, ω ∼Ft ω if and only if ωu = ωu

for each u ≤ t; and ω ∼σ(Z) ω if and only if Z(ω) = Z(ω).We only consider trading strategies γ which are of finite variation. This allows,

similarly to [19, 22], to define stochastic integrals pathwise simply via the integration byparts formula:

∫ t

0γ(u)dω(u) := γ(t)ω(t) − γ(0)ω(0) −

∫ t

0ω(u)dγ(u), ω ∈ Ω.

Definition 2.2. (i) We say that a mapping γ : Ω → D([0, T ],Rd+K) is G-adapted if forevery ω, ω ∈ Ω and every t ∈ [0, T ] we have

ω ∼Gt ω implies γ(ω)t = γ(ω)t. (1)

(ii) Let M ∈ L0(Ω,G0), the space of G0-measurable random variables. We say that γ :Ω → D([0, T ],Rd+K) is (G,M)-admissible if it is G-adapted and of finite variation,satisfying ∫ t

0γ(ω)udSu(ω) ≥ −M(ω) ∀ t ∈ [0, T ], ω ∈ Ω . (2)

The set of all (G,M)-admissible strategies is denoted AM (G). The set of all G-admissiblestrategies is defined by

A(G) :=⋃

M∈L0(Ω,G0)

AM (G). (3)

Finally, when the admissibility (2) is only required on some time interval [T1, T2] we writeAM (G, [T1, T2]) and A(G, [T1, T2]) for the respective sets of trading strategies.(iii) A (G,M)-admissible semi-static strategy is a pair (X, γ) where

X = a0 +m∑

i=1

aiXi and γ ∈ AM(G)

5

for some m ∈ N, Xi ∈ X and G−1-measurable random variables ai, i = 0, 1, ...,m. Initialcost of such a strategy is P(X) = a0 +

∑mi=1 aiP(Xi).

The set of all (G,M)-admissible semi-static strategies is denoted by AMX (G) and all G-

admissible semi-static strategies are given by

AX (G) :=⋃

M∈L0(Ω,G0)

AMX (G). (4)

3 Pathspace approach to information quantification

We are now in a position to define the main quantities of interest: the robust pricingand hedging prices of an option. We work with a general filtration G and this inducesfurther difficulties, as compared with the case of the natural filtration F.

We start with superhedging problem.

Definition 3.1. Let A ⊂ Ω. The G-superhedging cost of ξ on A is given by

V GX ,P,A(ξ)(ω) := infP(X)(ω) : ∃ (X, γ) ∈ AX (G) such that

X(ω(1), ..., ω(d)) +

∫ T

0γ(ω)udSu(ω) ≥ ξ(ω) ∀ω ∈ A.

Thus the G-superhedging cost of ξ on A is the infimum over all initial costs P(X)of G-admissible strategies (X, γ) ∈ AX (G) which super–replicate ξ on A over [0, T ] i.e.,X + γ ST ≥ ξ on A. When a different time horizon is considered, e.g. [T1, T2], we makeit explicit. We have the following obvious inequalities:

V G+

X ,P,A(ξ) ≤ V GX ,P,A(ξ) ≤ V G−

X ,P,A(ξ) ≤ V FX ,P,A(ξ).

We now show that the superhedging cost is constant on the atoms on G−1.

Proposition 3.2. The G-superhedging cost of ξ on Ω, defined in Definition 3.1, is con-stant on atoms of G−1. Specifically, for any ω ∈ Ω we have

V GX ,P,Ω(ξ)(ω) = V G

X ,P,Ω(ξ)(ω′) = V G

X ,P,Aω(ξ)(ω′) ∀ω′ ∈ Aω,

where Aω denotes the G−1-atom containing ω. In particular,

∀ω′ ∈ Aω V GX ,P,Aω(ξ)(ω) = V G

X ,P,Aω(ξ)(ω′).

Proof: Fix ω ∈ Ω and ω′ ∈ Aω. It follows from G−1–measurability that P(X)(ω) =P(X)(ω′) for any G-admissible strategy (X, γ) ∈ AX (G) which super–replicates ξ on Aω.This in turn implies that

V GX ,P,Aω(ξ)(ω) = V G

X ,P,Aω(ξ)(ω′).

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It remains to argue that these are also equal to V GX ,P,Ω(ξ)(ω

′). Clearly the latter canonly be larger. As for the reverse inequality, note that any G-admissible semi-staticstrategy (X, γ) which super–replicates ξ on Aω can be extended to a strategy (X, γ)super–replicating on Ω by taking X = X on Aω and a0 = ||ξ|| otherwise, γ = γ11Aω .

Note that P(X) is G−1–measurable and hence V GX ,P,A(ξ) is random but its measur-

ability is not clear a priori. However if G−1 is generated by a discrete random variablethen it has at most countably many atoms. Since V G

X ,P,Ω(ξ) is constant on atoms of G−1,it then follows that it is G−1-measurable. With a slight abuse of notation, we denote

V GX ,P,Aω(ξ) := V G

X ,P,Ω(ξ)|Aω ,

the superhedging price on an atom Aω in G−1.We now turn to the pricing problem. In the classical approach markets with no-

arbitrage are modelled using martingale measures. We denote by MG probability mea-sures on (Ω,FT ) such that S is a G–martingale. For a given A ∈ FT , we look at possibleclassical market models which calibrate to market prices of options and are supportedon A:

MGX ,P,A :=

P ∈ MG : P(A) = 1 and EP(X|G−1) = P(X) for all X ∈ X , P-a.s.

.

Now we would like to give a precise meaning to the expression "supP∈MG

X ,P,ΩEP(ξ|G−1)".

Unless G−1 is a trivial σ-field, the conditional expectation EP(ξ|G−1) is a random variablewhich is determined only P-a.s.. As measures in the set MG

X ,P,Ω may be mutually singularwe have to be careful about choosing a good version of these conditional expectations.

Proposition 3.3. Assume each element of G is countably generated. Let P ∈ MGX ,P,Ω.

Then, there exists a set ΩP ∈ G−1 with P(ΩP) = 1 and a version Pω of the regular con-ditional probabilities of P with respect to G−1 such that for each ω ∈ ΩP, Pω ∈ MG

X ,P,Aω

where Aω is the G−1-atom containing ω.

