Arbitrage in markets with bid-ask spreads

In this paper a finite discrete time market with an arbitrary state space and bid-ask spreads is considered. The notion of an equivalent bid-ask martingale measure (EBAMM) is introduced and the fundamental theorem of asset pricing is proved using (EBAMM) as an equivalent condition for no-arbitrage. The Cox-Ross-Rubinstein model with bid-ask spreads is presented as an application of our results.


Introduction
The fundamental theorem of asset pricing, often called the Dalang-Morton-Willinger theorem, states that for the standard discrete-time finite horizon model of security market there is no arbitrage if and only if the price process is a martingale with respect to an equivalent probability measure. However, the equivalent conditions for the absence of arbitrage in markets without friction were proposed and fully proved up to the 90s the general problem of the equivalent conditions for the absence of arbitrage even in markets with proportional transaction costs is still open. On the other hand many great and significant work was done in this topic. We shortly recall some of papers devoted to multi-asset discrete-time models with friction.
In paper [7] of Kabanov, Rásonyi, Stricker the equivalent conditions for the absence of so-called weak arbitrage opportunities (i.e. strict no-arbitrage) were given under the assumption of efficient friction. The general version of this theorem was proved by Kabanov and Stricker in [10] but in the model with finite state space Ω. Soon after Schachermayer in his famous paper [13] gave the equivalent conditions for the absence of the so-called robust no-arbitrage. The general theorem states that the robust no-arbitrage is equivalent to the existence of a strictly consistent price system. Moreover, the robust no-arbitrage cannot be replaced be the strict no-arbitrage due to Schachermayer's counterexample presented in [13]. Going further in this direction very interesting theorem was proved by Grigoriev in [4]. The general result of [10] was extended for an arbitrary Ω in the special case of 2 assets.
One of the corollaries of that theorem states that in the market with bid and ask scalar processes S b , S a and with a money account the absence of arbitrage is equivalent to the existence of a processS, which is a martingale under an equivalent probability measure and satisfies the inequalities S b ≤S ≤ S a . In our terminology (which we introduce in a similar way as in [6]) it simply means that no arbitrage in the case of one risky asset and a money account is equivalent to the existence of a bid-ask consistent price system. For arbitrary d risky assets this problem remains open. However, the author of [4] suggested that its solution seems to be negative. One of purposes of our paper is to analyse the general model of market with bid-ask prices and a money account in order to research Grigoriev's question.
Market model with bid and ask price processes was mainly developed in the famous paper of Jouini and Kallal [6] where the main result states that the absence of the so-called no free lunch is equivalent to the existence of a bid-ask consistent price system. It is noteworthy that the result of Grigoriev [4] also strengthens the one of Jouini and Kallal in the case of one risky asset. In our paper we introduce the notion of an equivalent bid-ask martingale measure and prove that in the model with bid-ask spreads and an arbitrary state space Ω the existence of such a measure is equivalent to no arbitrage. On the other hand we show that the existence of (EBAMM) is equivalent to the existence of both supermartingale as well as submartingale consistent price systems under the same equivalent probability measure. We hope that the main theorem of our paper contributes to the solution of Grigoriev's hypothesis. In some sense it develops the results of [6] as well as [4]. Moreover, the notion of (EBAMM) can be seen as a generalization of an equivalent martingale measure which is successfully used in markets without friction. It can also bypass the condition of the existence of a processS, which evolves between bid and ask price processes and is a martingale under some equivalent probability measure. It is rather obvious that such a process cannot exist in real and is only a useful tool for the pricing. We believe that this paper gives a contribution to the arbitrage theory in markets with friction and may reply in a comprehensible way to some of the questions.
The paper is organized as follows. In section 2 a model of a financial market and some basic definitions are introduced. In section 3 the main theorem is presented and proved. The last chapter consists of examples as well as the applications of achieved results. Mainly the Cox-Ross-Rubinstein model with bid-ask spreads is presented.

