Hedging of game options under model uncertainty in discrete time

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DOI:10.1214/ECP.v19-2714 ISSN:1083-589X

COMMUNICATIONS in PROBABILITY

Hedging of game options under model uncertainty in discrete time

Yan Dolinsky

^∗

Abstract

We introduce a setup of model uncertainty in discrete time. In this setup we derive dual expressions for the super–replication prices of game options with upper semicontinuous payoffs. We show that the super–replication price is equal to the supremum over a special (non dominated) set of martingale measures, of the corresponding Dynkin games values. This type of results is also new for American options.

Keywords: Dynkin games; game options; super–replication; volatility uncertainty; weak convergence.

AMS MSC 2010:Primary: 91G10 Secondary: 60F05, 60G40.

Submitted to ECP on March 31, 2013, final version accepted on December 8, 2013.

1 Introduction

A game contingent claim (GCC) or game option, which was introduced in [11], is defined as a contract between the seller and the buyer of the option such that both have the right to exercise it at any time up to a maturity date (horizon)T. If the buyer exercises the contract at timetthen he receives the paymentYt, but if the seller exercises (cancels) the contract before the buyer then the latter receivesXt. The difference

∆t =Xt−Ytis the penalty which the seller pays to the buyer for the contract cancellation. In short, if the seller will exercise at a stopping timeσ ≤T and the buyer at a stopping timeτ≤T then the former pays to the latter the amountH(σ, τ)where

H(σ, τ) =XσI^σ<τ+YτI^τ≤σ and we setIQ = 1if an eventQoccurs andIQ = 0if not.

A hedge (for the seller) against a GCC is defined as a pair(π, σ)that consists of a self financing strategyπ and a stopping timeσ which is the cancellation time for the seller. A hedge is called perfect if no matter what exercise time the buyer chooses, the seller can cover his liability to the buyer.

Until now there is quite a good understanding of pricing game options in the case where the probabilistic model is given. For details see [12] and the references therein.

However, for the case of volatility uncertainty, there are only few papers which deal with American options and game options (see [14] and [15]). For European options the topic of super–replication under volatility uncertainty was widely studied (see for instance, [3], [7], [8], [16], [17] and [22]). In the papers (see, [8], [17] and [22]) the authors established a connection between G–expectation which was introduced by Peng (see

∗Hebrew University, Israel. E-mail:[email protected]

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[19] and [20]), and super–replication under volatility uncertainty in continuous time models.

In this paper we introduce a discrete setup of volatility uncertainty. We consider a simple model which consists of a savings account and of one risky asset, and we assume that the payoffs are upper semicontinuous. Our main result says that the super–

replication price is equal to the supremum over a special (non dominated) set of martingale measures, of the corresponding Dynkin games values. In continuous time models, the problem remains open for American options and game options.

Main results of this paper are formulated in the next section. In Section 3 we prove the main results of the paper for continuous payoffs. This proof is quite elementary and does not use advanced tools. In section 4 we extend the main results for upper semicontinuous payoffs. This extension is technically involved and requires the establishment of some stability results for Dynkin games under weak convergence. In Section 5 we prove an Auxiliary Lemma that we use in Section 4.

2 Preliminaries and main results

First we introduce a discrete time version of volatility uncertainty. LetN∈N^,s >0 andI= [a, b]⊂R⁺. Define the setK⊂R^N+1++ by

K={(x0, ..., xN) :x0=s, |lnxi+1−lnxi| ∈I, i < N}.

The financial market consists of a savings accountBand a risky assetS(stock). The stock price process isSk,k= 0,1, ..., N, whereN <∞is the maturity date or the total number of allowed trades. By discounting, we normalize B ≡1. We assume that the stock price process satisfies(S₀, ..., S_N)∈ K. Namely the initial stock price isS₀ =s and for anyi < N we have|lnS_i+1−lnS_i| ∈I. This is the only assumption that we make on our financial market and we do not assume any probabilistic structure.

