Consistency of option prices under bid–ask spreads

Abstract Given a finite set of European call option prices on a single underlying, we want to know when there is a market model that is consistent with these prices. In contrast to previous studies, we allow models where the underlying trades at a bid–ask spread. The main question then is how large (in terms of a deterministic bound) this spread must be to explain the given prices. We fully solve this problem in the case of a single maturity, and give several partial results for multiple maturities. For the latter, our main mathematical tool is a recent result on approximation by peacocks.


Introduction
Calibrating martingales to given option prices is a central topic of mathematical finance, and it is thus a natural question which sets of option prices admit such a fit, and which do not.Note that we are not interested in approximate model calibration, but in the consistency of option prices, and thus in arbitrage-free models that fit the given prices exactly.Put differently, we want to detect arbitrage in given prices.We do not consider continuous call price surfaces, but restrict to the (practically more relevant) case of finitely many strikes and maturities.Therefore, consider a financial asset with finitely many European call options written on it.In a frictionless setting, the consistency problem is well understood: Carr and Madan [4] assume that interest rates, dividends and bid-ask spreads are zero, and derive necessary and sufficient conditions for the existence of arbitrage free models.Essentially, the given call prices must not admit calendar or butterfly arbitrage.Davis and Hobson [6] include interest rates and dividends and give similar results.They also describe explicit arbitrage strategies, whenever arbitrage exists.Concurrent related work has been done by Buehler [2].Going beyond existence, Carr and Cousot [3] present practically appealing explicit constructions of calibrated martingales.More recently, Tavin [16] considers options on multiple assets and studies the existence of arbitrage strategies in this setting.Spoida [14] gives conditions for the consistency of a set of prices that contains not only vanillas, but also digital barrier options.See [10] for many related references.
As with virtually any result in mathematical finance, robustness with respect to market frictions is an important issue in assessing the practical appeal of these findings.Somewhat surprisingly, not much seems to be known about the consistency problem in this direction, the single exception being a paper by Cousot [5].He allows positive bid-ask spreads on the options, but not on the underlying, and finds conditions on the prices that determine the existence of an arbitrage-free model explaining them.
The novelty of our paper is that we allow a bid-ask spread on the underlying.Without any further assumptions on the size of this spread, it turns out that there is no connection between the quoted price of the underlying and those of the calls: Any strategy trying to exploit unreasonable prices can be made impossible by a sufficiently large bid-ask spread on the underlying (see Example 2.3 and Proposition 6.1).In this respect, the problem is not robust w.r.t. the introduction of a spread on the underlying.However, an arbitrarily large spread seems questionable, given that spreads are usually tight for liquid underlyings.We thus enunciate that the appropriate question is not "when are the given prices consistent", but rather "how large a bid-ask spread on the underlying is needed to explain them?"We thus put a bound ≥ 0 on the (discounted) spread of the underlying and want to determine the smallest such that leads to a model explaining the given prices.We then refer to the call prices as -consistent (with the absence of arbitrage).
We assume discrete trading times and finite probability spaces throughout; no gain in tractability or realism is to be expected by not doing so.In the case of a single maturity, we obtain necessary and sufficient conditions for -consistency.The multi-period problem, on the other hand, seems to be challenging.We provide two partial results: necessary (but presumably not sufficient) conditions for -consistency, and necessary and sufficient conditions under simplifying assumptions.The latter, in particular, drop the bid-ask spread on the options, retaining only the spread on the underlying.
Recall that the main technical tool used in the papers [4,5,6] mentioned above to construct arbitrage-free models is Strassen's theorem [15], or modifications thereof.In the financial context, this theorem essentially states that option prices have to increase with maturity.This property breaks down if a spread on the underlying is allowed.We will therefore employ a recent generalization of Strassen's theorem, obtained in [9].It gives necessary and sufficient conditions for the existence of martingales within a prescribed distance, measured in terms of the infinity Wasserstein distance.This generalized Strassen theorem will be the key to obtain the -consistency conditions under simplified assumptions mentioned above.
The structure of the paper is as follows.In Section 2 we will describe our setting and give a precise formulation of our problem.Then, in Section 3 we will present necessary and sufficient conditions for the existence of arbitrage free models with bounded bid-ask spreads for a single maturity.Necessary conditions for multiple maturities are found in Section 4, while Section 5 contains necessary and sufficient conditions under simplifying assumptions.There, we invoke the main result from [9].In Section 6 we will discuss the case where models with unbounded spread are allowed; again, this gives an opportunity to apply results from [9].We argue, though, that studying the consistency problem with unbounded spread seems to be unnatural.Section 7 concludes.

Notation and Preliminaries
Our time index set will be T = {0, . . ., T } throughout, where 1 ≤ T ∈ N. Whenever we talk about "the given prices" or similarly, we mean the following data: A positive deterministic bank account (B(t)) t∈T with B(0) = 1, (2.1) corresponding call option bid and ask prices (at time zero) and the current bid and ask price of the underlying 0 < S 0 ≤ S 0 . (2.4) We write D(t) = B(t) −1 for the time zero price of a zero-coupon bond maturing at t, and k t,i = D(t)K t,i for the discounted strikes.The symbol C t (K) denotes a call option with maturity t and strike K.