A mathematical model of a financial market
Let (Ω, F, P) be a complete probability space equipped with a discrete-time filtration F = (F t ) T t=0 such that F T = F and T is a finite time horizon. Assume that in the market there are two processes , which are d-dimensional adapted to F and have strictly positive components, i.e. S i t > 0 and S i t > 0, P-a.e. Furthermore we assume that S i t ≤ S i t for any t = 0, 1, . . . , T and i = 1, . . . , d. These processes model prices of shares for selling and buying respectively, i.e. at every moment t the investor can buy or sell unlimited amounts of i-th shares at prices S i t and S i t respectively. We will call S the bid price process and S the ask price process. Analogously the pair (S, S) will be called the bid-ask price process. Let us assume that there exists a money account or a bond in the market, which is a strictly positive predictable process B = (B t ) T t=0 and all transactions are calculated in units of this process. For simplicity we assume that B t ≡ 1 for all t = 0, . . . , T and to avoid technical ambiguity we put F −1 := {∅, Ω}. It is noteworthy that this assumption do not restrict the generality of our model thanks to the discounting procedure. This procedure in details was described in chapter 2.1 of [3] for the case of markets without transaction costs.
A trading strategy in the market is an d-dimensional process H = (H t ) T t=1 = (H 1 t , . . . , H d t ) T t=1 , which is predictable with respect to F. We denote the set of all such strategies by P T . Let us also define its subsets P + T := {H ∈ P T | H ≥ 0}, t=1 be a value process in the market with bid-ask spreads for the strategy H starting from 0 units in bank and stock accounts, i.e. x t is defined as follows . . , d and j = 1, . . . , t. Especially we put ∆H i 1 = H i 1 and we will usually skip the symbol of the inner product. The random variable x t models the gain or loss occurred up to time t. The first sum is the aggregate purchase of assets up to time t despite the second sum, which corresponds to the aggregate sales. Notice that at time t we liquidate all positions in risky assets and the following equality is satisfied t j=1 ∆H j = H t . It can be interpreted as follows. If we want to know the real value of our portfolio at time t we should calculate it in units of a money account. To do this we should liquidate all positions in risky assets. Actually this procedure must not be carry out in real. We can use it only for calculating the value of our portfolio. In literature it is known as the immediate liquidation value of the portfolio (see e.g. [1] where on the other hand the marked-to-market value is considered).
Remark 2.1. Notice that all changes in units of assets must be obtained by borrowing or investing in a money account. Hence our position in the money account is uniquely determined by the strategy, which actually is self-financing. We will use the notation L 0 (R d , F t ) for the set of F t -measurable random vectors taking values in R d with the convention that L 0 (R d ) stands for L 0 (R d , F T ). In the case of random variables (i.e. d = 1) we will simply use the abbreviations L 0 (F t ) := L 0 (R, F t ) and L 0 := L 0 (R). Moreover let L 0 + (R d , F t ) denotes the subspace of L 0 (R d , F t ) consisting of only non-negative random vectors. To simplify the notation we will use the same convention as previous, i.e. we will write L 0 + (R d ), L 0 + (F t ), L 0 + in an appropriate situation. Furthermore the standard spaces L 1 and L ∞ are treated in the same way.
To make our reasoning much more clear we introduce for any 1 ≤ t ≤ t + k ≤ T and H ∈ L 0 (R d , F t−1 ) the following random variable: Let us now define R T := {x T (H) | H ∈ P T } and the set of hedgeable claims, which is of the form A T := R T − L 0 + . By A T we denote the closure of A T in probability. The following definition is crucial.
Definition 2.2. We say that there is no arbitrage in the market with bid-ask spreads if and only if . Now we introduce another sets what simplify and make possible to prove the main theorem. Let us define for any 0 ≤ j < t ≤ T : . As counterparts of sets R t and A t we define for any t = 1, . . . , T the following sets Consequently we also introduce Notice that if H is an arbitrage strategy in a model with the time horizon t (so at time t we liquidate all positions in stock) then it is also an arbitrage strategy in a model with a larger time horizon, especially with the time horizon T . It suffices to take the same strategy H up to time t and later 0.
We now introduce the definition of a consistent price system, similarly as it was done in [6].
Definition 2.5. We say that a pair (S,P ) is a consistent price system (CPS) in the market with bid-ask spreads whenP is a probability measure equivalent to P andS = (S t ) T t=0 is an d-dimensional process adapted to the filtration F, which is ã P -martingale and the following inequalities are satisfied e. for all i = 1, . . . , d and t = 0, . . . , T . If the processS is aP -supermartingale (P -submartingale) then we say that a pair (S,P ) is a supermartingale consistent price system (supCPS) (a submartingale consistent price system (subCPS) respectively).