For anyk= 0,1, ..., N letFk, Gk :K →R⁺ be upper semicontinuous functions with the following properties, for anyu, v ∈ K, F_k(u) = F_k(v)andG_k(u) =G_k(v)ifu_i =v_i for alli= 0,1, ..., k. Furthermore, we assume thatF_k≤G_k.

Consider a game option with the payoff function

H(k, l, S) =Gk(S)I^k<l+Fl(S)Il≤k, k, l= 0,1, ..., N. (2.1) Observe thatH(k, l, S)is the reward that the buyer receives given that his exercise time island that the seller cancelation time isk. Furthermore, the rewardH(k, l, S)depends only on the stock history up to the momentk∧l.

In our setup a portfolio with initial capitalxis a pairπ= (x, γ)whereγ:{0,1, ..., N− 1} ×K → Ris a progressively measurable process, namely for anyk = 0,1, ..., N−1 andu, v∈K,γ(k, u) =γ(k, v)ifui =vifor alli= 0,1, ..., k. The portfolio value at timek is given by

V_k^π(S) =x+

k−1

X

i=0

γ(i, S)(Si+1−Si), S∈K, k= 0,1, ..., N. (2.2)

A stopping time is a measurable functionσ : K → {0,1, ..., N} which satisfies the following, for anyu ∈ K and k = 0,1, ..., N if σ(u) = k thenσ(v) = k for any v with vi=uifor alli= 0,1, ..., k.

A pair (π, σ) of a self financing strategy π and a stopping time σ will be called a hedge. A hedge(π, σ)will called perfect if

V_σ(S)∧l^π (S)≥H(σ(S), l, S), ∀S∈K, l= 0,1, ..., N. (2.3)

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The super–replication price is given by

V= inf{V₀^π|there exists a stopping timeσsuch that(π, σ)is a perfect hedge}. (2.4) Observe that we do not have any underlying probability measure, and we require to construct a super–hedge for any possible values of the stock prices. Similar setup (but not the same) was studied in [7] for European options.

We make some preparations before we formulate the main result of the paper. Let Z = (Z0, ..., ZN)be the canonical process on the Euclidean space R^N⁺¹. Namely for anyz = (z0, ..., zN)∈ R^N⁺¹ ^andk ≤N we haveZk(z) = zk. A probability measureP supported onKis called a martingale law if for anyk < N

EP(ZN|Z0, ..., Zk) =Zk P ^a.s. ^(2.5) whereEP denotes the expectation with respect toP. Denote byMthe set of all martingale laws. Clearly,M 6= ∅. For instance the probability measure P^b which is given by

P^b(Z0=s) = 1 and

P^b(lnZi+1−lnZi =b) = 1−P^b(lnZi+1−lnZi=−b) =_e^1−e_b_−e^−b−b, i < N, is an element inM.

Let Fk =σ(Z0, ..., Zk),k ≤N be the canonical filtration, and letT be the set of all stopping times (with respect to the above filtration) with values in the set{0,1, ..., N}.

The following theorem is the main result of the paper.

Theorem 2.1. The super–replication price is given by

V= inf_σ∈T sup_P_∈Msup_τ∈T EPH(σ, τ, Z) =

sup_P∈Minfσ∈T sup_τ∈T EPH(σ, τ, Z) = sup_P∈Msup_τ∈Tinfσ∈T EPH(σ, τ, Z).

It is well known that inf sup ≥ sup inf, thus in order to prove Theorem 2.1 it is sufficient to prove the following relations

V≤ sup

P∈M

sup

τ∈T

σ∈Tinf EPH(σ, τ, Z) (2.6) and

V≥ inf

σ∈T sup

P∈M

sup

τ∈TEPH(σ, τ, Z). (2.7)

The first inequality is the difficult one and it will be proved in Sections 3–4. The second inequality is simpler and we show it by the following argument.

