LetT>0 be afinite time horizon and setT ={0,1, . . .,T}: thefinancial market model consists of one risk-less asset with price process normalized to one and d risky assets with price process S = {St = (S1t, ...,Sdt)∗,t = 0, ...,T}valued in (0,∞)d. Here the notation∗is for the transposition. The stochastic price process (St)t∈T is defined on a complete probability space (Ω,F,P) equipped with thefiltrationF = {Ft,t ∈ T }, where the σ-field Ftis generated by the random variablesS0,S1,ã ã ã,St. We make the usual assumption thatF0is trivial andFT=F.
A trading portfolio is a Rd-valued F-adapted process φ = {φt = (φ1t, . . . , φdt)∗,t = 0, ...,T−1}, where φit represents the amount of wealth invested in thei-th risky asset at timet. TheR-valuedF-adapted process C ={Ct,t ∈ T }represents the cumulative consumption process. We as- sume thatC0 = 0 and thatCis non-decreasing. We also use the notation
∆St=St−St−1and∆Ct =Ct−Ct−1, fort=1, ...,T.
Given an initial wealthx∈ R, a trading portfolioφ,and a cumulative consumption processC, the wealth processXx,φ,Cis governed by :
Xx0,φ,C=x,
Xxt,φ,C=Xxt−,φ,1C+φ∗t−1∆St−∆Ct, fort=1, . . . ,T.
(1)
The induction equation (1) leads to Xx,φ,Ct =x+
t u=1
φ∗u−1∆Su−Ct, t∈ T.
The conditionC= 0 means that the portfolioφis self-financed. We now impose some constraints on the trading portfolios. LetK be a closed convex cone ofRdwith vertex in 0. For anyx∈[0,∞), we say that a trading strategy (x, φ,C) is admissible, and we denote (x, φ,C)∈ A,if for allt= 0, . . . ,T−1, φt ∈ K a.s. Such constraints cover in particular the case of incomplete markets (K ={k∈ Rd :ki =0,i=1, ...,n}: it is impossible to trade in thenfirst risky assets) and short-sales constraints (K=[0,∞)d).
LetHbe an European contingent claim, i.e., aFT-measurable random variable. Following F¨ollmer and Kramkov [11], we introduce the notion of minimal hedging strategy forH. First, an EuropeanHhedging strategy is a strategy (x, φ,C)∈ Asuch thatXxT,φ,C≥H a.s. We will denote byAeHthe set of EuropeanHhedging strategies. Then, ( ˆx,φ,ˆ C)ˆ ∈ AeH is minimal if for all (x, φ,C)∈ AeHXxt,φ,C≥Xtxˆ,φ,ˆCˆa.s. for allt∈ T. Note that ˆxis then the so-called super-replication costpe(H) ofH, i.e. the minimal initial capital needed for hedging without riskH:
pe(H)=inf{x∈R : ∃(φ,C) s.t. (x, φ,C)∈ AeH}.
It is straightforward that ˆx ≥pe(H). Conversely, setx∈ Rsuch that there exists (Φ,C) with (x,Φ,C) ∈ AeH, then by minimality ofXxˆ,φ,ˆCˆ,x ≥ xˆand taking the infimum over suchx, we get the reverse inequality.
We now define the same notion for American contingent claim (Ht)t∈T. An AmericanHhedging strategy is some (x, φ,C) ∈ A such that for all t ∈ T, Xxt,φ,C ≥ Ht a.s. We will denote byAaH the set of American H hedging strategies. Then ( ˆx,φ,ˆ C)ˆ ∈ AaHis minimal if for all (x, φ,C)∈ AaH, Xxt,φ,C≥Xxtˆ,φ,ˆCˆ a.s., for allt∈ T. Again ˆxis the super-replication costpa(H) ofH, i.e.
pa(H)=inf{x∈R : ∃(φ,C) s.t. (x, φ,C)∈ AaH}.