In the presence of a bid-ask spread on the underlying, it is not obvious how to define the payoff of an option; this issue seems to have been somewhat neglected in the transaction costs literature.Indeed, suppose that an agent holds a call option with strike $100, and that at maturity T = 1 bid and ask are S 1 = $99 resp.S 1 = $101.Then, the agent might wish to exercise the option to obtain a security that would cost him $1 more in the market, or he may forfeit the option on the grounds that spending $100 would earn him a position whose liquidation value is only $99.The exercise decision cannot be nailed down without making further assumptions.
In the literature on option pricing under transaction costs, it is usually assumed that bid and ask of the underlying are constant multiples of a mid-price (often assumed to be geometric Brownian motion).This mid-price is then used as trigger to decide whether an option should be exercised, followed by physical delivery [1,7,17].The assumption that such a constantproportion mid-price triggers exercise seems to be rather ad-hoc, though.To deal with this problem in a parsimonious way, we assume that call options are cash-settled, using a reference price process S C .This process evolves within the bid-ask spread.It is not a traded asset by itself, but just serves to fix the call option payoff (S C t − K) + for strike K and maturity t.This payoff is immediately transferred to the bank account without any costs.Definition 2.1.A model consists of a finite probability space (Ω, F, P) with a discrete filtration (F t ) t∈T and three adapted stochastic processes S, S, and S C , satisfying 1 Clearly, S t and S t denote the bid resp.ask price of the underlying at time t.As for the reference price process S C , we do not insist on a specific definition (such as, e.g., S C = 1 2 (S +S)), but allow any adapted process inside the bid-ask spread.We now give a definition for consistency of option prices, allowing for (arbitrarily large) bid-ask spreads on both the underlying and the options.
Definition 2.2.The prices (2.1)-(2.4)are consistent with the absence of arbitrage, if there is a model (in the sense of Definition 2.1) such that • There is a consistent price system for the underlying, i.e., a process S * such that S t ≤ S * t ≤ S t for t ∈ T and such that (D(t)S * t ) t∈T is a P-martingale. 2he process S * is also called a shadow price.According to Kabanov and Stricker [12] (see also [13]), these requirements yield an arbitrage free model comprising bid and ask price processes for the underlying and each call option.Indeed, for the call with maturity t and strike K t,i , one may take r t,i 1 {s=0} + B(s)E[(D(t)S C t − k t,i ) + |F s ]1 {s>0} s∈T as bid price process (and similarly for the ask price), and B(s)E[(D(t)S C t − k t,i ) + |F s ] s∈T as consistent price system.As mentioned in the introduction, if consistency is defined according to Definition 2.2, then there is no interplay between the current prices of the underlying and the options, which seems to make little sense.As an illustration, the following example shows how frictionless arbitrage strategies may fail in the presence of a sufficiently large spread; a general result is given in Section 6 below.
Example 2.3.Let c > 0 be arbitrary.We set k := k 1,1 = k 2,1 = 1 and assume Thus C 1 (k) is "too expensive", and without frictions, buying C 2 (k)−C 1 (k) would be an arbitrage opportunity (upon selling one unit of stock if C 1 (k) expires in the money).In particular, the first condition from Corollary 4.2 in [6] and equation ( 5) in [5] are violated: they both state that r 1 ≤ r 2 is necessary for the absence of arbitrage strategies.
But with spreads we can choose c as large as we want and still the above prices would be consistent with no-arbitrage.Indeed, we can define a deterministic model as follows: This model is free of arbitrage (see Proposition 6.1 below).In particular, consider the portfolio C 2 (k) − C 1 (k): the short call −C 1 (k) finishes in the money with payoff −(c + 1).This cannot be compensated by going short in the stock, because its bid price stays at 2. The payoff at time t = 2 of this strategy, with shorting the stock at time t = 1, is Our focus will thus be on a stronger notion of consistency, where the discounted spread on the underlying is bounded.Hence, our goal becomes to determine how large a spread is needed to explain given option prices.Definition 2.4.Let ≥ 0. Then the prices (2.1)-(2.4)are -consistent with the absence of arbitrage, or simply -consistent, if they are consistent (Definition 2.2) and the following conditions hold: The bound (2.7) is an additional mild assumption on the reference price S C , made for tractability, and makes sense given the actual size of market prices and spreads (recall that S ≤ S C ).With the same justification, in our main results on -consistency we will assume that all discounted strikes k t,i are larger than .If = 0 and the bid and ask prices in (2.3) and (2.4) agree, then we recover the frictionless consistency definition from [6].
As mentioned above, we do not insist on any specific definition of the reference price S C .However, it is not hard to show that choosing S C = 1 2 (S + S) yields almost the same notion of -consistency: Proposition 2.5.Assume that we are only interested in arbitrage free models where, in addition to the requirements of Definition 2.4, we have that where p ∈ [0, 1] and p = 1 2 .We now define semi-static trading strategies in the bank account, the underlying asset, and the call options.Here, semi-static means that the position in the call options is fixed at time zero.The definition is model-independent; as soon as a model (in the sense of Definition 2.1) is chosen, the number of risky shares at time t, e.g., becomes Definition 2.6.(i) A semi-static portfolio, or semi-static trading strategy, is a triple where φ 0 0 ∈ R, φ 0 t : (0, ∞) 3t → R is Borel measurable for 1 ≤ t ≤ T , analogously for φ 1 , and φ t,i ∈ R for t ∈ T , i ∈ {1, . . ., N t }.Here, φ 0 t denotes the investment in the bank account, φ 1 t denotes the number of stocks held at time t ∈ T , and φ t,i ∈ R is the number of options with maturity t ∈ T and strike K t,i which the investor buys at time zero.