From (2.4) it follows that for any >0 there exists a perfect hedge(˜π,σ)˜ with an initial capitalV₀^π^˜=V+. From (2.2) we get that for anyP∈ Mthe stochastic process {V_k^π^˜(Z)}^N_k=0is a martingale with respect toP. Observe thatσ(Z)˜ ∈ T, and so from (2.3) we obtain that for anyτ∈ T

V+=V₀^π^˜=EPV_˜_σ(Z)∧τ^π^˜ ≥EPH(˜σ(Z), τ, Z).

The termsP∈ Mandτ ∈ T are arbitrary, thus we conclude that V+≥ sup

P∈M

sup

τ∈TEPH(˜σ(Z), τ, Z)≥ inf

σ∈T sup

P∈M

sup

τ∈TEPH(σ, τ, Z).

By letting↓0we derive (2.7).

Remark 2.2. From Theorem 2.1 we obtain the following probabilistic corollary.

σ∈Tinf sup

P∈M

sup

τ∈TEPH(σ, τ, Z) = sup

P∈M

sup

τ∈T

σ∈Tinf EPH(σ, τ, Z).

This corollary is not obvious since the set M is a set of non dominated probability measures, and so it does not follow from the results in [13].

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3 Proof of the main result

This section is devoted to the proof of (2.6), for the case where the functionsFk, Gk : K→R⁺^,k≤N are continuous.

3.1 Discretization of the space Letn∈N. Introduce the set

Kn :={(x0, ..., xN) :x0=s and

|lnxi+1−lnxi| ∈ {a, a+ (b−a)/n, a+ 2(b−a)/n, ..., b}}.

Consider a multinomial model for which the stock priceS = (S₀, ..., S_N)lies in the set K_n. As before the savings account is given byB ≡1. In this model a portfolio with an initial capitalxis a pairπ= (x, γ)whereγ:{0,1, ..., N−1} ×Kn→Ris a progressively measurable process. A hedge is a pair(π, σ)which consists of a portfolio strategyπand a stopping timeσ. A stopping time is a map σ:Kn → {0,1, ..., N} which satisfies that ifσ(u) = kthenσ(v) =kfor anyvwithvi =ui for alli= 0,1, ..., k. A hedge(π, σ)will called perfect if

V_σ(S)∧l^π (S)≥H(σ(S), l, S), ∀S ∈Kn, l= 0,1, ..., N (3.1) where the portfolio value is given by the same formula as (2.2).

Let

Vn= inf{V₀^π|there exists a stopping timeσsuch that(π, σ)is a perfect hedge} (3.2) be the super–replication price in the multinomial model. Next, we introduce a modified super–replication price. Let M > 0 and let ΓM be the set of all portfolio strategies π= (x, γ)whereγ : {0,1, ..., N−1} ×Kn → [−M, M]. Namely, we consider portfolios for which the absolute value of the number of stocks is not exceedingM. Consider the super–replication price

V^Mn = inf

π∈ΓM

{V₀^π|there exists a stopping timeσsuch that(π, σ)is a perfect hedge}. We will need the following technical lemma.

Lemma 3.1. There exists a constantM >0(which is independent ofn) such that V^Mn =Vⁿ.

Proof. Clearly,V^Mn ≥Vⁿ.Thus its sufficient to show thatV^Mn ≤Vⁿ.Set A= max

0≤k≤N sup

x∈K

Fk(x).

Clearly there exists a perfect hedge with an initial capitalA(in this case the investor does not trade and stop only at the maturity). Let(π, σ)be a perfect hedge in the sense of (3.1). We will assume (without loss of generality) that the initial capital V₀^π is no bigger than A > 0. Furthermore, since the option is exercised no later than in the momentσ(S), we can assume (without loss of generality) thatγ(k, S)≡0fork≥σ(S).

