Hedging of Game Options under Model Uncertainty in Discrete Time

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.


Introduction
A game contingent claim (GCC) or game option, which was introduced in [10], 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 time t then he receives the payment Y t , but if the seller exercises (cancels) the contract before the buyer then the latter receives X t . The difference ∆ t = X t − Y t is the penalty which the seller pays to the buyer for the contract cancellation. In short, if the seller will exercise at a stopping time σ ≤ T and the buyer at a stopping time τ ≤ T then the former pays to the latter the amount H(σ, τ ) where H(σ, τ ) = X σ I σ<τ + Y τ I τ ≤σ and we set I Q = 1 if an event Q occurs and I Q = 0 if not.
A hedge (for the seller) against a GCC is defined as a pair (π, σ) that consists of a self financing strategy π and a stopping time σ 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 [11] and the references therein. However, so far super-replication of American options and game options was not studied in the case of volatility uncertainty. In fact, super-replication under volatility uncertainty was studied only for European options (see, [6], [7], [13], [14] and [19]). In the papers (see, [7], [14] and [19]) the authors established a connection between G-expectation which was introduced by Peng (see [16] and [17]), 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.

Preliminaries and main results
First we introduce a discrete time version of volatility uncertainty. Let N ∈ N, The financial market consists of a savings account B and a risky asset S (stock). The stock price process is S k , k = 0, 1, ..., N , where N < ∞ 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 (S 0 , ..., S N ) ∈ K. Namely the initial stock price is S 0 = s and for any i < N we have | ln S i+1 − ln S i | ∈ I. This is the only assumption that we make on our financial market and we do not assume any probabilistic structure.
For any k = 0, 1, ..., N let F k , G k : K → R + be upper semicontinuous functions with the following properties, for any u, v ∈ K, F k (u) = F k (v) and G k (u) = G k (v) if u i = v i for all i = 0, 1, ..., k. Furthermore, we assume that F k ≤ G k .
Observe that H(k, l, S) is the reward that the buyer receives given that his exercise time is l and that the seller cancelation time is k. Furthermore, the reward H(k, l, S) depends only on the stock history up to the moment k ∧ l. In our setup a portfolio with initial capital x is a pair π = (x, γ) where γ : {0, 1, ..., N − 1} × K → R is a progressively measurable process, namely for any The portfolio value at time k is given by A stopping time is a measurable function σ : K → {0, 1, ..., N } which satisfies the following, for any u ∈ K and k = 0, 1, ..., N if σ(u) = k then σ(v) = k for any v with v i = u i for all i = 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 The super-replication price is given by (2.4) V = inf {V π 0 | there exists a stopping time σ such that (π, σ) is a perfect hedge} . 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 [6] for European options.
We make some preparations before we formulate the main result of the paper. Let Z = (Z 0 , ..., Z N ) be the canonical process on the Euclidean space R N +1 . Namely for any z = (z 0 , ..., z N ) ∈ R N +1 and k ≤ N we have Z k (z) = z k . A probability measure P supported on K is called a martingale law if for any k < N where E P denotes the expectation with respect to P. Denote by M the set of all martingale laws. Clearly, M = ∅. For instance the probability measure P b which is given by Let F k = σ(Z 0 , ..., Z k ), k ≤ N be the canonical filtration, and let T 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 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 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 ǫ > 0 there exists a perfect hedge (π,σ) with an initial capital Vπ 0 = V + ǫ. From (2.2) we get that for any P ∈ M the stochastic process {Vπ k (Z)} N k=0 is a martingale with respect to P. Observe thatσ(Z) ∈ T , and so from (2.3) we obtain that for any τ ∈ T The terms P ∈ M and τ ∈ T are arbitrary, thus we conclude that By letting ǫ ↓ 0 we derive (2.7). Remark 2.2. From Theorem 2.1 we obtain the following probabilistic corollary.
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 [12].

Proof of the main result
This section is devoted to the proof of (2.6), for the case where the functions 3.1. Discretization of the space. Let n ∈ N. Introduce the set Consider a multinomial model for which the stock price S = (S 0 , ..., S N ) lies in the set K n . As before the savings account is given by B ≡ 1. In this model a portfolio with an initial capital x is a pair π = (x, γ) where γ : {0, 1, ..., N − 1} × K n → R is a progressively measurable process. A hedge is a pair (π, σ) which consists of a portfolio strategy π and a stopping time σ. A stopping time is a map σ : .., N where the portfolio value is given by the same formula as (2.2). Let (3.2) V n = inf {V π 0 | there exists a stopping time σ such that (π, σ) is a perfect hedge} be the super-replication price in the multinomial model. Next, we introduce a modified super-replication price. Let M > 0 and let Γ M be the set of all portfolio . Namely, we consider portfolios for which the absolute value of the number of stocks is not exceeding M . Consider the super-replication price there exists a stopping time σ such that (π, σ) is a perfect hedge} .
We will need the following technical lemma.
Clearly there exists a perfect hedge with an initial capital A (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 V π 0 is no bigger than A > 0. Furthermore, since the option is exercised no later than in the moment σ(S), we can assume (without loss of generality) that γ(k, S) ≡ 0 for k ≥ σ(S).
First let us prove by induction that for any S ∈ K n and k = 0, 1, ..., N , If σ ≡ 0 then the statement is clear. Thus we assume that σ(S) > 0 for any (σ is a stopping time) S ∈ K n . Choose S ∈ K n . Clearly, the portfolio value at time 1 should be non negative, for any possible growth rate of the stock. In particular we have, . Thus (3.1) holds for k = 0. Next, assume that (3.3) holds for k, and we prove it for k + 1. From the induction assumption we get , as required. Next, if σ(S) ≤ k + 1 then γ(k + 1, S) = 0. If σ(S) > k + 1, then the portfolio value at time k + 2 should be non negative, for any possible growth rate of the stock. Thus, This completes the proof of (3.3). Finally, observe that S k ≥ se −bk and so, we conclude that for M := As e bN 1−e −b 1 + e b −1 , we have |γ(k, S)| ≤ M for all k, S. Now, we can easily prove the following lemma.
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 F k , G k , k = 0, 1, ..., N it follows that for sufficiently large n (3.5) where we denote ||(z 0 , ..., z N )|| = max 0≤i≤N |z i |. Let S ∈ K. Set Y (n) = ψ n (S). From (3.5) and the fact that γ ∈ [−M, M ] it follows that (for sufficiently large n) for any l ≤ N we get Thus for sufficiently large n, V ≤ 2ǫ + V n . Since ǫ > 0 was arbitrary this concludes the proof.