Recall that the call options are cash-settled.Therefore, φ 0 t includes the payoffs of all options with maturity t.
(iii) For prices (2.1)-(2.4), the initial portfolio value of a semi-static portfolio Φ is given by This is the cost of setting up the portfolio Φ.
Having defined semi-static portfolios, we can now formulate two useful notions of arbitrage.
Definition 2.7.Let ≥ 0. The prices (2.1)-( 2.4) admit model-independent arbitrage with respect to spread-bound , if we can form a self-financing semi-static portfolio Φ in the bank account, the underlying asset and the options, such that the initial portfolio value r Φ is negative and the following holds: For all real numbers (2.5), (2.6), and (2.7)), we have where s is defined in (2.10).
In particular, if Φ is such a strategy, then for any model satisfying (2.6) and (2.7), we have Definition 2.8.Let ≥ 0. The prices (2.1)-(2.4)admit a weak arbitrage opportunity with respect to spread-bound if there is no model-independent arbitrage strategy (with respect to spread-bound ), but for any model satisfying (2.6) and (2.7), there is a semi-static portfolio Φ such that the initial portfolio value r Φ is non-positive, and Most of the time we will fix ≥ 0 and write only model-independent arbitrage, meaning model-independent arbitrage with respect to spread-bound , and similarly for weak arbitrage.The notion of weak, i.e. model-dependent, arbitrage was first used in [6], where the authors give examples to highlight the distinction between weak arbitrage and model-independent arbitrage.The crucial difference is that a weak arbitrage opportunity may depend on the null sets of the model.E.g., suppose that we would like to use two different arbitrage strategies according to whether a certain call will expire in the money with positive probability or not.Such portfolios could serve to exhibit weak arbitrage (Definition 2.8), but will not show model-independent arbitrage (Definition 2.7).
Note that the process (D(t)S C t ) t∈T does not have to be a martingale, since S C is not traded on the market.The option prices give us some information about the marginals of the process S C , though.On the other hand, the process (D(t)S * t ) t∈T has to be a martingale, but we have no information about its marginals, except that . This is equivalent to where W ∞ denotes the infinity Wasserstein distance, and L the law of a random variable.We define W ∞ on M, the set of probability measures on R with finite mean, by The infimum is taken over all probability spaces (Ω, F, P) and random pairs (X, Y ) with marginals given by µ and ν.Clearly, for ≥ 0 and random variables X and Y , we have that See [9] for some references on W ∞ .Definition 2.9.Let µ, ν be two measures in M. Then we say that µ is smaller in convex order than ν, in symbols µ ≤ c ν, if for every convex function φ : R → R we have that φ dµ ≤ φ dν, as long as both integrals are well-defined.A family of measures (µ t ) t∈T in M is called a peacock, if µ s ≤ c µ t for all s ≤ t in T (see Definition 1.3 in [11]).
For µ ∈ M and x ∈ R we define the call function of µ.The mean of a measure µ will be denoted by Eµ = y µ(dy).These notions are useful for constructing models for -consistent prices, as made explicit by the following lemma.
As is evident from its proof, the sequence (µ t ) consists of the marginals of a (discounted) reference price, whereas (ν t ) gives the marginals of a (discounted) consistent price system.
Lemma 2.10.For ≥ 0 the prices (2.1)-(2.4)are -consistent with the absence of arbitrage, if there are sequences of measures (µ t ) t∈T and (ν t ) t∈T in M such that: ] for all t ∈ T and i ∈ {1, . . ., N t }, and t∈T is a peacock and its mean satisfies Eν ∈ [S 0 , S 0 ], and Proof.Let (µ t ) t∈T and (ν t ) t∈T be as above.Recall that Strassen's theorem [9,15] asserts that any peacock is the sequence of marginals of a martingale.Therefore, there is a finite probability space (Ω, F, P) with a martingale ( S t ) t∈T such that ν t is the law of S t for t ∈ T .Let S C be an adapted (to the filtration generated by S) process such that D(t)S C t ∼ µ t for t ∈ T .By the definition of the infinity Wasserstein distance, we then have for t ∈ T yields an arbitrage free model.