First let us prove by induction that for anyS∈K_nandk= 0,1, ..., N,

V_k∧σ(S)^π (S)≤A 1 +e^b^k

and |γ(k, S)| ≤ A 1 +e^b^k

(1−e^−b)S_k. (3.3)

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Ifσ≡0 then the statement is clear. Thus we assume thatσ(S)>0for anyS ∈K_n (σ is a stopping time, and so eitherσ≡0orσ(S)>0∀S ∈K_n). ChooseS ∈K_n. Clearly, the portfolio value at time1should be non negative, for any possible growth rate of the stock. In particular we have,

V₀^π(S) +γ(0, S)s(e^b−1)≥0 and V₀^π(S) +γ(0, S)s(e^−b−1)≥0

and we conclude that|γ(0, S)| ≤ _s(1−e^A−b). Thus (3.1) holds fork= 0. Next, assume that (3.3)holds fork, and we prove it fork+ 1. From the induction assumption we get

V_{(k+1)∧σ(S)}^π (S) =V_k∧σ(S)^π (S) +γ(k∧σ(S), S)(S_{(k+1)∧σ(S)}−S_k∧σ(S))≤

A 1 +e^bk

+ ^A(^1+e^b)^k

(1−e^−b)S_kSk(e^b−1)≤A 1 +e^bk+1

,

as required. Next, ifσ(S)≤k+ 1thenγ(k+ 1, S) = 0. Ifσ(S)> k+ 1, then the portfolio value at timek+ 2should be non negative, for any possible growth rate of the stock.

Thus,

V_k+1^π (S) +γ(k+ 1, S)S_k+1(e^b−1)≥0 and V_k+1^π (S) +γ(k+ 1, S)S_k+1(e^−b−1)≥0 and so,

|γ(k+ 1, S)| ≤ V_k+1^π (S)

(1−e^−b)S_k+1 ≤ A 1 +e^b^k+1 (1−e^−b)S_k+1.

This completes the proof of (3.3). Finally, observe thatSk ≥se^−bkand so, we conclude that forM :=As_1−e^e^bN−b 1 +e^bN

, we have|γ(k, S)| ≤M for allk, S. Now, we can easily prove the following lemma.

Lemma 3.2.

V≤lim inf

n→∞Vⁿ.

Proof. Fix > 0. Let n ∈ N. Consider the multinomial model for which the stock price processS = (S0, ..., SN)lies in the setKn. Let(π, σ)be a perfect hedge for this multinomial model such thatπ = (Vn +, γ). From lemma 3.1 it follows that we can assume that|γ(k, S)| ≤M for anyk, S. Consider the mapψ_n :K →K_n which is given byψn(y0, ..., yN) = (x0, ..., xN)where

x₀=y₀ and for k >0 lny_i+1= lny_i

+sgn(lnxi+1−lnxi)(a+ (b−a)[n(|lnxi+1−lnxi| −a)/(b−a)]/n)

where[v]is the integer part ofvandsgn(v) = 1forv >0and=−1otherwise. For the original financial market define a hedge(˜π,σ)˜ by the following relations,π˜ = (Vⁿ+2,˜γ) where

˜

γ(k, S) =γ(k, ψn(S)) and σ(S) =˜ σ(ψn(S)), k < N, S∈K. (3.4) Observe thatγ˜is a progressively measurable map andσ˜ is a stopping time. Thus(˜π,σ)˜ is indeed a hedge for the original financial market. From the continuity of the functions Fk, Gk,k= 0,1, ..., Nit follows that for sufficiently largen

||S−ψ_n(S)||+|Fk(S)−F_k(ψ_n(S))|+|Gk(S)−G_k(ψ_n(S))|<

2M N, S∈K, k≤N, (3.5) where we denote||(z0, ..., zN)|| = max_0≤i≤N|zi|. Let S ∈ K. SetY⁽ⁿ⁾ = ψn(S). From (3.5) and the fact thatγ∈[−M, M]it follows that (for sufficiently largen) for anyl≤N

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we get

V_l∧˜^π^˜_σ(S)(S) =+V_l∧σ(Y^π _(n)₎(Y⁽ⁿ⁾) + Pl∧˜σ(S)−1

k=0 γ(k, Y⁽ⁿ⁾)((Sk+1−Sk)−(Y_k+1⁽ⁿ⁾−Y_k⁽ⁿ⁾))≥ +H(σ(Y⁽ⁿ⁾), l, Y⁽ⁿ⁾)−2N M||S−Y⁽ⁿ⁾| ≥H(˜σ(S), l, S).