Proof: Existence of Pω and its properties are all classical results, see Stroock &Varadhan [31, pp. 12–16]. The property that P-a.s. Pω(A

ω) = 1 and that ΩP can betaken to be in G−1 come from the fact that G−1 is countably generated.Fix t ≥ s and G ∈ Gs. Then, since G0 := EPω((St − Ss)11G) > 0 ∈ G−1, we get

0 = EP((St − Ss)11G∩G0) = EP(EP((St − Ss)11G|G−1)11G0) = EP(EPω((St − Ss)11G)11G0),

which implies that P-a.s. EPω((St − Ss)11G) ≤ 0. In the same way we prove that P-a.s.EPω((St − Ss)11G) ≥ 0. So finally, P-a.s. EPω((St − Ss)11G) = 0. To conclude thatP-almost every Pω is a G-martingale measure we use continuity of paths of S and theassumption that each element Gs is countably generated. To end the proof we note that

P-a.s. ∀X ∈ X P(X) = EP(X|G−1) = EPω(X).

Suppose now that we fix a representation ΩP in Proposition 3.3 for each P ∈ MGX ,P,Ω and

use these to define the market model price as follows:

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Definition 3.4. Let A ∈ FT . The G-market model price of ξ on A is defined by

PGX ,P,A(ξ)(ω) := sup

P∈MG

X ,P,A

EPω(ξ), ω ∈ Ω,

where EPω(ξ) = EPω(ξ) for ω ∈ ΩP and EPω(ξ) = −∞ for ω ∈ Ω\ΩP.

We note the following obvious inequalities:

PG+

X ,P,A(ξ) ≤ PGX ,P,A(ξ) ≤ PG−

X ,P,A(ξ) ≤ P FX ,P,A(ξ).

We now show that, as in the case of superhedging, the model price is constant on atomsof G−1 and is uniquely defined, irrespective of the choice of ΩP above.

Proposition 3.5. The G-market model price of ξ on Ω is uniquely defined on Ω and isconstant on atoms of G−1. Specifically, for ω ∈ Ω we have

PGX ,P,Ω(ξ)(ω) = PG

X ,P,Ω(ξ)(ω′) = PG

X ,P,Aω(ξ)(ω′) = supP∈MG

X ,P,Aω

EP(ξ), ∀ω′ ∈ Aω,

where Aω is the G−1-atom containing ω.

Proof: Note that if P ∈ MGX ,P,Aω , then Aω ⊂ ΩP and Pω′ = P for all ω′ ∈ Aω.

In particular, PGX ,P,Aω(ξ)(ω′) = supP∈MG

X ,P,AωEP(ξ) for all ω′ ∈ Aω, i.e. PG

X ,P,Aω(ξ) is

well defined and constant on Aω. Further, the obvious inclusion MGX ,P,Aω ⊂ MG

X ,P,Ω

combined with Proposition 3.3, show that the set MGX ,P,Aω is composed of conditional

probabilities of measures in MGX ,P,Ω. In consequence,

PGX ,P,Ω(ξ)(ω

′) = PGX ,P,Aω(ξ)(ω′), ω′ ∈ Aω

are required. In particular, PGX ,P,Ω(ξ) is well defined and constant on atoms and does not

depend on the choice of ΩP for P ∈ MGX ,P,Ω. Finally, if for some ω ∈ Ω, MG

X ,P,Aω = ∅

then, for any P ∈ MGX ,P,Ω, ΩP ∩Aω = ∅ and the definitions agree giving

MGX ,P,Ω(ω) = −∞.

Note that in general the measurability of PGX ,P,Ω(ξ) is not clear. However, when G−1 is

generated by a discrete random variable then it has at most countably many atoms andPGX ,P,Ω(ξ) is G−1-measurable. With a slight abuse of notation, we denote

PGX ,P,Aω(ξ) := sup

P∈MG

X ,P,Aω

EP(ξ) = PGX ,P,Ω(ξ)|Aω

its value on an atom Aω ∈ G−1. This representation implies that it is sufficient to focuson measures supported on a single atom. We note that these are not necessarily theextreme measures in MG

X ,P,Ω.

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Corollary 3.6. When computing the G-market model price of ξ on Ω, or on some setin G−1, it is sufficient to maximise over measures supported on a single atom of G−1.

We turn now to the easy inequality in the pricing–hedging duality.

Lemma 3.7. The G-superhedging cost V GX ,P,Ω(ξ) and the G-market model price PG

X ,P,Ω(ξ)of ξ on Ω satisfy

V GX ,P,Ω(ξ)(ω) ≥ PG

X ,P,Ω(ξ)(ω) ∀ω ∈ Ω .

Proof: Using Propositions 3.2 and 3.5, it is enough to show the asserted inequalityseparately on each atom Aω of the σ-field G−1. The proof then follows by a classicalargument. Take any G-super–replicating portfolio (X, γ) ∈ AM(G) on Aω and anymeasure P ∈ MG

X ,P,Aω . Let Pv denote regular conditional probabilities of P with respect

to G0. Thus, by Proposition 3.3, P-a.s., Pv ∈ MGBv where Bv are G0-atoms containing v.

Note that Bv form finer partition than Aω. Then, since P-a.s., Pv(M ≡ const) = 1,we deduce that

EPv(ξ) ≤ EPv

(X +

∫ T

0γudSu

)≤ EPv(X) P-a.s.

since∫ ·0 γudSu is a G-local martingale bounded from below and thus a G-supermartingale.

Since Pv are regular conditional probabilities of P, taking expectations under P, wehave

EP(ξ) ≤ EP

(X +

∫ T

0γudSu

)≤ EP(X) = P(X)

which completes the proof.

4 Pricing–hedging duality under initially enlarged filtration

We turn to the main question in this paper: namely when a pricing–hedging dualityholds in the enlarged filtration? That is, when the inequality in Lemma 3.7 is in factan equality? We start with the case when G is an initial enlargement of F. Then, γis G-progressively measurable if and only if γ(ω)t = γ(ω)t whenever ω|[0,t] = ω|[0,t] andω ∼G0 ω.

