A complement to the Grigoriev theorem for the Kabanov model

We provide an equivalent characterisation of absence of arbitrage opportunity for the bid and ask financial market model analog to the Dalang–Morton–Willinger theorem formulated for discrete-time financial market models without friction. This result completes and improves the Grigoriev theorem for conic models in the two dimensional case by showing that the set of all terminal liquidation values is closed.


Introduction
The Dalang-Morton-Willinger theorem [5] asserts, for the discrete-time final horizon frictionless market models, that the no-arbitrage property (NA) is equivalent to the existence of an equivalent martingale measure and any of these properties ensures that the set of super-replicated claims A T is closed in probability.
The models with friction were first considered in the pioneering paper [10] and, later, were extensively studied, e.g. in the papers [12], [8], [16], [7], [14].With proportional transaction costs, it is classical to express the portfolio processes as stochastic vectors of R d , d ≥ 1.Indeed, in presence of transaction costs, the exchanges are allowed between the assets at different rates so that it is not possible to describe them directly through the liquidation values (see [13,Ch. 3]).
In the theory of markets with proportional transaction costs, the closest analog of NA is the property NA w , i.e. the absence of strong arbitrage opportunities usually denoted.With the latter concepts, the DMW theorem can be extended but only for two-asset models and only partially.The corresponding result is known as the Grigoriev theorem, claiming that NA w holds iff there is a consistent price system, accompanied by unexpected examples where the set of all vector-valued terminal portfolio processes is not closed under NA w , see [13,Ex. 1,Sect. 3.2.4 ].Closedness is only proved under a strong absence of arbitrage opportunity, i.e. a robust no-arbitrage property NA r , see [13,Lem. 3.2.8].Actually, one can mention that closedness is the important property to characterize the super-hedging prices (see [2] and [4]).
Here, we prove that the set of all terminal liquidation values is closed under NA w for the two dimensional conic models.We then deduce a dual characterization of the prices super hedging a contingent claim when they are only expressed in the first asset.

Notations.
We define e 1 := (1, 0) ∈ R 2 , R 2 + is the set of all vectors in R 2 having only non negative components.For a random set E, L p (E, F), p ∈ [1, ∞) (resp.p = ∞ or p = 0), is the normed space of all measurable selections of the random set E and the superscript p denotes the selections belonging to the corresponding Lebesgue space L p (E, F).
The Kabanov model.Let (Ω, IF := (F t ) t≤T , P) be a discrete-time complete stochastic basis.We consider a risk-free asset S 0 = 1 and a risky asset defined by bid and ask adapted prices S b > 0 and S a > 0. This is the two-dimensional Kabanov model, [13,Ch. 3], equivalently defined by a IFadapted set-valued process with values in the set of closed sectors (convex cones) (G t ) t≤T of the real plane R 2 which are measurable in the sense that: In finance, G t is interpreted as the set of all positions x ∈ R 2 it is possible to liquidate at time t without any debt.We have where the liquidation value process introduced in [1] is It is easy to check that the liquidation value satisfies the following: Lemma 2.1.Let L t be defined by (2.1).
1.The mapping The boundary We introduce the set of all terminal values at time t ≤ T of the portfolio processes starting from the zero initial endowment at time u ≤ t, i.e.
The corresponding set of terminal liquidation values is: We introduce a condition E satisfied by classical examples of markets: We say that condition E holds if the following implications hold for all t ≤ T − 1, for all u ≥ t + 1 and for all F u ∈ F u : Let us present some examples where condition E holds: Example 1: This first example generalizes the model [8].Let (S t ) t≤T be a mid-price adapted process and consider a process ( t ) t≤T of proportional transaction cost rates with values in [0, 1).We suppose that (S t ) t≤T and ( t ) t≤T are two independent processes and for every t < u, one of the random variables S u /S t and (1 + t )/(1 − u ) does not admit any atom.The bid and ask prices are given by S b t := S t (1 − t ), S a t := S t (1 + t ).Then, we show that P(S a t = S b u ) = 0 if u > t so that condition E trivially holds.Example 2: We consider the Cox-Ross-Rubinstein model with bid-ask spreads of [11,Sect. 4].The bid and ask prices are Example 3: As in [8], we suppose that the bid and ask prices are S b t = S t − t and S a t = S t + t , t ≤ T. Here, S and are two positive adapted processes such that S b > 0.Then, condition E trivially holds when S and are independent and one of them does not admit any atom.