We now recall the usual notion of No-Arbitrage, which characterization is meaningful for super-replication theorem 2.2.
Definition 2.1. We say that there is no arbitrage opportunity if, for all trading strategiesΦsuch that (0,Φ,0)∈ A, we have
X0T,Φ,0 ≥ 0 a.s. =⇒ X0T,Φ,0 = 0 a.s.
In Pham and Touzi [22], a characterization of this no-arbitrage condition is provide and to state it, we introduce the following two sets:
Kˆ =
x∈Rd :φ∗x≤0,∀φ∈ K .
P=
Q∼P : dQ
dP ∈ L∞, ∆St ∈ L1(Q)
and EQ[∆St|Ft−1] ∈ Kˆ, 1≤t≤T P−a.s.
. We also need a non-degeneracy assumption. This assumption is essential to prove Theorem 2.1 below: if it fails to hold, the set offinal dominated payoffs may not be closed, see Brannath [2].
Assumption 2.1. Let t=1, . . . ,T. Then for allFt−1-measurable random vari- ablesϕvalued inK,
ϕ∗∆St(ω) = 0=⇒ϕ(ω) = 0 for a.e. ω∈Ω. Models studied in Section 4 fulfill the above assumption.
Theorem 2.1. (Pham-Touzi [22]).
Under Assumption 2.1, the no arbitrage condition is equivalent toP∅.
Without cone constraints, see also Dalang-Morton-Willinger [8]. LetSt,T
be the set of all stopping w.r.t. thefiltrationFsuch thatt≤τ≤T.
2.2 Super-replication Theorem
Our starting point to derive closed formulae for super-replication prices is the dual formulation of the super-replication theorem. It states that the super-replication cost of an European (resp. American) contingent claim, H(resp (Ht)t∈T), is essentially the supremum over any probability measure Qin given setP(resp. and every stopping timeτless thanT) ofEQ(H) (respEQ(Hτ)): this is given by Theorem 2.2. We give the proof of this non surprising result, since to our knowledge, it has not been done before in the American case.
Indeed, F¨ollmer and Kramkov [11] obtain, via an Optional Decom- position Theorem, for continuous time asset price process and convex constrained the super-replication Theorem (this is no longer the expecta- tion ofHbut of a modification ofHwhich takes into account the convex constraints). But to deal with this great generality, they have to assume first that the wealth process is non negative; second, the strategyφhas to be chosen so that the set{(t
u=1φ∗u−1∆Su)t=1...T}is locally bounded from below: in a discrete setup, with say T = 1, this boundedness Assump- tion implies to choose φ0 ≥ 0 or S1 bounded, which is rather restrictive.
Kabanov-R`asonyi-Stricker [16] and Schachermayer [23] derive a general version of super-replication theorem for European claims, while Sch¨al [24]
have studied the American context for aL2-setup without constraints on the strategy.
Our proof is different since the result for American claims is obtained thanks to that for European ones.
Theorem 2.2. Suppose that Assumption 2.1 and the no arbitrage condition hold.
Let H be an European contingent claim, assume that sup
Q∈P
EQ[H]<∞.
Then, there exists a minimal hedging strategy( ˆx,φ,ˆ C)ˆ ∈ AeHsuch that Xxtˆ,φ,ˆCˆ =ess sup
Q∈P
EQ[H| Ft]. In particular,
pe(H)=xˆ=sup
Q∈P
EQ[H]. Let(Ht)t∈T be an American contingent claim, assume that,
sup
τ∈S0,T,Q∈P
EQ[Hτ]<∞,
Then, there exists a minimal hedging strategy( ˆx,φ,ˆ C)ˆ ∈ AaHsuch that Xx,tˆφ,ˆCˆ =ess sup
τ∈St,T,Q∈P
EQ[Hτ| Ft]. In particular,
pa(H)=xˆ= sup
τ∈S0,T,Q∈P
EQ[Hτ]. Proof.See Appendix.