3.2.
Analysis of the multinomial models. Fix n ∈ N. Let Ω = R N +1 . Define the piecewise constant stochastic processes t=0 be the filtration which is generated by the process S (n) . The set K n ⊂ Ω is finite, and so, there exists a probability measure P n on Ω which is supported on K n and gives to any element in K n a positive probability. Thus we can apply Theorem 2.2 in [12] for a market with one risky asset S (n) which lives on the probability space (Ω, F (n) 1 , P n ), and a game option with the payoffs Y (n) ≤ X (n) . In this case the super-replication price coincides with V n which is given by (3.2). Thus let M n ⊂ M be the set of all martingale laws which are supported on the set K n and T be the set of all stopping times (with respect to the filtration {F  [12] and the fact that the processes Y (n) , X (n) are piecewise constant we obtain Since M n ⊂ M, we conclude that for any n ∈ N, This together with Lemma 3.2 completes the proof of (2.6). Our conjecture is that for regular enough payoffs the limit behavior of the super-replication prices V := V(N ) as N → ∞ is equal to a stochastic game version of G-expectation, defined on the canonical space C[0, T ]. For European options the limit is the standard G-expectation, this follows from [5] and [9]. It seems that the tool which was employed in [5] 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.

Extension for upper semicontinuous payoffs
In this section we prove (2.6) for the case where the functions F k , G k : K → R + , k ≤ N are upper semicontinuous (and not necessarily continuous).
Let A = max 0≤k≤N sup x∈K G k (x) < ∞. By using similar arguments as in Lemma 5.3 in [8] it follows that for any k = 0, 1, ..., N there are two sequences of continuous functions {F which satisfy the following: k , for all n. (ii).
for every x ∈ K and every sequence {x n } ∞ n=1 ⊂ K with lim n→∞ x n = x. (iii). Furthermore, for any n ∈ N and u, v ∈ K, F (n) Let V be the super-replication price which corresponds to the payoff functions F, G, and for any n ∈ N let V n be the super-replication price which corresponds to the payoff functions F (n) , G (n) .
From (i), it follows that for any n ∈ N, V ≤ V n . Thus from Theorem 2.1 (for continuous payoffs) it follows that where H (n) (k, l, S) = G We conclude that in order to establish (2.6) we need to prove the following lemma. Thus we will prove that (infact this is the inequality that we need) For any n ∈ N, let P n ∈ M and ρ n ∈ T be such that Consider the set Π of all probability measures on K, induced with the topology of weak convergence. Observe that Π is 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 [18] page 227) we obtain that there exist measurable functions h (n) k : R k+1 → Π, k < N , such that for any Borel set A ⊂ K and n ∈ N P n ((Z 0 , ..., Z N ) ∈ A|Z 0 , ..., Z k ) = h Since the space [0, N ] × K × Π N is compact then by Prohorov's theorem there is a subsequence which for simplicity we still denote by N −1 (Z 0 , ..., Z N −1 )), n ∈ N which converges weakly. Thus from the Skorohod representation theorem (see [3]) we obtain that we can redefine the sequence (ρ n , Z 0 , ..., Z N , h (II). For any k, the conditional distribution of (U 0 , ..., U N ) given U 0 , ..., U k equals to W k . (III). For any k, the event {τ = k} and G N are independent given G k .
Denote by E the expectation with respect to P . From Lebesgue's dominated convergence theorem if follows that for any k ≤ N and continuous bounded func- where the last equality follows the fact that P n ∈ M is a martingale distribution. From the definition of the topology on Π, we also have E(f (U 0 , ..., U k )g(U 0 , ..., U N )) = (4.6) = E(f (U 0 , ..., U k ) g(y)W k (dy)).
By applying standard density arguments we obtain that (4.5) implies (I) and (4.6) implies (II). Next, fix k. From (II) and the fact that ρ n is a stopping time we obtain that E(I ρ=k E(g(U 0 , ..., U N )|G k )) = E(I ρ=k g(y)W k (dy)) = lim n→∞ E(I ρn=k g(y)h and (III) follows. Property (III) is important because it implies the following. For any stochastic process (L 0 , ..., L N ) which is adapted to the filtration G k , k ≤ N , we have The proof of this implication can be done in the same way as in Lemma 3.3 in [4], and so we omit it. Now we arrive to the final step of the proof. Choose 0 < ǫ < 1. Letσ ∈ T U be such that where U = (U 0 , ..., U N ). For any k there exists a continuous function f k : R k+1 → R such that P (Iσ =k = f k (U 0 , ..., U k )) < ǫ 2 k+1 . For any n ∈ N defineσ n = N ∧ min{k|f k (Z N . Let C be the following set C = {ω ∈ Ω|∃m := m(ω) such that ∀n > mσ n (ω) =σ(ω)}.
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.