Single maturity: -consistency
In this section, we characterize -consistency (according to Definition 2.4) in the special case that all option maturities agree.The consistency conditions for a single maturity are similar to those derived in Theorem 3.1 of [6] and Proposition 3 of [5].In addition to the conditions given there, we have to assume that the mean of S C t is "close enough" to S 0 .We fix t = 1 ∈ T and often drop the time index for notational convenience, i.e. we write r i instead of r 1,i etc.In the frictionless case the underlying can be identified with an option with strike k = 0.Here we will do something similar: in the formulation of the next theorem we set k 0 = , as if we would introduce an option with strike B(1), but we think of C( B(1)) as the underlying.The choices for r 0 = S 0 − 2 and r 0 = S 0 made in Theorem 3.1 can be motivated as follows: in every model which is -consistent with the absence of arbitrage, (2.7) implies that the discounted expected payoff of an option with strike B(1) has to satisfy − .Furthermore, to guarantee the existence of a consistent price system, D(1)E[S C  1 ] has to lie in the closed interval [S 0 − , S 0 + ], which implies that the price of an option with strike B(1) has to lie in the interval [S 0 − 2 , S 0 ].Therefore, in the proof of Theorem 3.1 (given in Appendix A) we will use the symbol C t ( B(t)) as a reference to the underlying and −C t ( B(t)) is a reference to a short position in the underlying plus an additional deposit of 2 in the bank account.
Theorem 3.1.Let ≥ 0 and consider prices as at the beginning of Section 2, with T = 1 and k 1 > (see the remarks after (2.7)).Moreover, for ease of notation (see the above remarks) we set k 0 = , r 0 = S 0 − 2 , and r 0 = S 0 .Then the prices are -consistent (see Definition 2.4) if and only if the following conditions hold: (i) All butterfly spreads have non-negative time-0 price, i.e.
(ii) Call-spreads must not be too expensive, i.e.
(iv) If a call-spread is available for zero cost, then the involved options have zero bid resp.ask price, i.e.
Moreover, there is a model-independent arbitrage, as soon as any of the conditions (i)-(iii) is not satisfied.
This theorem is proved in Appendix A. Supplementing Theorem 3.1, we now show that there is only weak arbitrage if (3.4) fails, i.e., no model-independent arbitrage.This is the content of the following proposition; its proof is a modification of the last part of the proof of Theorem 3.1 from [6], and is presented in Appendix B. We conclude that the trichotomy of consistency/weak arbitrage/model-independent arbitrage, which was uncovered by Davis and Hobson [6] in the frictionless case, persists under bid-ask spreads (at least in the one-period setting).Proposition 3.2.Under the assumptions of Theorem 3.1, there is a weak arbitrage opportunity if (3.1), (3.2) and (3.3) hold, but (3.4) fails, i.e. there exist i < j such that r i = r j > 0. For = 0 and r i = r i = r i , the conditions from Theorem 3.1 simplify to and r i = r i−1 implies r i = 0, for i ∈ {1, . . ., N }.
These are exactly the conditions required in Theorem 3.1 of [6].
Remark 3.3.Note that in contrast to the frictionless case, we do not have to require that bid or ask prices decrease as the strike increases, in order to get models which are -consistent with the absence of arbitrage.This means that we do not have to require r i ≥ r j or r i ≥ r j for i < j, as shown in the following example.Consider two call options, where = 0 (no spread on the underlying), and the prices are given by S 0 = S 0 = 4, We assume that the bank account is constant until maturity.The prices and a possible choice of shadow prices are shown in Figure 1.(Note that, in the proof of Theorem 3.1 in Appendix A, e i denotes the shadow price of the option with strike k i .)Clearly all conditions from Theorem 3.1 are satisfied, and therefore there exists an arbitrage free model.For example we can choose µ = δ 5 , where δ denotes the Dirac delta.
This example shows that, in our setting, prices which are admissible from a no-arbitrage point of view do not necessarily make economic sense: Since the payoff of C(K 2 ) at maturity never exceeds the payoff of C(K 1 ), the utility indifference ask-price of C(K 2 ) should not be higher than the utility indifference ask-price of C(K 1 ).

Multiple maturities: necessary conditions for -consistency
The -consistency conditions for a single maturity (Theorem 3.1) are a generalisation of the frictionless conditions in [5,6].They guarantee that for each maturity t ∈ T the option prices can be associated to a measure µ t , such that Eµ t ∈ [S 0 , S 0 ] (cf.Lemma 2.10).In this section we state necessary conditions for multiple periods.Our conditions (see Definition 4.1 and Theorem 4.3) are fairly involved, and we thus expect that it might not be easy to obtain tractable equivalent conditions, instead of just necessary ones.In the case where there is only a spread on the options, but not on the underlying, it suffices to compare prices with only three or two different maturities (see equations ( 4), ( 5) and ( 6) in [5] and Corollary 4.2 in [6]) to obtain suitable consistency conditions.These conditions ensure that the family of measures (µ t ) t∈T is a peacock.
If we consider a bid-ask spread on the underlying and want to check for -consistency according to Definition 2.4 ( > 0), it turns out that we need conditions that involve all maturities simultaneously (this will become clear by condition (5.2) below).We thus introduce calendar vertical baskets (CVB), portfolios which consist of various long and short positions in the call options.We first give a definition of CVBs.Then, in Lemma 4.2 we will study a certain trading strategy involving a short position in a CVB: this strategy will then serve as a base for the conditions in Theorem 4.3, which is the main result of this section.Note that our definition of a CVB depends on ≥ 0: the contract defined in Definition 4.1 only provides necessary conditions in markets where the bid-ask spread is bounded by ≥ 0. Definition 4.1.Fix u ∈ T and ≥ 0 and assume that vectors σ = (σ 1 , . . ., σ u ), x = (x 1 , . . ., x u ), I = (i 1 , . . ., i u−1 ) and J = (j 1 , . . ., j u ) are given, such that (i ) x t ∈ R for all t ∈ {1, . . ., u}, (ii ) σ 1 ∈ {−1, 1} and σ t = sgn(x t−1 − x t ) for all t ∈ {2, . . ., u}, (iii ) i t ∈ {0, . . ., N t+1 } and k t+1,it ≤ x t + σ t+1 for all t ∈ {1, . . ., u − 1}, (iv ) j t ∈ {0, . . ., N t } and k t,jt = x t + σ t for all t ∈ {1, . . ., u}.