4.1 Preliminaries: pricing–hedging duality with beliefs

We start by recalling notions and results from Hou & Obłój [22]. As explained before,we use their setup with beliefs to zoom on the part of the pathspace considered by aninformed agent. For P ∈ FT let V F

X ,P,P(ξ) be the approximate F-superhedging cost of ξon P, i.e.,

V FX ,P,P(ξ) := infP(X) : ∃(X, γ) ∈ AX (F) such that

X(ω(1), ..., ω(d)) +

∫ T

0γ(ω)udSu(ω) ≥ ξ(ω) ∀ ω ∈ P(ε) for some ε > 0,

9

whereP(ε) = ω ∈ Ω : inf

ω∈P||ω − ω|| ≤ ε. (5)

Similarly, let P FX ,P,P(ξ) be the approximate F-market model price of ξ, i.e.,

P FX ,P,P(ξ) := lim

ηց0sup

P∈MF,ηX ,P,P

EP(ξ)

with MF,ηX ,P,P := P ∈ MF

Ω : P(P(η)) > 1− η and |EP(X)−P(X)| < η for all X ∈ X .The following assumption says that the set X is not too large, and initial prices ofdynamically traded options are not "on the boundary of no-arbitrage region".

Assumption 4.1. (i) Lin1(X ) is a compact subset of C(Ω,R), where

LinN (X ) := a0 +m∑

i=1

aiXi : m ∈ N, Xi ∈ X ,m∑

i=0

|ai| ≤ N. (6)

(ii) Either K = 0 or there exists an ε > 0 such that for any (pk)k≤K with |P(X(c)k )−pk| ≤

ε for all k ≤ K, MΩ6= ∅ where

Ω = ω ∈ Ω : S(d+i)T (ω) = X

(c)i (ω)/pi) ∀i ≤ K.

Theorem 4.2 (Hou & Obłój [22]). Suppose that Assumption 4.1 holds. Let P ∈ FT besuch that MF,η

X ,P,P 6= ∅ for any η > 0. Then, for any uniformly continuous and boundedξ, the approximate pricing–hedging duality holds:

V FX ,P,P(ξ) = P F

X ,P,P(ξ).

We also isolate the following general fact which extends the pricing–hedging dualitywithout static hedging to the general case using a min–max type of argument.

Theorem 4.3. Let G be an arbitrary filtration such that G−1 is a trivial σ-field. LetX satisfies Assumption 4.1 and assume that MG

X ,P,Ω 6= ∅. Moreover assume that there

is pricing–hedging duality for the model without options, i.e., PGΩ (ξ) = V G

Ω (ξ) for alluniformly continuous and bounded ξ. Then there is pricing–hedging duality for the modelwith options, i.e.,

PGX ,P,Ω(ξ) = V G

X ,P,Ω(ξ) for all uniformly continuous and bounded ξ.

Proof: We use a calculus of variations argument similarly to Sections 4.1 and 4.2 of[22]. We have, with LinN (X ) defined in (6) and Lin(X ) = ∪NLinN (X ), that

V GX ,P,Ω(ξ) = infP(X) : ∃(X, γ) ∈ AX (G) such that X + γ S ≥ ξ on Ω

= infX∈Lin(X )

P(X) + infx : ∃ γ ∈ A(G) such that x+ γ S ≥ ξ −X on Ω

= infX∈Lin(X )

P(X) + V GΩ (ξ −X)

= infX∈Lin(X )

P(X) + PGΩ (ξ −X)

= limN→∞

infX∈LinN (X )

supP∈MG

Ω

EP(ξ −X) + P(X).

10

Define the function f : LinN (X )×MGΩ → R by f(X,P) := EP(ξ−X)+P(X). Note that

f is linear and continuous in the first variable due to bounded convergence theorem. Itis also linear and continuous in the second variable. Then, since LinN (X ) is convex andcompact, minmax Theorem (Corollary 2 in Terkelsen [32]) implies that

infX∈LinN (X )

supP∈MG

Ω

(EP(ξ −X) + P(X)) = supP∈MG

Ω

infX∈LinN (X )

(EP(ξ −X) + P(X)).

Since ξ is bounded, it follows that for P ∈ MGΩ\M

GX ,P,Ω the quantity infX∈LinN (X )(EP(ξ−

X)+P(X)) can be made arbitrarily small by taking large N . Since MGX ,P,Ω is not empty,

we obtain the desired equality.

4.2 Duality in an enlarged filtration: case of G+

From Theorem 4.2, taking P to be atoms in G+−1 = G0, we can deduce a pricing–hedging

duality in the enlarged filtration. We isolate the following assumption which is ofteninvoked.

Assumption 4.4. The set P ⊂ Ω is such that MFX ,P,P 6= ∅, and

P FX ,P,P(ξ) = V F

X ,P,P(ξ) for any bounded uniformly continuous ξ. (7)

Theorem 4.5. Suppose Assumption 4.1 holds and let Z be a random variable and G :=F∨σ(Z). Assume that for each value c ∈ Z(Ω) either Z = c satisfies Assumption 4.4 orMF

X ,P,Z=c = ∅. Then, there is no duality gap in G+ on the set ω : MFX ,P,Z=Z(ω) 6=

∅, i.e., V G+

X ,P,Ω(ξ)(ω) = PG+

X ,P,Ω(ξ)(ω) holds for any ω such that MFX ,P,Z=Z(ω) 6= ∅ and

any bounded uniformly continuous ξ.

Proof: In the first step we prove that for each value c ∈ Z(Ω) such that MFX ,P,Z=c 6=

∅ we have

P FX ,P,Z=c(ξ) = PG+

X ,P,Z=c(ξ) ≤ V G+

X ,P,Z=c(ξ) = V FX ,P,Z=c(ξ). (8)

First let us prove the last equality. To this end note that any G+-progressively measurableprocess γ is of the form γ(ω)t = Γ(Z(ω), t, ω) where for each t ∈ [0, T ] the mappingΓ : R × [0, t] × Ω → Rd+K

+ is B(R) ⊗ B([0, t]) ⊗ Fs-measurable. So there exists F-progressively measurable process γF given by γF(t, ω) = Γ(c, t, ω) which is equal to γ onZ = c thus the last equality in (8) follows.To show the first equality, it suffices to show that MG+

X ,P,Z=c = MFX ,P,Z=c. For any

P ∈ MFX ,P,Z=c and 0 ≤ s ≤ t ≤ T we have

EP(St|G+s ) = EP(St|Fs ∨ σ(Z)) = EP(EP(St|F

Ps )|Fs ∨ σ(Z)) = EP(Ss|Fs ∨ σ(Z)) = Ss,

where FPs is a P-completion of Fs, showing P ∈ MG+

X ,P,Z=c. The reverse inclusion is

clear. Finally, the middle inequality in (8) is implied by Lemma 3.7.

11

Finally, by representations given in Propositions 3.2 and 3.5, the proof is completedsince we get that V G+

X ,P,Z=c(ξ) = PG+

X ,P,Z=c(ξ) for any value c ∈ Z(Ω) for which

MFX ,P,Z=c 6= ∅ and any bounded uniformly continuous claim ξ.