Thus for sufficiently largen,V≤2+Vⁿ^{. Since} >0was arbitrary this concludes the proof.

3.2 Analysis of the multinomial models

Fixn∈N^{. Let}Ω =R^N⁺¹. Define the piecewise constant stochastic processes S_t⁽ⁿ⁾(z₀, ..., z_N) :=z_[nt], Y_t⁽ⁿ⁾(z₀, ..., z_N) =F_[nt](z₀, ..., z_N)

and X_t⁽ⁿ⁾=G_[nt](z₀, ..., z_N), z∈Ω, t∈[0,1].

Let{F_t⁽ⁿ⁾}¹_t=0be the filtration which is generated by the processS⁽ⁿ⁾. The setK_n ⊂Ωis finite, and so, there exists a probability measurePn onΩwhich is supported onK_nand gives to any element inKna positive probability. Thus we can apply Theorem 2.2 in [13]

for a market with one risky assetS⁽ⁿ⁾which lives on the probability space(Ω,F₁⁽ⁿ⁾,Pⁿ), and a game option with the payoffsY⁽ⁿ⁾≤X⁽ⁿ⁾. In this case the super–replication price coincides withVnwhich is given by (3.2). Thus letMn ⊂ Mbe the set of all martingale laws which are supported on the setK_n and T be the set of all stopping times (with respect to the filtration {F_t⁽ⁿ⁾}¹_t=0) with values in the set[0,1]. From Theorem 2.2 in [13] and the fact that the processesY⁽ⁿ⁾, X⁽ⁿ⁾are piecewise constant we obtain

Vⁿ = sup

P∈Mn

sup

τ∈T

σ∈infTEP(X_σ⁽ⁿ⁾I^σ<τ+Y_τ⁽ⁿ⁾Iσ≥τ) = sup

P∈Mn

sup

τ∈T

SinceMn⊂ M, we conclude that for anyn∈N^, Vn≤ sup

P∈M

sup

τ∈T

This together with Lemma 3.2 completes the proof of (2.6).

Remark 3.3. An interesting question which remains open is the limit behavior where the maturity dateN goes to infinity. Namely, for a givenN ∈ N consider the interval I := I(N) = h

√a N,^√^b

N

i

. Our conjecture is that for regular enough payoffs the limit behavior of the super–replication pricesV:=V(N)asN → ∞is equal to a stochastic game version ofG–expectation, defined on the canonical spaceC[0, T]. For European options the limit is the standardG–expectation, this follows from [6] and [10]. It seems that the tool which was employed in [6] can work for the American options case. In this case the limit of the super–replication prices is equal to an optimal stopping version of G–expectation. However for game options the problem is more complicated.

4 Extension for upper semicontinuous payoffs

In this section we prove (2.6) for the case where the functions Fk, Gk : K → R⁺^, k≤N are upper semicontinuous (and not necessarily continuous).

Let A = max_0≤k≤Nsup_x∈KGk(x) < ∞. By using similar arguments as in Lemma 5.3 in [9] it follows that for anyk = 0,1, ..., N there are two sequences of continuous functions{F_k⁽ⁿ⁾}^∞_n=1and{G⁽ⁿ⁾_k }^∞_n=1which satisfy the following:

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(i).A ≥G⁽ⁿ⁾_k ≥Gk,A ≥F_k⁽ⁿ⁾≥Fk andG⁽ⁿ⁾_k ≥F_k⁽ⁿ⁾, for alln. (ii).

lim sup

n→∞

G⁽ⁿ⁾_k (xn)≤Gk(x) and lim sup

n→∞

F_k⁽ⁿ⁾(xn)≤Fk(x) (4.1) for everyx∈Kand every sequence{xn}^∞_n=1⊂Kwithlim_n→∞xn=x.