We introduce the notion of the so-called equivalent bid-ask martingale measure, which will play the similar role as an equivalent martingale measure in markets without friction. Definition 2.6. We shall say that a probability measure Q is an equivalent bid-ask martingale measure (EBAMM) for the bid-ask price process (S, S) if Q ∼ P, all S t are integrable and the following inequalities are satisfied e. for any t = 1, . . . , T and i = 1, . . . , d.
The interpretation of this measure is quite obvious. Let us consider (EBAMM) in the context of a stock market. If we buy shares at any time t − 1 at price S i t−1 we shouldn't expect, on average, that at time t we sell shares at better price (i.e. at price S i t ) than we've bought them previous. On the other hand the analogous situation is if we short sale shares. The following lemma presents the straightforward relation between (CPS) and (EBAMM). Proof. Let (S, Q) be a consistent price system. Then for any t = 1, . . . , T and i = 1, . . . , d we have the following inequalities: Remark 2.8. The notion of (EBAMM) can be seen as a generalization of an equivalent martingale measure (EMM) in markets without friction. Indeed, when we assume that S = S, then our model comes down to a finite discrete time market model without transaction costs and (EBAMM) is actually the same as (EMM). Hence intuitively the notion suggests that if we could consider process (S, S) as a whole then such a process should behave similarly to a martingale under an equivalent probability measure with precision to bid-ask spreads.

Main results
At the beginning of this chapter we present the sufficient condition for the absence of arbitrage, which is actually the existence of a consistent price system. This result is standard and well-known but we will prove it in our model using a slightly weaker assumption.
and we have the absence of arbitrage in our model, i.e.
Proof. Notice that the conditionR T ∩ L 0 + = {0} is equivalent to the absence of arbitrage for any one-step model, i.e. in our notation for any t = 1, . . . , T . We will use the analogous reasoning as in [8], see chapter 2.1.1. Assume that we have the absence of arbitrage in any one-step model. We show that there is no arbitrage. Take the smallest t ≤ T such thatR t ∩ L 0 + (F t ) = {0} and notice that 1 < t < T . Hence there exist two strategiesĤ ∈ P + t ,Ȟ ∈ P − t such that Due to the choice of t, either the set Γ : . In any case we have a contradiction. Therefore we can assume that there exists H t ∈ L 0 (R d , F t−1 ) satisfying the following conditions (3.5) H t ∆S t ≥ 0, P-a.e. and P(H t ∆S t > 0) > 0.
It suffices to show that there existsH t ∈ L 0 (R d , F t−1 ), which is bounded and satisfies the condition (3.5). One can takẽ It is also possible to use the arguments from Hence X = 0, Q-a.e. and from the equivalence of measures X = 0, P-a.e. We show now that A T ∩ L 0 + = {0}. Take any ξ ∈ A T ∩ L 0 + . Then the following inequalities are satisfied: It means that we split the strategy into another two strategies, which consist of long and short positions only. Moreover, when ∆H t ≥ 0 then ∆Ĥ i t , ∆Ȟ i t ≥ 0 and on the other hand if ∆H t < 0 then ∆Ĥ i t , ∆Ȟ i t < 0. Hence we always have Notice that the following inequalities are satisfied e. for any t = 0, . . . , T and i = 1, . . . , d. Hence we can write the next inequality, i.e.
Due to the conditionR T ∩ L 0 + = {0} we get (Ĥ ·Ŝ) T + (Ȟ ·Š) T = 0, P-a.e. and hence ξ = 0, P-a.e.  Proof. Notice that it suffices to show that F t ⊂ A t where F t is defined as in (2.4). Take any Π ∈ F t . By definition we can assume that is of the form where Θ = (θ j ) t j=1 ,Θ = (θ j ) t j=1 are predictable and non-negative processes. Notice that there exist maybe another predictable and non-negative processes ϑ = (ϑ j ) t j=1 , ϑ = (θ j ) t j=1 such that for any j = 1, . . . , t we have {ϑ j > 0,θ j > 0} = ∅, a.e. and the following inequality is satisfied Let us define the strategy H = (H j ) t j=1 ∈ P t as follows Moreover, we put H 1 := ∆H 1 and H j := ∆H j + H j−1 for j > 1. Notice that H is a well defined strategy. Furthermore

By the previous observation
Obviously Ξ + r = x t (H) ∈ R t and Π ≤ x t (H) − r. Furthermore there exists a random variabler ∈ L 0 + (F t ) such that Π = x t (H) − r −r. It suffices to definẽ r := Ξ − Π. Hence we get that Π ∈ A t .