Remark 4.6. Under Assumption 4.1, thanks to Theorem 4.2, we have

V FX ,P,Z=c(ξ) = P F

X ,P,Z=c(ξ). (9)

Combining inequality (9) with general following inequalities

P FX ,P,Z=c(ξ) ≤ V F

X ,P,Z=c(ξ) ≤ V FX ,P,Z=c(ξ)

we conclude that in Theorem 4.5, instead of assuming that (7) holds, it is enough toassume that

P FX ,P,Z=c(ξ) ≤ P F

X ,P,Z=c(ξ).

We present two examples where the informed agent has some additional knowledgeabout the price process at the terminal date. We consider two extreme situations: firstwhen the informed agent gets rather detailed information and there is a continuum ofatoms, second when the informed agent gets binary information (and G−1 only has twoatoms). In both cases we only consider the situation with no static trading: X = ∅ andsuch that pricing–hedging duality in G+ holds.

Example 4.7. We consider a one-dimensional setting with no statically or dynamicallytraded options: d = 1, K = 0 and X = ∅. The informed agent acquires detailedknowledge of the stock prices process: namely she knows the maximum deviation overtime interval [0, T ] of the stock price from the initial price. Naturally, the agent does notknow the sign of the deviation as this would give an instant arbitrage. This situationcorresponds to taking Z = supt∈[0,T ] |St − 1|.

For c > 1, the market model price PG+

Z=c(ξ) = −∞ as any martingale measure P ∈

MG+

Z=c would satisfy

1 = P

(sup

t∈[0,T ]≥ 1 + c

)≤

1

1 + c< 1,

thus the set of martingale measures MG+

Z=c must be empty. Likewise the super–hedging

cost V G+

Z=c(ξ) = −∞ as a long position in the stock generates arbitrage.

Fix c ≤ 1. Our proof is similar to Example 3.12 in [22]. Firstly observe that underany P ∈ MG+

Z=c, S is uniformly integrable martingale with S = Sτc P-a.s. where

τc := inft : St ∈ 1 + c, 1 − c.

For each N there exists a measure PN ∈ MF,1/NZ=c such that

EPN (ξ) ≥ supP∈M

F,1/NZ=c

EP(ξ)−1

N.

12

Since PN is a martingale measure for S, Doob’s martingale inequality implies:

PN(||S|| > M) ≤1

M,

and then, defining τM := inft : St = M for an arbitrary large M , leads to:

|EPN (ξ(S) − ξ(SτM ))| ≤2||ξ||

M.

Let πN be the distribution of SτMT under PN , for N ∈ N. Since πN ([0,M ]) = 1 for each

N , this is a tight family of probability measures and therefore a converging subsequence(πNk)k exists. Denote the limit of (πNk)k by π and note that, since each measure πN hasmean equal to one so does π.For each ε > 0, weak convergence of measures and Portemanteau Theorem imply that

π([1− c− ε, 1 + c+ ε]) ≥ lim supk→∞

πNk([1− c− ε, 1 + c+ ε]) = 1 (10)

since, letting UN := [1− c− 1N , 1 + c+ 1

N ], one has

PN (SτMT ∈ UN ) = PN (||S|| ≤ M, ST ∈ UN ) ≥ PN (||S|| ≤ b+ 1, ST ∈ UN ) ≥ 1−

1

N.

It holds for each ε > 0 hence π([1 − c, 1 + c]) = 1. Finally,

P FZ=c = lim

N→∞sup

P∈MF,1/NZ=c

EP(ξ) ≤ limN→∞

(EPN (ξ) +

1

N

)

≤ lim supk→∞

supP∈MF

s.t. L(ST )=πNk

EP(ξ) +2||ξ||

M

≤ supP∈MF

s.t. L(ST )=π

EP(ξ) +2||ξ||

M≤ sup

P∈MF

Z=c

EP(ξ) +2d||ξ||

M

where the fourth inequality holds by Lemma 4.4 in [22] and the fifth since under anyP ∈ MF

Z=c the distribution of ST equals π. Since M was arbitrary we obtain that

P FZ=c ≤ P F

Z=c and by Theorem 4.5 and Remark 4.6 we conclude that there is no

duality gap in G+.

Example 4.8. Assume X = ∅, d = 1 and K = 0. Take Z = 11St∈(a,b) ∀t∈[0,T ] wherea < 1 < b. We use Theorem 4.5 to show that the pricing–hedging holds in G+.Equality P F

Z=0(ξ) = P FZ=0(ξ) follows by the same arguments used in the previous

example and the proof of Example 3.12 in [22] since the set [0, a] ∪ [b,M ] is compact forany M . The only difference lies in arguing analogous inequality as in (10). Here we have

π([0, a + ε] ∪ [b− ε,∞)) ≥ lim supk→∞

πNk([0, a+ ε] ∪ [b− ε,∞)) = 1

13

since

PN (SτMT ∈ [0, a + 1/N ] ∪ [b− 1/N,∞)) ≥ PN (ST ∈ [0, a+ 1/N ] ∪ [b− 1/N,∞)) ≥ 1−

1

N.

It holds for each ε > 0 and hence π([0, a] ∪ [b,∞)) = 1.To prove second equality that P F

Z=1(ξ) = P FZ=1(ξ) we first note that, again by the

same type of arguments as in the proof on Example 3.12, for the interval [a, b] we have

P FSt∈[a,b] ∀t∈[0,T ](ξ) = P F

St∈[a,b] ∀t∈[0,T ](ξ) = P FZ=1.

Thus, in order to show that P FZ=1(ξ) = P F

Z=1(ξ) it is enough to prove that

P FSt∈[a,b] ∀t∈[0,T ](ξ) = P F

Z=1(ξ).

Take P ∈ MFSt∈[a,b] ∀t∈[0,T ]. For each k ∈ (0, 1) define Sk as

Skt (ω) := kωt + (1− k) for ω ∈ St ∈ [a, b] ∀t ∈ [0, T ].

Then

P (Sk)−1 ∈ MFST∈(a,b) and ||Sk(ω)− ω|| ≤ (1− k)[(b − 1) ∨ (1− a)].

Thus|EP(ξ)− EP(ξ S

k)| ≤ eξ((1− k)[(b − 1) ∨ (1− a)])

and, since k is arbitrary close to 1, there exists sequence of measures PN ∈ MFST∈(a,b)

such that EP(ξ) = limN→∞ EPN (ξ).