Then we define a calendar vertical basket with these parameters as the contract The market ask resp.bid-price of CV B u (σ, x, I, J ) are given by We will refer to u as the maturity of the CVB.
Lemma 4.2.Fix ≥ 0. For all parameters u, σ, x, I, J as in Definition 4.1, there is a selffinancing semi-static portfolio Φ whose initial value is given by r Φ = −r CV B u (σ, x, I, J ), such that for all models satisfying (2.6) and (2.7) and for all t ∈ T one of the following conditions holds: (i) φ 0 t ≥ 0 and φ 1 t = 0, or The arguments of φ 0 t , φ 1 t are of course the same as in (2.9), and are omitted for brevity.In the proof of Lemma 4.2, we define the functions φ 0 t , φ 1 t inductively.As we are defining a modelindependent strategy, we could also use the real dummy variables (2.10) from Definition 2.6 as arguments.It seems more natural to write (S u ) u≤t , (S C u ) u≤t , (S u ) u≤t , though.We just have to keep in mind that φ 0 t , φ 1 t have to be constructed as functions of (S u ) u≤t , (S C u ) u≤t , (S u ) u≤t , without using the distribution of these random vectors.
Proof of Lemma 4.2.Assume that we buy the contract thus we are getting an initial payment of r CV B u (σ, x, I, J ).We have to keep in mind that if i t = 0 for some t ∈ {1, . . ., u − 1}, then the corresponding expression in (4.2) denotes a long position in the underlying, and if j t = 0 for some t ∈ {1, . . ., u}, then the expression −C t (K t,jt ) in (4.2) denotes a short position in the underlying plus an additional deposit of 2 in the bank account at time 0 (see the beginning of Section 3).To ease notation, we will write K t,i instead of K t,it−1 and K t,j instead of K t,jt .
We will show inductively that at the end of each period t ∈ {1, . . ., u} we can end up in one of two scenarios: either the investor holds a non-negative amount of bank units (i.e.φ 0 t ≥ 0), we will call this scenario A, or we have one short position in the underlying (i.e.φ 1 t = −1) and φ 0 t ≥ k t,j − σ t ; we will refer to this as scenario B. Note that scenarios A and B are not disjoint, but this will not be a problem.
We will first deal with the case where σ 1 = −1 and afterwards with the case σ 1 = 1.We start with t = 1 and first assume that j 1 > 0. If C 1 (K 1,j ) expires out of the money, then we do not trade at time 1 and obtain φ 0 1 = 2 ≥ 0, so we are in scenario A. Otherwise we sell one unit of the underlying, and thus yielding scenario B. Recall from Section 2 that D(t) = B(t) −1 .If j 1 = 0 then k 1,j = .We do not close the short position in this case and we get that φ 0 = 4 ≥ k 1,j − σ 1 , so we also get to scenario B.
For the induction step we split the proof into two parts.In part A we will assume that at the end of period t we are in scenario A, and in part B we will assume that at the end of period t we are in scenario B.
Part A: We will show that at the end of period t + 1 we can end up either in situation A or B. First we assume that j t , i t−1 > 1, and so both expressions in (4.2) with maturity t denote options (and not the underlying).Under these assumptions φ 0 t satisfies Clearly, if K t,i ≤ K t,j or if both options expire out of the money, then φ 0 t ≥ 0, and we are in situation A at the end of period t + 1.So suppose that K t,i > K t,j and that S C t > K t,j .This also implies that σ t = 1.If this is the case, we go short in one unit of the underlying, and φ 0 t can be bounded from below as follows: This corresponds to situation B. Next assume that j t = 0 and i t > 0. Then we have that k t,j = .At time t we end up in scenario B: We proceed with the case that j t > 0 and i t−1 = 0. Since k t,j > , we can close the long position in the underlying and end up in scenario A at time t: The case where j t = i t−1 = 0 is easily handled since the long and the short position simply cancel out.We are done with part A. Part B: Assume that at the end of period t we are in scenario B, and thus φ 0 t−1 = k t−1,j − σ t−1 .First we will consider the case where j t , i t−1 > 1.If at time t the option with strike K t,j expires in the money, we do not close the short position and have which means that we end up in scenario B. Now we distinguish two cases according to x t−1 ≤ x t and x t−1 > x t , and always assume that C t (K t,j ) expires out of the money.If x t−1 ≤ x t , then we also have that k t,i ≤ k t,j and that σ t = −1.We close the short position to end up in scenario A: If on the other hand x t−1 > x t and σ t = 1, we do not trade at time t to stay in scenario B: We proceed with the case where j t = 0 and i t > 0. As before, we have k t,j = , and we can close one short position to stay in scenario B: If j t > 0 and i t−1 = 0, then we distinguish two cases: either C t (K t,j ) expires out of the money, in which case we cancel out the long and short position in the underlying and have: which corresponds to scenario A. Or, C t (K t,j ) expires in the money.Then we sell one unit of the underlying and hence we end up in scenario B: In the last inequality we have used that k t−1,j − σ t−1 = x t−1 ≥ k t,i − σ t , and that k t,i = .