Remark 3.5. It is not clear whether Λ T ⊂ A T or not. We only know that Λ
Remark 3.6. Notice that for any Π ∈ F T (respectively F T ) there exists a strategy H ∈ P T and a random variable r ∈ L 0 + such that Π = x T (H) − r.
Lemma 3.7. For any 1 ≤ t ≤ t + k ≤ T the following inclusions hold F t−1,t+k ⊂ F t+k ⊂ A t+k and for any x ∈ F t−1,t+k there exists H t ∈ L 0 (R d , F t−1 ) and r ∈ L 0 + (F t+k ) such that x = x t−1,t+k (H t ) − r.

Now let us define the random vector
Moreover the random variablel := Ξ−Π ∈ L 0 + (F t+k ) and we get the equality x = Π − l = Ξ − l −l. Let r := l +l ∈ L 0 + (F t+k ). Then we have Also as we see x ∈ F t+k and by Lemma 3.4 F t+k ⊂ A t+k .
The following theorem presents the equivalent conditions for the absence of arbitrage in markets with bid-ask spreads and a money account.
for any 1 ≤ t ≤ t + k ≤ T ; (f ) there exists an equivalent bid-ask martingale measure Q for the bid-ask process (S, S) such that dQ dP ∈ L ∞ (EBAMM); (g) there exists supCPS (Ŝ, Q) and subCPS (Š, Q) such that dQ dP ∈ L ∞ . In the proof of Theorem 3.8 the following results will be used. Their proofs can be found e.g. in [9]. Lemma 3.9. Let X n be a sequence of random vectors taking values in R d such that for almost all ω ∈ Ω we have lim inf X n (ω) < ∞. Then there is a sequence of random vectors Y n taking values in R d satisfying the following conditions: (1) Y n converges pointwise to Y almost surely where Y is a random vector taking values in R d , (2) Y n (ω) is a convergent subsequence of X n (ω) for almost all ω ∈ Ω.
Remark 3.10. The above claim can be formulated as follows: there exists an increasing sequence of integer-valued random variables σ k such that X σ k converges a.s. Lemma 3.11 (Kreps-Yan). Let K ⊇ −L 1 + be a closed convex cone in L 1 such that K ∩ L 1 + = {0}. Then there is a probability P ∼ P with d P dP ∈ L ∞ such that EP ξ ≤ 0 for all ξ ∈ K.