4.3 Duality in an enlarged filtration: case of G−

Theorem 4.10 concerns the pricing–hedging duality in G−. We recall that G−−1 = ∅,Ω

which models the situation when the static position in options has to be determinedbefore acquiring any additional information. In order to formulate the results we needto introduce further conditions on Z, namely:

Assumption 4.9. There exists a countable subset cnn of Z(Ω) such that for eachsequence ηnn of positive real numbers there exists a family Σnn of σ(Z)-measurablesets satisfying: Ω =

⋃n Σ

n and that for each ω ∈ Σn there exists an ω ∈ Z = cn suchthat ||ω − ω|| ≤ ηn.

Remark that Assumption 4.9 is satisfied when:

• Z is a discrete random variable since it is enough to take cnn = Z(Ω) andΣn = Z = c;

14

• Z is a continuous random variable then it is enough to take cn a countable densesubset of Z(Ω) containing levels c such that there exists ω ∈ Z = c and r > 0such that v : ||v − ω|| ≤ r ⊂ Z = c (level sets with non empty interior) andΣn = cn − εn ≤ Z ≤ εn + cn where εn ≥ 0 are chosen regarding ηn. Note thatabove is possible since there can be only countable number of level sets with notempty interior.

Theorem 4.10. Let Z be a random variable such that for each c ∈ Z(Ω) the set Z = csatisfies Assumption 4.4 with no options (X = ∅) and that Assumption 4.9 holds. DefineG = F∨σ(Z). Assume moreover that MG−

X ,P,Ω 6= ∅ and that Assumption 4.1 holds. Then

pricing–hedging duality holds in G−:

V G−

X ,P,Ω(ξ) = PG−

X ,P,Ω(ξ) ∀ bounded uniformly continuous ξ.

Proof: In the first step of the proof we assume that X = ∅ and we prove the followingsequence of equalities

V G−

Ω (ξ) = supc∈Z(Ω)

V G+

Z=c(ξ) = supc∈Z(Ω)

PG+

Z=c(ξ)

= supP∈

⋃c∈Z(Ω) M

G+

Z=c

EP(ξ) = supP∈MG−

Ω

EP(ξ) = PG−

Ω .(11)

Let us start with first equality. Note that, under Assumption 4.4, one has

V G−

Z=c(ξ) = V G−

Z=c(ξ).

Let cnn be a subset from Assumption 4.9. Fix ε > 0. Then, for each n, there existsηn > 0 such that

V G−

Σn (ξ)− V G−

Z=cn(ξ) ≤ ε/2.

Next, there exist xn ∈ R, Mn and γn ∈ AMn(G) such that

xn − V G−

Σn ≤ ε/2 and xn + γn S ≥ ξ on Σn.

We build up the following strategy (X, γ) ∈ AMχ (G) where M :=

∑∞n=1M

n11Σn\

⋃n−1k=1 Σk(ω):

X(ω) :=

∞∑

n=1

xn11Σn\⋃n−1

k=1 Σk(ω) and γt(ω) :=

∞∑

n=1

γnt (ω)11Σn\⋃n−1

k=1 Σk(ω).

Note that (X, γ) satisfies, for each ω ∈ Ω, X(ω) +∫ T0 γ(ω)tdS(ω)t ≥ ξ(ω) and hence

supω

X(ω) +

∫ T

0γ(ω)tdS(ω)t ≥ ξ(ω).

Thus

V G−

Ω (ξ) ≤ supω

X(ω) = supn

xn ≤ supn

V G−

Z=cn+ ε ≤ sup

c∈Z(Ω)V G−

Z=c + ε = supc∈Z(Ω)

V G+

Z=c + ε,

15

where the last equality follows by definition since Z = c is an atom of G+−1 and hence

V G+

Z=c = V G−

Z=c. As ε was arbitrary one has supc∈Z(Ω) VG+

Z=c(ξ) ≥ V G−

Ω (ξ). Since evi-

dently for each c ∈ Z(Ω), V G+

Z=c(ξ) ≤ V G−

Ω (ξ), we finally get that supc∈Z(Ω) VG+

Z=c(ξ) =

V G−

Ω (ξ) thus first equality in (11) holds. By our assumption ω : MFZ=Z(ω) 6= ∅ = Ω

and pricing–hedging duality holds on Z = c for each c ∈ Z(Ω). Thus the secondequality in (11) follows. Whereas the third and fifth equalities in (11) hold by defini-tion. To show the fourth one, note that, one inequality is immediate since for any c,MG+

Z=c ⊂ MG−

Ω and the other inequality then follows by Proposition 3.3.The general case with non–empty set of options X follows by Theorem 4.3.

5 Dynamic programming principle and pricing–hedging du-

ality

To extend the initial enlargement perspective we study here the case where the additionalinformation is disclosed at time T1 ∈ (0, T ), i.e. the filtration G is of the form: Gt = Ft

for t ∈ [0, T1) and Gt = Ft ∨ σ(Z) for t ∈ [T1, T ]. We divide our problem into two timeintervals using the results from the previous sections. First we look at the pricing andhedging problems on [T1, T ] and then on [0, T1]. Along this section we assume that thereare no dynamically traded options, i.e., K = 0. Moreover we assume that Z is of aspecial form, namely, Z satisfies

Z(ω) =

Z(ω|[T1,T ]

ωT1

)if ωT1 > 0

1 if ωT1 = 0for r.v. Z on Ω|[T1,T ]. (12)

This condition encodes the idea that the additional information only pertains to theevolution of prices after time T1 irrespectively of the prices on, or before, time T1.

We begin with two propositions where we develop the dynamic programming principlefor two cases: superhedging cost and market model price. Note that the case Z ≡ const.and F = G is also of interest, as it gives the dynamic programming principle under F.

Proposition 5.1. Let Bω denote the FT1-atom containing ω. Then for a bounded uni-formly continuous ξ the following hold.

(i) The mapping VG,[T1,T ]Ω (ξ) : Ω → R defined as

VG,[T1,T ]Ω (ξ)(ω) := infx ∈ R : ∃ γ ∈ A(G, [T1, T ]) such that x+

∫ T

T1

γtdSt ≥ ξ on Bω

is uniformly continuous and FT1-measurable.(ii) The dynamic programming principle holds in the form:

VG,[0,T ]Ω (ξ) = V

F,[0,T1]Ω

(V

G,[T1,T ]Ω (ξ)

).