The case where j t = i t−1 = 0 is again easy to handle, since the long and the short position cancel out and we are in scenario B at the end of the (t + 1)-st period.
Thus at time u we are either in scenario A or scenario B, which proves the assertion if σ 1 = −1.
The proof for σ 1 = 1 is similar.We will first show that at the end of the second period we can either be in scenario A or scenario B, and the statement of the proposition then follows by induction exactly as in the case σ 1 = −1.
First we assume that j 1 > 0.Then, if the option C 1 (K 1,j ) expires out of the money, we are in scenario A; otherwise we go short in the underlying and have which corresponds to scenario B. If j 1 = 0, then we also have that k j,1 = , and hence we are in scenario B.
According to Lemma 4.2, there is a semi-static, self-financing trading strategy Φ for the buyer of the contract −CV B u (σ, x, I, J ), such that (φ 0 u , φ 1 u ) only depends on σ u , k u,j (the investor might have some surplus in the bank account).In the following we will use this strategy and only write x, I, J ).In the case where φ 0 u ≥ 0 and φ 1 u = 0 we will say that the calendar vertical basket expires out of the money; otherwise we will say that it expires in the money.
The next theorem states necessary conditions for the absence of arbitrage in markets with spread-bound ≥ 0. Theorem 4.3.Let ≥ 0, s, t, u ∈ T such that s < t and s < u and i ∈ {0, . . ., N t }, j ∈ {0, . . ., N s }, l ∈ {0, . . ., N u }.Fix prices as at the beginning of Section 2, with k t,1 > for all t ∈ T .Then, for all calendar vertical baskets with maturity s ∈ T and parameters k s,j and σ s , the following conditions are necessary for -consistency: (iv) Proof.We will assume that s < t ≤ u and that i, l > 0. The other cases can be dealt with similarly.In all four cases (i )-(iv ) we will assume that until time s we followed the trading strategy described in Lemma 4.2.(i ) If (4.3) fails we set and buy θC t (K ), making an initial profit.If the calendar vertical basket CV B s (σ s , K s,j ) expires out of the money, then we have model-independent arbitrage.Otherwise we have a short position in the underlying at time s.In order to close the short position, we buy θ units of the underlying at time t, and we buy 1 − θ units of the underlying at time u.The liquidation value of this strategy at time u is then non-negative: (ii ) Next, assume that (4.4) fails.Then buying the contract earns an initial profit.If CV B s (σ s , K s,j ) expires out of the money, then we leave the portfolio as it is.Otherwise we immediately enter a short position and close it at time u.The liquidation value is then non-negative: (iii ) If (4.5) fails, then we buy the contract C t (K t,i )−CV B s (σ s , k s,j ) for negative cost.Again we can focus on the case where CV B s (σ s , k s,j ) expires in the money.We sell one unit of the underlying at time s and close the short position at time t.The liquidation value of this strategy at time t is non-negative: (iv ) We will show that there cannot exist an -consistent model, if (4.6) fails.In every model where the probability that CV B s (σ s , k s,j ) expires in the money is zero, we could simply sell CV B s (σ s , k s,j ) and follow the trading strategy from Lemma 4.2, realising (model-dependent) arbitrage.On the other hand, if CV B s (σ s , k s,j ) expires in the money with positive probability, then we can use the same strategy as in the proof of (iii ).At time t the liquidation value of the portfolio is positive with positive probability.
Note that if = 0 then CV B s (σ s , k s,j ) has the same payoff as −C s (K s,j ).Keeping this in mind, it is easy to verify that the conditions from Theorem 4.3 are a generalisation of equations ( 4), ( 5) and ( 6) in [5].

Multiple maturities: Necessary and sufficient conditions under simplifying assumptions
It remains open whether the conditions from Theorem 4.3 together with the conditions for each single maturity (as given in Theorem 3.1) are sufficient for the existence of -consistent models or not.As a first step towards this problem, in Theorem 5.3 we will state necessary and sufficient conditions which guarantee the existence of -consistent models under simplifying assumptions: Assumption 5.1.(i) For all maturities 1 ≤ t ≤ T , options with all (discounted) strikes k ∈ R are traded.
(ii) For each option, bid and ask price agree.We will write R t (k) for the price of an option with strike B(t)k and maturity t.
(iii) For all t ∈ T the function k → R t (k) is a call function and the associated measure µ t has finite support which is contained in [ , ∞).