(iii). Furthermore, for anyn∈N^andu, v ∈K,F_k⁽ⁿ⁾(u) =F_k⁽ⁿ⁾(v)andG⁽ⁿ⁾_k (u) =G⁽ⁿ⁾_k (v) ifui=vifor alli= 0,1, ..., k.

LetVbe the super–replication price which corresponds to the payoff functionsF, G, and for anyn∈N^letVnbe the super–replication price which corresponds to the payoff functionsF⁽ⁿ⁾, G⁽ⁿ⁾.

From (i), it follows that for anyn∈N^,V≤Vn. Thus from Theorem 2.1 (for continuous payoffs) it follows that

V≤lim inf

n→∞ sup

P∈M

sup

τ∈T

σ∈Tinf EPH⁽ⁿ⁾(σ, τ, Z)

where

H⁽ⁿ⁾(k, l, S) =G⁽ⁿ⁾_k (S)Ik<l+F_l⁽ⁿ⁾(S)Il≤k, k, l= 0,1, ..., N, S∈K.

We conclude that in order to establish (2.6) we need to prove the following lemma.

Lemma 4.1.

sup

P∈M

sup

τ∈T

σ∈Tinf EPH(σ, τ, Z) = lim inf

n→∞ sup

P∈M

sup

τ∈T

σ∈Tinf EPH⁽ⁿ⁾(σ, τ, Z).

Proof. From (i), sup

P∈M

sup

τ∈T

inf

σ∈TEPH(σ, τ, Z)≤lim inf

n→∞ sup

P∈M

sup

τ∈T

inf

σ∈TEPH⁽ⁿ⁾(σ, τ, Z).

Thus we will prove that (infact this is the inequality that we need) sup

P∈M

sup

τ∈T

σ∈Tinf EPH(σ, τ, Z)≥lim inf

n→∞ sup

P∈M

sup

τ∈T

σ∈Tinf EPH⁽ⁿ⁾(σ, τ, Z). (4.2) For anyn∈N^{, let}Pn∈ Mandρ_n∈ T be such that

sup

P∈M

sup

τ∈T

σ∈Tinf EPH⁽ⁿ⁾(σ, τ, Z)< 1 n+ inf

σ∈TEPnH⁽ⁿ⁾(σ, ρn, Z). (4.3) Consider the setΠof all probability measures onK, induced with the topology of weak convergence. Observe thatΠis a compact set (this follows from Prohorov’s theorem, see [2] Section 1 for details). From the existence of the regular distribution function (for details see [21] page 227) we obtain that there exist measurable functions h⁽ⁿ⁾_k : R^k+1→Π,k < N, such that for any Borel setA⊂Kandn∈N

Pn((Z₀, ..., Z_N)∈A|Z0, ..., Z_k) =h⁽ⁿ⁾_k (Z₀, ..., Z_k)(A), Pn a.s. For anyn∈Nconsider the distribution of (under the measurePn)

(ρ_n, Z₀, ..., Z_N, h⁽ⁿ⁾₀ (Z₀), ..., h⁽ⁿ⁾_N−1(Z₀, ..., Z_N₋₁))

on the space{0,1, ..., N} ×K×Π^N with the product topology.

Since the space{0,1, ..., N} ×K×Π^N is compact then by Prohorov’s theorem there is a subsequence which for simplicity we still denote by

(ρn, Z0, ..., ZN, h⁽ⁿ⁾₀ (Z0), ..., h⁽ⁿ⁾_N₋₁(Z0, ..., ZN−1)), n∈N

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which converges weakly. Thus from the Skorohod representation theorem (see [4]) we obtain that we can redefine the sequence

(ρn, Z0, ..., ZN, h⁽ⁿ⁾₀ (Z0), ..., h⁽ⁿ⁾_N₋₁(Z0, ..., Z_N−1)), n∈N on a probability space(Ω,F, P)such that we haveP a.s convergence

(ρn, Z₀⁽ⁿ⁾, ..., Z_N⁽ⁿ⁾, h⁽ⁿ⁾₀ (Z₀⁽ⁿ⁾), ..., h⁽ⁿ⁾_N₋₁(Z₀⁽ⁿ⁾, ..., Z_N⁽ⁿ⁾₋₁))→(ρ, U0, ..., UN, W1, ..., WN).