16

Proof: In the proof we denote ξ := VG,[T1,T ]Ω (ξ).

(i) Let v := (v1, ..., vd) ∈ Ω and v := (v1, ..., vd) ∈ Ω. Note that

∣∣∣ξ((v1, .., vd))− ξ((v1, .., vd))∣∣∣ ≤

d∑

k=1

∣∣∣ξ((v1, .., vk−1, vk, .., vd))− ξ((v1, .., vk, vk+1, .., vd))∣∣∣ .

Thus, to establish uniform continuity of ξ, it is enough to consider v and v which differon one coordinate only and, without loss of generality, we may assume that d = 1.

Consider a small δ > 0. Suppose that ||v − v||[0,T1] ≤ δ, |vT1 − vT1 | = D ≥ 0, vT1 > 0and vT1 > 0.

In the first step we show that ξ(v) ≤ ξ(v) + ε for an appropriately chosen ε, dependingonly on ξ and δ. For each η > 0 there exists a strategy γ such that

ξ(v) +

∫ T

T1

γtdSt ≥ ξ − η on Bv.

Let λ = vT1/vT1 ∈ (0,∞) and define the path modification mapping αv,v by

α(ω) := αv,v(ω) :=

v|[0,T1] ⊗vT1vT1

ω|[T1,T ] ω ∈ Bv

v|[0,T1] ⊗vT1vT1

ω|[T1,T ] ω ∈ Bv

ω ω /∈ Bv ∪Bv

(13)

where λω is a multiplicative modification of ω by λ in Ω and v|[0,T1] ⊗ λω|[T1,T ] meansthat the path is equal to v on [0, T1] and to λω| on [T1, T ]. Note that α is a bijectionsatisfying α = α−1. Introduce a stopping time

τ(ω) := τv,v(ω) :=

inft > T1 : ωt − vT1 ≥ vT1D− 1

2 ∧ T ω ∈ Bv

inft > T1 : ωt − vT1 ≥ vT1D− 1

2 ∧ T ω ∈ Bv

T1 ω /∈ Bv ∪Bv

. (14)

To show that ξ(v) ≤ ξ(v) + ε we will consider a strategy λγ α + D14

vT11[T1,τ) on Bv.

The second term of this strategy is clearly G-adapted. To show that the first term isG-adapted as well, it is enough to show that Z α is σ(Z)-measurable. The last is truesince

Z α(ω) = Z

(α(ω)|[T1,T ]

α(ω)T1

)= Z

(ω|[T1,T ]

ωT1

)= Z(ω).

Then, we obtain

ξ(v) + λ

∫ T

T1

γ α(ω)tdSt(ω) +D

14

vT1

(ωτ − vT1)

= ξ(v) +

∫ T

T1

γ α(ω)tdSt α(ω) +D

14

vT1

(ωτ − vT1)

≥ ξ α(ω)− η +D

14

vT1

(ωτ − vT1)

17

where the first equality is due to our definition of integration. In the case that τ(ω) = Tone has

D14

vT1

(ωτ − vT1) ≥ −D

14

vT1

vT1 = −D1/4

and and‖α(ω)− ω‖ ≤ δ ∨ (λ− 1)vT1(D

− 12 + 1) ≤ 2δ1/2. (15)

Thus, for τ(ω) = T , it follows that

ξ α(ω) − η +D

14

vT1

(ωτ − vT1) ≥ ξ(ω)− eξ(2δ1/2)− η −D1/4

where eξ is modulus of continuity of ξ. Hence, for τ(ω) = T , we deduce ξ(v) ≤ ξ(v) +eξ(2δ

1/2) +D1/4. In the case that τ(ω) < T one has

D14

vT1

(ωτ − vT1) =D

14

vT1

vT1D− 1

2 = D−1/4

and

ξ α(ω)− η +D

14

vT1

(ωτ − vT1) ≥ −||ξ|| − η +D−1/4

which, for D small enough (D ≤ (2||ξ||)−4), majorates ξ(ω). We deduce that ξ(v) ≤ξ(v) + eξ(2δ

1/2) +D1/4.

Therefore, ξ is uniformly continuous on ω ∈ Ω : ‖ω‖ > 0.To complete the proof, we now consider the case where vT1 = 0. Let, for some small

δ > 0, ||v − v||[0,T1] ≤ δ and vT1 = D > 0. Firstly notice that v must satisfy ξ(v) =ξ(v|[0,T1] ⊗ 0|[T1,T ]) since we can buy any amount of stock at price 0 at time T1 thus only

constant path is relevant, and therefore ξ(v) ≤ ξ(v) + eξ(δ). Now consider the strategyγ for ω ∈ Bv defined as

γ(ω) := δ−1/21[T1,σ(ω)) where σ(ω) := inft > T1 : ωt − vT1 ≥ δ1/4.

Then, whenever σ(ω) < T ,

ξ(v) +

∫ T

T1

γ(ω)tdSt(ω) = ξ(v) + δ−1/2(ωσ − vT1) = ξ(v) + δ−1/4

which, for d small enough, majorates ξ(ω). Otherwise, if σ(ω) = T

ξ(v) +

∫ T

T1

γ(ω)tdSt(ω) ≥ ξ(v)− δ1/2 ≥ ξ(ω)− eξ(2δ1/4)− δ1/2

since ||v − ω|| ≤ 2δ1/4. Therefore, ξ(v) ≤ ξ(v) + eξ(2δ1/4) + δ1/2.

18

(ii) Let V 1 := VG,[0,T ]Ω (ξ) and V 2 := V

F,[0,T1]Ω (ξ). For each η > 0 there exist γ1 ∈

A(F, [0, T1]) and γ2 ∈ A(G, [T1, T ]) such that

V 1 +

∫ T1

0γ1t dSt +

∫ T

T1

γ2t St ≥ ξ − η on Ω.

In particular, for a fixed ω ∈ Ω, the superhedging holds on Bv. Since V 1 +∫ T1

0 γ1t dSt is

constant on Bω, we deduce that V 1 +∫ T1

0 γ1t dSt ≥ ξ on Ω|[0,T1] and therefore V 1 ≥ V 2.To prove the reverse inequality take z > V 2. Then there exists γ1 ∈ A(F, [0, T1]) such that

z+∫ T1

0 γ1t dSt ≥ ξ on Ω. Hence, for each η > 0, there exists a strategy γ2 ∈ A(G, [T1, T ])

such that z +∫ T1

0 γ1t dSt +∫ TT1

γ2t St ≥ ξ − η. Then, clearly, z ≥ V 1.