Note that, for convenience, we have slightly changed the notation from Sections 2 and 3: the option prices are now denoted by R t (k) instead of r t,i .If k → R t (k) is not a call function, then the prices cannot be consistent with the absence of arbitrage (see Theorem 3.1).Semi-static portfolios are defined as in Section 2, involving finitely many call options.In order to construct arbitrage-free models under Assumption 5.1, we now recall the main result of [9], which gives a criterion for the existence of the peacock (ν t ) from Lemma 2.10.Recall also the notation W ∞ , M introduced before Definition 2.9.According to Proposition 3.2 in [9], for > 0, a measure µ ∈ M, and m ∈ [Eµ − , Eµ + ], the set {ν ∈ M : W ∞ (µ, ν) ≤ , Eν = m} has a smallest and a largest element, and their respective call functions can be expressed by the call function R µ of µ (see (2.12)) as follows: where conv denotes the convex hull.The main theorem of [9] gives an equivalent condition for the existence of a peacock within W ∞ -distance of a given sequence of measures: Theorem 5.2.(Theorem 3.5 in [9]) Let > 0 and (µ n ) n∈N be a sequence in M such that is not empty.Denote by (R n ) n∈N the corresponding call functions.Then there exists a peacock if and only if for some m ∈ I and for all N ∈ N, x 1 , . . ., x N ∈ R, we have Here, σ n = sgn(x n−1 − x n ) depends on x n−1 and x n .In this case it is possible to choose Assumption 5.1 allow us to circumvent many problems, as we only have to check whether the family (µ t ) t∈T satisfies the condition from Theorem 5.2.The conditions in Theorem 5.3 can be directly derived from (5.2).Note that in our case m = S 0 and Eµ t = R t ( ) + .
Proof.We will first show that there is model-independent arbitrage with respect to spreadbound if any of the above conditions fail.We will assume that u = T .Throughout the first part of the proof we fix k 1 , . . .k T −1 ∈ R and set K t = B(t)k t and t = B(t) , for t ∈ T .(i ) If (5.3) fails, we buy the contract where k 0 = k T = 0 and σ t = sgn(k t−1 −k t ), making an initial profit.Therefore, we will construct a semi-static trading strategy Φ, and φ 0 t will again denote the number of bank account units held by the investor at time t ∈ T .We will show inductively that for all t ∈ {1, . . ., T } we either have that φ 0 t ≥ k t -which we will refer to as scenario A -or we can have a long position the underlying asset S and φ 0 t ≥ 0, to which we will refer as scenario B .We start with t = 1.If both options with maturity t = 1 expire in the money, then we do not trade and hence which corresponds to scenario A .If on the other hand S C 1 ≤ K 1 − 1 , then we get the amount S C 1 + 1 transferred to our bank account, which is enough to buy the underlying asset for S 1 ≤ S C 1 + 1 .Now suppose that at the end of the t-th period we are in scenario A .If both options with maturity t expire in the money, then φ 0 t ≥ k t , and we finish the t + 1-th period in scenario A .So assume that k t−1 ≤ k t , and that at least one option is out of the money.Then S C t ≤ K t − t , in which case we have enough money to buy the underlying for S t and end up in scenario B : Next, if k t−1 ≥ k t and both options with maturity t expire out of the money, then φ 0 t ≥ k t−1 ≥ k t .So we only have to check the case where k t−1 + ≤ D(t)S C t ≤ k t + : Now assume that at the end of the t-th period we are in situation B .If k t−1 ≤ k t , then the cashflow generated by the two options with maturity t is non-negative, and we stay in scenario B .If on the other hand k t−1 > k t , then we sell the underlying for S t and obtain φ 0 t ≥ k t , which can be seen similarly as in the previous step by differentiating cases.This completes the first part of the proof.
(ii ) If (5.4) is violated, then buying the contract where k 0 = 0, earns an initial profit.Following the same strategy as in (i ) yields a modelindependent arbitrage strategy.
(iii ) If (5.5) fails, then we buy the contract where k T = 0, making an initial profit.Whenever C 1 (K 1 + 1 ) expires out of the money, we still have the underlying, whereas if S C 1 ≥ K 1 + 1 , then we sell one unit of the underlying, yielding φ 0 1 ≥ k 1 .The rest can be done by induction, as in part (i ).
(iv ) If (5.6) fails, then buying the contract earns an initial profit.We will show inductively that at the end of the t + 1-th period, t ∈ {1, . . ., T }, it is possible to have either φ 0 t ≥ 0, or to have φ 0 t ≥ k t and be short one unit of the underlying asset.These two scenarios are exactly scenario A resp. scenario B from the proof of Lemma 4.2.We start with t = 1.If C 1 (K 1 + 1 ) expires out of the money, then the former condition is satisfied; otherwise we short-sell one unit of the asset.Thus, we get Now suppose that at the end of the t-th period we are in scenario A. If k t−1 ≤ k t , then the cashflow generated by the two options with maturity t is non-negative and we stay in scenario A. If on the other hand k t ≤ k t−1 , we sell one unit of the underlying at the end of period t + 1.The corresponding trading strategy satisfies Now assume that at the end of the t-th period we are in situation B, meaning that we have a short position in the underlying and hold at least k t−1 units of the bank account.Then, if both options with maturity t are in the money, we do not trade, and φ 0 t ≥ k t .Otherwise we close the short position and finish in scenario A. At time T we have model-independent arbitrage, which can be seen similarly as in the previous steps.