(4.4) Redefining means that for anyn∈Nthe distribution of

(ρn, Z0, ..., ZN, h⁽ⁿ⁾₀ (Z0), ..., h⁽ⁿ⁾_N−1(Z0, ..., ZN−1))

underPn is equal to the distribution of

(ρn, Z₀⁽ⁿ⁾, ..., Z_N⁽ⁿ⁾, h⁽ⁿ⁾₀ (Z₀⁽ⁿ⁾), ..., h⁽ⁿ⁾_N₋₁(Z₀⁽ⁿ⁾, ..., Z_N⁽ⁿ⁾₋₁))

underP. LetGk=σ{U0, ..., Uk},k≤Nbe the filtration which is generated byU0, ..., UN. Denote byTU the set of all stopping times (with respect to this filtration) with values in the set {0,1, ..., N}. From Lemma 5.1 (property (III)) it follows that for any stochastic process(L0, ..., LN)which is adapted to the filtrationGk,k≤N, we have

ELρ≤ sup

τ∈TU

ELτ (4.5)

whereE denotes the expectation with respect toP. The proof of this implication can be done in the same way as in Lemma 3.3 in [5], and so we omit it.

Next, choose0< <1. Letσ˜∈ TU be such that

σ∈TinfU

EH(σ, ρ, U)> EH(˜σ, ρ, U)−, (4.6) whereU = (U0, ..., UN). For anykthere exists a continuous functionfk :R^k+1→R^such thatP(I^˜^σ=k 6=fk(U0, ..., Uk))<₂_k+1 . For anyn∈N^defineσ˜n=N∧min{k|fk(Z₀⁽ⁿ⁾, ..., Z_k⁽ⁿ⁾)

>¹₂}. Clearlyσ˜_nis a stopping time with respect to the filtration generated byZ₀⁽ⁿ⁾, ..., Z_N⁽ⁿ⁾. LetCbe the following set

C={ω∈Ω|∃m:=m(ω) such that∀n > mσ˜n(ω) = ˜σ(ω)}.

From (4.4) and the fact thatf_k,k≤N are continuous functions, it follows that

P(C)≥1−

N

X

i=0

2ⁱ⁺¹ ≥1−. (4.7)

Observe that (4.4) also implies that a.s. ρ_n(ω) = ρ(ω)for sufficiently large n (which depends onω). Thus from (4.4) we get

H(˜σ, ρ, U)I^C≥lim sup

n→∞H⁽ⁿ⁾(˜σn, ρn, Z⁽ⁿ⁾)I^C

whereZ⁽ⁿ⁾ = (Z₀⁽ⁿ⁾, ..., Z_N⁽ⁿ⁾). SinceH^andH⁽ⁿ⁾are uniformly bounded byAthen from Fatou’s lemma we derive

EH(˜σ, ρ, U)I^C≥lim sup

n→∞

EH⁽ⁿ⁾(˜σn, ρn, Z⁽ⁿ⁾)I^C. (4.8) Finally, letQbe the distribution of(U0, ..., UN). From Lemma 5.1 (property (I)) it follows that Q ∈ Mis a martingale distribution. It is well known that for Dynkin games the

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inf and thesupcan be exchanged (for details see [18]). Thus from (4.3)–(4.4), (4.5) for L_k=H(σ, k, U)and (4.6)–(4.8) we get

sup_P_∈Msup_τ∈T inf_σ∈T EPH(σ, τ, Z)≥sup_τ∈T

Uinf_σ∈T_UEH(σ, τ, U)

= inf_σ∈T_Usup_τ∈T

UEH(σ, τ, U)≥inf_σ∈T_UEH(σ, ρ, U)

≥EH(˜σ, ρ, U)I^C−≥lim sup_n→∞EH⁽ⁿ⁾(˜σn, ρn, Z⁽ⁿ⁾)I^C−

≥lim sup_n→∞EH⁽ⁿ⁾(˜σn, ρn, Z⁽ⁿ⁾)− A−

≥lim sup_n→∞sup_P∈Msup_τ∈T inf_σ∈T EPH⁽ⁿ⁾(σ, τ, Z)−(A+ 1),

and sincewas arbitrary we obtain (4.2) as required. The reason that we havelim sup in the above equation and notlim infas in (4.2), is because we passed to a subsequence, but left the same notations.