Define the set of measures MG,[T1,T ]A concentrated on A ∈ FT as follows:

MG,[T1,T ]A := P : S is a G-martingale on [T1, T ] and P(A) = 1.

Proposition 5.2. Let Bω denote the FT1-atom containing ω and assume that MG,[T1,T ]Bω 6=

∅ for each ω. Then for a bounded uniformly continuous ξ the following hold.

(i) The mapping PG,[T1,T ]Ω (ξ) defined as P

G,[T1,T ]Ω (ξ)(ω) := sup

P∈MG,[T1,T ]Bω

EP(ξ) is uni-

formly continuous and FT1-measurable.(ii) The dynamic programming principle holds in the form:

PG,[0,T ]Ω (ξ) = P

F,[0,T1]Ω (P

G,[T1,T ]Ω (ξ)).

Proof: In the proof we denote ξ := PG,[T1,T ]Ω (ξ).

(i) Let v := (v1, ..., vd) ∈ Ω and v := (v1, ..., vd) ∈ Ω. Note that

∣∣∣ξ((v1, .., vd))− ξ((v1, .., vd))∣∣∣ ≤

d∑

k=1

∣∣∣ξ((v1, .., vk−1, vk, .., vd))− ξ((v1, .., vk, vk+1, .., vd))∣∣∣ .

Thus, to prove uniform continuity of ξ, it is enough to consider v and v which differ onone coordinate only and, without loss of generality, we may assume that d = 1.Suppose that ||v − v||[0,T1] ≤ δ and |vT1 − vT1 | = D ≥ 0. It is enough to show that

ξ(v) ≤ ξ(v) + ε for an appropriately chosen ε depending only on ξ and δ. Take P ∈

MG,[T1,T ]

Bv , i.e.: P(Bv) = 1, where Bv := ω : ωt = vt for t ∈ [0, T1], P(ST1 = vT1) = 1

and EP(St11G) = EP(Ss11G) for each T1 ≤ s ≤ t ≤ T and G ∈ Gs. Define measure P as

P = P α with path modification α given in (13). Then P is an element of MG,[T1,T ]Bv

since P(Bv) = P(α(Bv)) = P(Bv) = 1, it is a martingale measure on [T1, T ] as forT1 ≤ s ≤ t ≤ T and G ∈ Gs

EP(St11G) = EP((S α)t11α(G)) = EP((S α)s11α(G)) = EP(Ss11G),

where the second equality follow by α(G) ∈ Gs. The latter is true since, for any Borel setB, one has α(Z ∈ B∩Bv) = Bv∩Z ∈ A; the σ-field Fs coincides with trivial σ-field

19

up to P-null sets and up to P-null sets; the general case follows from the monotone classargument. Hence, with τ defined in (14),

|EP(ξ)− EP(ξ)| = |EP(ξ)− EP(ξ α)|

= EP((ξ − ξ α)1τ=T) + EP((ξ − ξ α)1τ<T)

≤ eξ(2δ1/2) + 2||ξ||

D

D +D1/2, (16)

where in the last inequality we used (15), Doob’s inequality and the fact that

P(τ < T ) = P

(sup

t∈[T1,T ]St ≥ vT1(1 +D−1/2)

)≤

vT1

vT1(1 +D−1/2)=

D

D +D1/2.

(ii) To prove that PG,[0,T ]Ω (ξ) ≤ P

F,[0,T1]Ω (P

G,[T1,T ]Ω (ξ)) it is enough to note that:

supP∈M

G,[0,T ]Ω

EP(ξ) = supP∈M

G,[0,T ]Ω

EP(EPω(ξ)) ≤ supP∈M

G,[0,T ]Ω

EP

sup

P∈MG,[T1,T ]

Bω

EP(ξ)

where Pω is regular conditional distribution with respect to FT1 and where in the laststep we used measurability implied by assertion (i).Now we will show the remaining inequality. Let ωnn be a countable dense subset ofΩ|[0,T1] and Bn := Bωn

. Define the path modification mapping αn,ω by αn,ω := αωn,ω

where αωn,ω is given in (13). Note that αn,ω is a bijection satisfying αn,ω = (αn,ω)−1.

For any Pn ∈ MG,[T1,T ]Bn the measure Pn αn,ω belongs to M

G,[T1,T ]Bω . Moreover, similarly

to (16), we obtain that

|EPn(ξ)− EPnαn,ω (ξ)| ≤ eξ(2δ1/2) + 2||ξ||

δ

δ + δ1/2

whenever ||ωn − ω||[0,T1] ≤ δ. Let us consider probability kernel Nn : Ω → MG,[T1,T ]

defined by Nn(ω) := Pn αn,ω. The kernel Nn is FT -measurable, i.e., Nn(ω,F ) =Pn α

n,ω(F ) is FT -measurable for any F ∈ FT , since (ω, ω) → 11F αn,ω(ω) is FT ⊗FT -measurable and bounded thus EPn(11F αn,ω) is FT -measurable (see [9, Section 3.3]).Then, since Nn is constant on atoms of FT1 , we deduce from Blackwell’s Theorem (see [13,Theorem 8.6.7] and/or [17, Ch III, §26, p.80-81]) that Nn is FT1-measurable probabilitykernel.Denoting the closed ball around ωn of radius δ by Bn(δ) := ω : supt∈[0,T1] |ωt−ωn

t | ≤ δ,we observe that

supP∈M

G,[T1,T ]

Bn(δ)

EP(ξ) = supω∈Bn(δ)

supP∈M

G,[T1,T ]

Bω

EP(ξ) = supω∈Bn(δ)

ξ(ω) ≤ ξ(ωn) + ε(δ)

where the second equality follows from uniform continuity of ξ.

20

Fix ε > 0. Then we can chose δ > 0 and family of measures Pεn ∈ M

G,[T1,T ]Bn for each

n such that

ε/2 + EPεn(ξ) ≥ ξ(ω) ∀ω ∈ Bn(δ) and eξ(2δ

1/2) + 2||ξ||δ/(δ + δ1/2) ≤ ε/2.

Let us now define the FT1-measurable probability kernel N ε as

N ε(ω) := 11Cn(ω) Pεn αn,ω where Cn := Bn\

n−1⋃

k=1

Bk.

Probability kernel N ε is constructed such that it satisfies

ε+ ENε(ω)(ξ) ≥ ξ(ω) ∀ω ∈ Ω and N ε(ω) ∈ MG,[0,T1]Bω .