Main result
Definition 3.1.We say that the market model G satisfies the weak noarbitrage property NA w if L T 0 ∩ L 0 (R + , F T ) = {0}.For the models with proportional transaction costs, G T strictly dominates R 2 + , i.e.R 2 + \ {0} ⊂ int G T , and we may easily show the following: Recall that a consistent price system (CPS) is a martingale (Z t ) t≤T satisfying Z t ∈ G * t \ {0} for all t ≤ T .
Theorem 3.3.( Grigoriev theorem, [6], [13, Th. 3.2.15])The following conditions are equivalent : In the following, we denote by M ∞ (P ) the set of all Q ∼ P such that dQ/dP ∈ L ∞ and E Q L T (V ) ≤ 0 for all L T (V ) ∈ L T 0 .For any contingent claim ξ ∈ L 1 (R, F T ), we define the set Γ ξ of all initial endowments of portfolio processes whose terminal liquidation values coincide with ξ, i.e.
The following is suggested by C. Kühn and confirms the necessity of E. Example 3.6.There exists a financial market model satisfying NA w but condition E fails and such that The bid and ask prices are defined by 2 The closure is taken in L 0 .
Moreover, we suppose that P ({ω k,1 }|F 1 ) = P ({ω k,2 }|F 1 ) for all k ≥ 1 so that E(S b 2 |F 1 ) = 1.We deduce that Z t = (1, S b t ) is a CPS so that NA w holds by the Grigoriev theorem.This is an example where condition E does not hold at time t = 0. Indeed, in the contrary case, as S a 0 = S b 1 a.s., we should have a.s. the existence of r ≥ 1 such that S a 0 ≥ S a r .Necessary r = 2 so that we should have 1 ≥ S a 2 a.s., which is not the case.Let us define We may show by contradiction that the payoff H = 1 H 1 −1 H 2 does not belong to L 2 0 .On the other hand, H = lim n H n where H n = 1 H 1n − 1 H 2n .We claim that H n ∈ L 2 0 .Indeed, it suffices to buy n + 1 risky assets at time t = 0, sell n + 1 − k ≥ 0 assets at time t = 1 on each {ω k,1 , ω k,2 } ∈ F 1 such that k ≤ n and sell the n + 1 assets otherwise.At last, liquidating the position at time t = 2, we finally get the payoff As H = lim n H n , we conclude that L T 0 is not closed.Closedness.It remains to show that (1) ⇒ (2), i.e.L T 0 is closed in probability.With one step, this is immediate as L T T = −L 0 (R + , F T ).We may show that, for any γ ∈ L T 0 , γe 1 = −g T 0 ∈ A T 0 where g t u , u ≤ t, is a general notation we introduce to designate a sum g t u = t r=u g r with g r ∈ L 0 (G r , F r ), r ≤ T .Moreover, we may suppose w.l.o.g. that g r ∈ ∂G t for all t ≤ T − 1.By the Grigoriev theorem, there exists a CPS Z.
We normalize the sequences by setting γn General case.Condition E is only used for 3 steps and more and we argue by induction.Suppose closedness holds between the dates t + 1 and T ≥ 2 and let us show the closedness holds between t and T .To do so, we suppose that lim n δ n T = δ T ∈ L 0 (Re 1 , F T ) where δ n T = −g T,n t ∈ A T t .We claim that δ n T = −ĝ
[11,t a.s.for all t ≤ T .In[11, Sect.4], ζ b and ζ a take two distinct values.Here, we suppose that P (ζ a t /S b t+u−1 .In that case, condition E holds.On the other hand, if we suppose that P (ζ a t = 0) = P (ζ b t = 0) = 0 for all t and ζ a t , ζ a t are independent of F t−1 , while the ratios S a t+r /S a t and S b t+r /S b t , r ≥ 1 admit densities, then condition E trivially holds.
{0} 2 ; 3 For any P ∼ P, there exists a bounded CPS under P .Suppose that condition E holds if T ≥ 2. The following conditions are equivalent: = g n t 1 {lim infn |g n t |<∞} .This holds for t = T −1 as shown above.(i)Wefirstworkon Λt := {lim inf n |g n t | < ∞} ∈ F t .Consider the smallest u ≥ t + 1 such that P (lim inf n |g n u | = +∞|Λ t ) > 0. As lim inf n |g n r | < ∞ a.s., we suppose that g n r → g r ∈ L 0 (∂G r , F r ) by [13, Lem.2.1.2]ifr≤ u−1.Then, we replace g n r by g r if r ≤ u − 1, letting aside a residual error n T → 0 a.s. and we only need to consider the case u ≤ T − 1.We split Λ t into Λ u ∈ F u and Γ u = Ω \ Λ u .On Γ u , we mormalize by dividing by |g n u | and we get that δn T = −g T,n := g n r /|g n u | for r ≤ T .As δn T and gn r , r ≤ u − 1, tend to 0, we may suppose that lim inf n |g n r | < ∞ by the induction hypothesis on Γ u if r ≥ u.By [13, Lem.2.1.2],we may suppose that gn r → g∞ r ∈ L 0 (G r , F r ) t r