for any t = 1, . . . , T and using Lemma 3.4 also F t ∩ L 0 + (F t ) = {0} for any t = 1, . . . , T . (b) ⇒ (c). Trivial. (Notice that also the implication (a) ⇒ (c) is obvious so we could skip the condition (b), which actually we put here to do our analysis more comprehensive.) (c) ⇒ (d) To prove this implication we will use the similar technique as in [9] and especially [12] (see Theorem 2.33). Take any t, k such 1 ≤ t ≤ t + k ≤ T . First notice that by Lemma 3.7 we have F t−1,t+k ∩ L 0 + (F t+k ) = {0}. We will show that the set F t−1,t+k is closed in topology generated by the convergence in probability of measure P. Take a sequence ξ n ∈ F t−1,t+k such that ξ n → ζ in probability. It suffices to show that ζ ∈ F t−1,t+k . The sequence ξ n contains a subsequence convergent to ζ a.s. Thus, at most restricting to this subsequence we can assume that ξ n → ζ, P-a.s. By Lemma 3.7 for any n there exists H n t ∈ L 0 (R d , F t−1 ) and , what simply means that x t−1,t+k (H n t ) → ζ, P-a.s. Consider first the situation on the set Ω 1 := {lim inf H n t < ∞} ∈ F t−1 . By Lemma 3.9 there exists an increasing sequence of integer-valued F t−1 -measurable stopping times τ n such that H τn t is convergent a.s. on Ω 1 and for almost all ω ∈ Ω 1 the sequence H τn(ω) t (ω) is a convergent subsequence of the sequence H n t (ω). Notice that H τn t ∈ L 0 (R d , F t−1 ) and r τn ∈ L 0 + (F t+k ) respectively. LetH t := lim Consider now the situation on the set Ω 2 := {lim inf H n t = ∞} ∈ F t−1 . Let us define G n t := and notice that G n t ∈ L 0 (R d , F t−1 ) and h n ∈ L 0 + (F t+k ). We get the convergence Similarly as on the set Ω 1 by Lemma 3.9 there exists an increasing sequence of integer-valued F t−1 -measurable stopping times σ n such that G σn t is convergent a.s. on Ω 2 and for almost all ω ∈ Ω 2 the sequence G σn(ω) t (ω) is a convergent subsequence of the sequence G n t (ω). LetG t := lim n→∞ G σn t . As previous, notice that by the convergence of the sequence G σn t also (G σn t ) + and (G σn t ) − are convergent. Moreover (G σn t ) + → (G t ) + and (G σn t ) − → (G t ) − . Hence also h σn is convergent a.s. on Ω 2 . Defineh := lim n→∞ h σn . We get the following equality By the condition F t−1,t+k ∩ L 0 P ∼ P such that dP dP ∈ L ∞ and η ∈ L 1 (P ). Property (d) is invariant under an equivalent change of probability. This consideration allows as to assume without loss of generality that all S t , S t are integrable. We will use induction on time horizon or equivalently on k. First let k = 0 and fix any t ∈ {1, . . . , T }. Define the set Ψ t−1,t := F t−1,t ∩ L 1 (F t ), which is a closed convex cone in L 1 (F t ). Since we have Ψ t−1,t ∩ L 1 + (F t ) = {0} then by Lemma 3.11 there exists a probability measure Q t ∼ P on (Ω, F t ) such that dQ t dP ∈ L ∞ (F t ) and E Q t ξ ≤ 0 for any ξ ∈ Ψ t−1,t . In particular for where H t = (0, . . . , 1 1 A , . . . , 0), P-a.e., A ∈ F t−1 and the value 1 1 A is on i-th position. For the case (3.10) it means that at time t − 1 if the event A holds we buy i-th asset at price S i t−1 and liquidate the portfolio at time t. For the case (3.11) the situation is opposite, i.e. first we short sale i-th asset at time t − 1 and then we buy it at time t. Hence we get the inequalities In conclusion, there exists (EBAMM) for the bid-ask process (S, S) where S = (S j ) t j=t−1 and S = (S j ) t j=t−1 . Assume now that the claim is true in a model with the time horizon k where k ≥ 1. We will show that it is true in a model with the time horizon k + 1. Fix any t, k such that 0 ≤ t ≤ t + k ≤ T . We show that there exists an equivalent bidask martingale measure in the market with the bid-ask process (S, S) where S = (S j ) t+k j=t−1 and S = (S j ) t+k j=t−1 . By the induction hypothesis there exists (EBAMM) Q t+k in the market with the bid-ask process ((S j ) t+k j=t , (S j ) t+k j=t ). Notice that the condition (d) is invariant under an equivalent change of probability. Hence we can apply the same method as in the previous part to the probability space (Ω, F t , Q t+k |Ft ) where Q t+k |Ft denotes the measure Q t+k with the restriction to the σ-algebra F t . Then there exists a probability measure Q t ∼ Q t+k |Ft such that dQ t dQ t+k |F t ∈ L ∞ and the following inequalities are satisfied Let us define a probability measure Q on a measure space (Ω, F t+k ) as follows (3.14) dQ dP := dQ t dQ t+k |Ft dQ t+k dP .