There as well exists a measure Pε ∈ MF,[0,T1]Ω such that ε+EPε(ξ) ≥ sup

P∈MF,[0,T1]Ω

EP(ξ).

The concatenation of measures Pε := Pε ⊗N ε (see Section 3.1 in [26]), defined, for eachF ∈ FT , as

Pε(F ) = EPε

(∑

n

11CnN ε(F ))

is a probability measure. Note that regular conditional probabilities of Pε w.r.t FT1 equalto N ε and d Pε|FT1

= dPε|FT1. Thus, for s ≤ t and Gs ∈ Gs, we have

EPε

((St − Ss)11Gs

)= EPε

(ENε((St − Ss)11Gs)

)

= EPε

(ENε((Ss∨T1 − Ss)11Gs)

)

= EPε

((Ss∨T1 − Ss)11Gs

)

= EPε

((Ss∨T1 − Ss)11Gs

)

= 0

which shows that S is a (Pε,G)-martingale. Moreover Pε satisfies

EPε(ξ) = EPε

(ENε(ξ)

)≥ EPε(ξ)− ε ≥ sup

P∈MF,[0,T1]Ω

EP(ξ)− 2ε.

The proof is completed.

Remark 5.3. The dynamic programming principle for P stated in Proposition 5.2 (ii)is linked to conditional sublinear expectations studied in [30, Theorem 2.3]. Since thereis more structure in our set-up we prove it relying on uniform continuity of ξ instead ofgeneral analytic selection argument.

Example 5.4. Consider Z = | lnST −lnST1 |11ST1=c∩ST>0. Then condition (12) is not

satisfied. It also well illustrates why we cannot demand uniform continuity of VG,[T1,T ]Ω (ξ)

or PG,[T1,T ]Ω (ξ) without assuming (12).

21

Using analogous argumentation as in Sections 3 and 4, we derive, in Theorem 5.5,that there is no duality gap in G. We treat independently each atom Bω of FT1 . Firstly,as in Theorem 4.5 for the corresponding G+ filtration, we look at the intersections witheach level set of r.v. Z, namely at the sets Bω ∩ Z = c with form the atoms of GT1 .Secondly we aggregate over Z, as in Theorem 4.10 for the corresponding G− filtration.The described operation reduces the problem to [0, T1] interval where G coincides withF where we conclude by dynamic programming principles.

Theorem 5.5. Let Z be a random variable such that for each c ∈ Z(Ω) and each FT1-atom Bω the set Z = c ∩ Bω satisfies Assumption 4.4 with no options (X = ∅) on[T1, T ] and that Assumption 4.9 holds. Assume moreover that MG

X ,P,Ω 6= ∅ and thatAssumption 4.1 holds. Then for any bounded uniformly continuous ξ

VG,[T1,T ]Ω (ξ)(ω) = P

G,[T1,T ]Ω (ξ)(ω) ∀ω and V G

X ,P,Ω(ξ) = PGX ,P,Ω(ξ).

Proof: In the first step of the proof we assume that X = ∅. We use the previoussubsection to show the following equalities:

VG,[0,T ]Ω (ξ) = V

F,[0,T1]Ω (V

G,[T1,T ]Ω (ξ)) = V

F,[0,T1]Ω (P

G,[T1,T ]Ω (ξ))

= PF,[0,T1]Ω (P

G,[T1,T ]Ω (ξ)) = P

G,[0,T ]Ω (ξ).

After applying Propositions 5.1 and 5.2, it remains to show the second and third equali-

ties. To prove the second equality that PG,[T1,T ]Ω (ξ) = V

G,[T1,T ]Ω (ξ) firstly we remark some

analogies with Section 3. Defining the (G+, [T1, T ])-superhedging cost VG+,[T1,T ]A as

VG+,[T1,T ]A := infx ∈ GT1 : ∃ γ ∈ A(G, [T1, T ]) such that x+

∫ T

T1

γudSu ≥ ξ on A,

analogously to Proposition 3.2 we have that

VG+,[T1,T ]Bω (ω′) = V

G+,[T1,T ]Bω∩Z=Z(ω′) for each ω and each ω′ ∈ Bω.

Analogously to Proposition 3.5, we deduce that

PG+,[T1,T ]Bω (ω′) = P

G+,[T1,T ]Bω∩Z=Z(ω′) for each ω and each ω′ ∈ Bω.

Then, by mimicking the proof of Theorem 4.10, we pass from "G+ to G−", and derivethat

VG,[T1,T ]Ω (ξ)(ω) = P

G,[T1,T ]Ω (ξ)(ω) ∀ω.

The third equality follows by general duality result on [0, T1].

The case of non-empty set of options X follows by Theorem 4.3.

22

Example 5.6. Analogously to Example 4.7 let us look at the additional informationin dynamic set-up which consists of a very detailed knowledge. We consider a one-dimensional setting with no statically or dynamically traded options: d = 1, K = 0 andX = ∅. The informed agent acquires at time T1 detailed knowledge of the stock pricesprocess of the form

Z =

supt∈[T1,T ] | lnSt − lnST1 | St > 0 ∀ t ∈ [T1, T ]

1 otherwise.

Note that for each c ∈ Z(Ω) and each FT1-atom Bω, one has that

MG,[T1,T ]Z=c∩Bω = M

F,[T1,T ]Z=c∩Bω 6= ∅

where the first equality holds since each Z = c ∩Bω is a GT1-atom. To show that

PF,[T1,T ]Z=c∩Bω = P

F,[T1,T ]Z=c∩Bω

we use the same arguments as in Example 4.7. Thus Assumption 4.4 with no options(X = ∅) on [T1, T ] is satisfied for each each set Z = c ∩Bω. Let cnn be a countablesubset of Z(Ω) such that c1 = 1. Then Assumption 4.9 is satisfied for cnn. Moreovernote that, by concatenation of measure argument, MG

Ω 6= ∅ and, since X = ∅, Assumption4.1 is also satisfied. We now apply Theorem 5.5 to show that there is no duality gap inG.

Example 5.7. Analogously to Example 4.8 we consider the additional information indynamic set-up of binary type. The same as before we consider a one-dimensional settingwith no statically or dynamically traded options: d = 1, K = 0 and X = ∅. The informedagent acquires at time T1 knowledge of of the form

Z = 11a<

StST1

<b ∀ t∈[T1,T ]

where a < 1 < b.

Along the lines of Example 5.6, applying Theorem 5.5, we deduce that there is no dualitygap in G.

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