If all four conditions hold, then we can use the fact that Eµ An inspection of the proof of Theorem 5.2 (i.e., Theorem 3.5 in [9]) easily shows that there exists a peacock (ν t ) t∈T with mean S 0 such that W ∞ (µ t , ν t ) ≤ , t ∈ T .
By Lemma 2.10 the prices are then -consistent with the absence of arbitrage.

Multiple maturities: consistency
As mentioned in the introduction, our main goal is to find the least bound on the underlying's bid-ask spread that allows to reproduce given option prices.The following result clarifies the situation if no such bound is imposed (see also Example 2.3).In our wording, we now seek conditions for consistency (Definition 2.2) and not -consistency (Definition 2.4).By enlarging the class of models, the no-arbitrage conditions become looser.In particular, we do not have any intertemporal conditions.Recall the notation used in Theorem 3.1, where i = 0 is allowed in (3.1)-(3.4),inducing a dependence of these conditions on S 0 and S 0 .In the following proposition, on the other hand, we require i, j, l ≥ 1, and therefore the current bid and ask prices of the underlying are irrelevant when checking consistency of option prices.Thus, the notion ofconsistency seems to make more sense than consistency.It turns out that the conditions of Theorem 3.1 are implied by an even weaker notion of no-arbitrage, where the spread bound has to hold only with a certain probability: Theorem 6.2.Let p ∈ (0, 1] and ≥ 0. For given prices (2.1)-(2.4) the following are equivalent: • The prices satisfy Definition 2.4 ( -consistency), but with (2.6) replaced by the weaker condition P S t − S t ≥ B(t) ≤ p, t ∈ T .
For the proof of Theorem 6.2 we employ a variant of Strassen's theorem for the modified Prokhorov distance.(To define the standard Prokhorov distance, replace p by h in the right-hand side.)A well known result, which was first proved by Strassen, and was then extended by Dudley [8,15], explains the connection of d P p to minimal distance couplings: Proposition 6.4.Given measures µ, ν on R, p ∈ [0, 1], and > 0, there exists a probability space (Ω, F, P) with random variables X ∼ µ and Y ∼ ν such that if and only if d P p (µ, ν) ≤ .(6. 2) The following theorem ("Strassen's theorem for the modified Prokhorov distance") is proved in Section 8 of [9].Theorem 6.5.Let (µ n ) n∈N be a sequence in M, > 0, and p ∈ (0, 1].Then, for all m ∈ R there exists a peacock (ν n ) n∈N with mean m such that where the last inequality follows from the fact that h 0,j (k 0 ) = g 0,i (k 0 ) = e 0 and that h 0,j = r j − e 0 k j − k 0 ≤ r i − e 0 k i − k 0 = g 0,i .
In the last step we used that e 0 ≥ f j,i (k 0 ).Now suppose we have already constructed e 1 , . . .The inequality f j,l (k s ) ≤ g s−1,i (k s ), i, j, l ∈ {s, . . ., N }, j < l, can be shown using the same arguments as before: first we note that f j,l (k s−1 ) ≤ e s−1 = g s−1,i (k s ) and then we distinguish between i < j, i = j and i > j.
We have now constructed a finite sequence (e s ) s∈{0,...,N } .Observe that for all s ∈ {0, . . ., N } the bounds on e s from above, namely (A.1) and (A.2) for s = 0, (A.4) and (A.5) for s = 1 and (A.Furthermore, by (A.4) Finally, L is strictly decreasing on {L > 0} which is most easily seen from e s ≤ g s−1,N (k s ).Therefore L can be extended to a call function R as follows (see Proposition 2.3 in [9]): shows that and so the portfolio C Φ in the modified market has negative cost.But its liquidation value at maturity is unchanged and hence non-negative, and we have thus constructed a model-independent arbitrage strategy for the modified set of prices, which is a contradiction.

Figure 1 :
Figure 1: This example shows that it is not necessary that the ask-prices resp.bid-prices decrease w.r.t.strike.The line represents the call function of δ 5 .
e l − e l ) ≤ r Φ − N l=l0+1 φ l (e l − e l ) ≤ r Φ + z N l=l0+1 |φ l | k l − k l0 N s=l0 Proof.By Theorem 3.1 these conditions are necessary.Now fix t ∈ T and assume that the conditions hold.Exactly as in the sufficiency proof of Theorem 3.1, we can construct e t,1 , e t,2 , . . ., e t,Nt such that e t,i ∈ [r t,i , r t,i ].The linear interpolation L t of the points (k t,i , e t,i ) i∈{1,...,Nt} can then be extended to a call function of a measure µ t (see the final part of the sufficiency proof of Theorem 3.1).K t,i ) + ] = e t,i ∈ [r t,i , r t,i ], i ∈ {1, . . .N t }.Furthermore, we pick s ∈ [S 0 , S 0 ] and set ν t = δ s (Dirac delta) for all t ∈ T .Clearly, (ν t ) t∈T is a peacock, and we set S * t = B(t)s, which implies D(t)S * t ∼ ν t .Finally, we define S t = S * t ∧ S C t and S t = S * t ∨ S C t , and have thus constructed an arbitrage free model.