Notice that the density dQ t dQ t+k |F t is bounded and F t -measurable hence for any j ∈ {t + 1, . . . , t + k} we have and on the other hand . By induction we conclude that there exists an equivalent bid-ask martingale measure for the bid-ask process ((S t ) T t=0 , (S t ) T t=0 ). (f) ⇒ (g) As in the previous implication we use induction on time horizon or equivalently on k. First let k = 0 and fix any t ∈ {1, . . . , T }. Let us define the processesŜ = (Ŝ j ) t j=t−1 andŠ = (Š j ) t j=t−1 as follows Notice that (Ŝ, Q) is (supCPS) and (Š, Q) is (subCPS).
Assume now that the claim is true in a model with the time horizon k where k ≥ 1. We will show that it is true in a model with the time horizon k + 1. Fix any t, k such that 0 ≤ t ≤ t + k ≤ T . We show that there exists (supCPS) (Ŝ, Q) and (subCPS) (Š, Q) in the market with the bid-ask process (S, S) where S = (S j ) t+k j=t−1 and S = (S j ) t+k j=t−1 . By the induction hypothesis there exists (supCPS) ((Ŝ j ) t+k j=t , Q t+k ) and (subCPS) ((Š j ) t+k j=t , Q t+k ) in the market with the bid-ask process ((S j ) t+k j=t , (S j ) t+k j=t ). Notice that the condition (e) is invariant under an equivalent change of probability. Hence we can apply the same method as in the previous part to the probability space (Ω, F t , Q t+k |Ft ) where Q t+k |Ft denotes the measure Q t+k with the restriction to the σ-algebra F t . Then there exists a probability measure Q t ∼ Q t+k |Ft such that dQ t dQ t+k |F t ∈ L ∞ and the processes (Ŝ j ) t j=t−1 , (Š j ) t j=t−1 defined as in (3.15), (3.16) are (supCPS), (subCPS). Let us define the stopping time τ := min{j ≥ t − 1 |Š j =Š t }. Then using the optimal stopping theorem the processŠ τ := (Š j∧τ ) t j=t−1 is a Q t -martingale. We now define the probability measure Q on a measure space (Ω, F t+k ) as follows (3.17) dQ dP := dQ t dQ t+k |Ft dQ t+k dP .
Remark 3.12. The conditions from Theorem 3.8 are also equivalent to the another one, i.e. for any 1 ≤ t ≤ t + k ≤ T there exists an equivalent bid-ask martingale measure Q t+k t−1 for the bid-ask process {(S t−1 , S t−1 ), (S t+k , S t+k )} such that Remark 3.13. Let us define for any t, k such that 1 ≤ t ≤ t + k ≤ T the following sets A t−1,t+k := R t−1,t+k − L 0 + (F t+k ) where R t−1,t+k := {x t−1,t+k (H t ) | H t ∈ L 0 (R d , F t−1 )}. Then under the assumption A t−1,t+k ∩ L 0 + (F t+k ) = {0} the set A t−1,t+k is closed in probability. It suffices to use the analogous reasoning as in the proof of the implication (c) ⇒ (d) of Theorem 3.8.  This model can be seen as a generalisation of the Cox-Ross-Rubinstein model to the case of bid-ask spreads. Notice that in our model we should also know that S t ≤ S t . In order to assure this we assume that St−1 By definition an equivalent bid-ask martingale measure (we shall denote it by q) should satisfy the following inequalities:  Hence from the inequalities (4.5) and (4.6) we can estimate (EBAMM) as follows: Notice that for the existence of (EBAMM) we need to know that the following inequalities are satisfied.
These conditions assure that there exists at least one q ∈ (0, 1). Then we get the necessary and sufficient conditions for the existence of (EBAMM), i.e.
It is noteworthy that by Theorem 3.8 these conditions are equivalent to the absence of arbitrage in the Cox-Ross-Rubinstein model with bid-ask spreads. Now we illustrate our model by some examples. Hence we can take any q ∈ (0, 1) to get an equivalent bid-ask martingale measure. Furthermore notice that d = − 3 4 , d = u = 0, u = 3 and the conditions from (4.10) are satisfied. Indeed, we have d = − 3 4 < 0 < 3 = u and d u = − 9 4 ≤ 0 = d u.   and by (4.7) q ∈ [ 1 13 , 3 5 ]. Notice that also the conditions from (4.10) are satisfied. Indeed, we have d = − 3 4 < 0 < 3 = u and d u = − 9 4 ≤ − 1 8 = d u.