Remark 4.2. Let us notice that in order to obtain Lemma 4.1 we used a stronger form of the standard weak convergence. Namely we also required a convergence of the conditional distributions. This is the discrete analog of the extended weak convergence which introduced by Aldous in [1] for continuous time processes. In general, the standard weak convergence is not sufficient for the convergence of the corresponding optimal stopping and Dynkin games values.

5 Auxiliary Lemma

In this section we establish several essential properties of the random vector(ρ, U0, ..., UN, W1, ..., WN)from (4.4).

Lemma 5.1.

(I). The distribution of(U0, ..., UN)(on the spaceK) is an element inM.

(II). For anyk, the conditional distribution of(U0, ..., UN)givenU0, ..., Uk equals toWk. (III). For anyk, the event{ρ=k}andGN are independent givenGk.

Proof.

(I). From Lebesgue’s dominated convergence theorem it follows that for anyk≤N and continuous bounded functionsf :R^k+1→R^,g:K→R^{we have}

E((UN−Uk)f(U0, ..., Uk)) = (5.1) lim_n→∞E((Z_N⁽ⁿ⁾−Z_k⁽ⁿ⁾)f(Z₀⁽ⁿ⁾, ..., Z_k⁽ⁿ⁾))

= limn→∞EPn((ZN −Zk)f(Z0, ..., Zk)) = 0,

where the last equality follows the fact that Pⁿ ∈ Mis a martingale distribution. By applying standard density arguments we obtain that (5.1) implies (I).

(II). From the definition of the topology onΠwe have

E(f(U₀, ..., U_k)g(U₀, ..., U_N)) = (5.2) limn→∞E(f(Z₀⁽ⁿ⁾, ..., Z_k⁽ⁿ⁾)g(Z₀⁽ⁿ⁾, ..., Z_N⁽ⁿ⁾))

= limn→∞E(f(Z₀⁽ⁿ⁾, ..., Z_k⁽ⁿ⁾)R

g(y)h⁽ⁿ⁾_k (Z₀⁽ⁿ⁾, ..., Z_k⁽ⁿ⁾)(dy))

=E(f(U0, ..., Uk)R

g(y)Wk(dy)).

Again, by applying standard density arguments we obtain that (5.2) implies (II).

(III). Next, fixk. The random variables ρn,n∈Ntake on values on the set{0,1, ..., N} and so from the factρn → ρit follows thatlimn→∞I^ρn=k =I^ρ=k. Thus, from (II), the

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definition of the topology onΠand the fact thatρ_nis a stopping time we obtain that E(I^ρ=kE(g(U0, ..., UN)|Gk)) =E(I^ρ=kR

g(y)Wk(dy)) = lim_n→∞E(Iρn=k

Rg(y)h⁽ⁿ⁾_k (Z₀⁽ⁿ⁾, ..., Z_k⁽ⁿ⁾)(dy)) = lim_n→∞E(Iρn=kE(g(Z₀⁽ⁿ⁾, ..., Z_N⁽ⁿ⁾)|Z₀⁽ⁿ⁾, ..., Z_k⁽ⁿ⁾)) = limn→∞E(g(Z₀⁽ⁿ⁾, ..., Z_N⁽ⁿ⁾)I^ρn=k) =E(g(U0, ..., UN)I^ρ=k), and again, from standard density arguments we conclude that

E(I^ρ=k|Gk) =E(I^ρ=k|GN) and (III) follows.

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Hedging of game options under model uncertainty in discrete time