Dynamic trading under integer constraints

In this paper we investigate discrete time trading under integer constraints, that is, we assume that the offered goods or shares are traded in integer quantities instead of the usual real quantity assumption. For finite probability spaces and rational asset prices this has little effect on the core of the theory of no-arbitrage pricing. For price processes not restricted to the rational numbers, a novel theory of integer arbitrage free pricing and hedging emerges. We establish an FTAP, involving a set of absolutely continuous martingale measures satisfying an additional property. The set of prices of a contingent claim is no longer an interval, but is either empty or dense in an interval. We also discuss superhedging with integral portfolios.


Introduction
Classical, frictionless no-arbitrage theory [8,15] makes several simplifying assumptions on financial markets. In particular, position sizes may be arbitrary real numbers, which allows trading strategies that cannot be implemented in practice. Even if brokers are receptive to fractional amounts of shares, there will be a smallest fraction that can be purchased or sold. Moreover, traders might wish to avoid odd lots because of additional brokerage fees and the usually poor liquidity of small positions. In this case, the smallest traded unit would be a round lot consisting of several (e.g., 100) shares. Both situations can be covered by assuming that integer amounts of a price process (S 1 t , . . . , S d t ) t∈T can be traded. The set of trading times T is assumed to be finite in this paper. For simplicity, we will call S i the price process of the i-th (risky) asset, although it may have the interpretation of a fraction or a round lot of an actual asset price. We assume that the amount of money in the risk-less asset may take arbitrary real values. On the one hand, this increases tractability; on the other hand, it makes economic sense, as the smallest possible modification of the bank account is usually several orders of magnitude smaller than that of the the risky positions. Thus, our integer trading strategies in a model with d risky assets take values in R × Z d at each time. For some results and proofs, we also consider rational strategies with values in R × Q d . By clearing denominators, the corresponding notions of freeness of arbitrage are equivalent (see Lemma 2.3).
To the best of our knowledge, the existing literature on arbitrage, pricing and hedging under trading constraints [7,11,12] invariably imposes convexity assumptions on the set of admissible strategies, which are unrelated to integrality constraints. The latter do feature prominently in the computational finance literature, e.g. in the papers [3,5,6], which employ mixed-integer nonlinear programming to solve the Markowitz portfolio selection problem. In the literature, other keywords Date: August 28, 2017. 2010 Mathematics Subject Classification. 91G10,91G20,11K60. 1 such as minimum lot restrictions, minimum transaction level, and integral transaction units are used with the same meaning as our integer constraints. Somewhat surprisingly, this kind of restriction seems to have received almost no attention from the viewpoint of no-arbitrage theory. One exception is a paper by Deng et al. [10], who show that deciding the existence of arbitrage in a one-period model under integer constraints is an NP-hard problem.
In our main results, we assume that the underlying probability space is finite (Assumption (F) of Section 2). This assumption is realistic, because actual asset prices move by ticks, and prices larger than 10 10 10 , say, will never occur. Still, extending our work to arbitrary probability spaces might be mathematically interesting, but is left for future work.
In Section 2, we introduce the notions of no integer arbitrage (NIA) and no integer free lunch (NIFL) in a straightforward way. It turns out (Theorem 2.5) that the latter property is equivalent to the classical no-arbitrage condition NA, and so we concentrate on NIA in the rest of the paper. Our first main result is a fundamental theorem of asset pricing (FTAP; Theorem 2.11) characterising NIA. It involves a set of absolutely continuous martingale measures satisfying an additional property. The latter amounts to explicitly avoiding integer arbitrage outside the support of the absolutely continuous martingale measure. The theorem is thus not as neat as the classical FTAP, but is still useful for establishing several of our subsequent results. In Section 3, we define the set Π Z (C) of NIA-compatible prices of a claim C. The integer variant of the classical representation using the set of equivalent martingale measures features only an inclusion instead of an equality (Proposition 3.4), and in fact Π Z (C) may be empty. Even if it is non-empty, it need not be an interval; however, Π Z (C) is then always dense in an (explicit) interval, which is the main result of Section 4. As regards methodology, many of our arguments just use the countability of Z d (and Q d ), or the density of Q d in R d . Still, at some places (such as Lemma 2.4, Example 5.3, and Theorem 5.4) we invoke non-trivial results from number theory, collected in Appendix A.
Readers who are mainly interested in the practical consequences of integer restrictions are invited to read (besides the basic definitions) Theorem 2.6, Theorem 3.6, and Section 5. In a nutshell, for the discrete-time models used in practice (finite probability space, floating-point -i.e., rational -asset values), the core of no-arbitrage theory does not change much. One exception is the fact that the supremum of claim prices consistent with no-integer-arbitrage need not agree with the smallest integer superhedging price (see Section 5). Still, this property holds in a limiting sense when superhedging a large portfolio of identical claims. That said, our work is by no means the last word on the practical consequences of integer restrictions in dynamic trading. Problems such as quantile hedging, hedging with risk measures, or hedging under convex constraints may well be worth studying under integer restrictions. In Section 6 we discuss a toy example of variance optimal hedging under integer constraints, which leads to the closest vector problem (CVP), a well-known algorithmic lattice problem.

Trading strategies and absence of arbitrage
We will work with a probability space (Ω, A, P ). Our main results use the following assumption: (F) Ω is finite, A is the power set of Ω, P [{ω}] > 0 for any ω ∈ Ω, and we choose an enumeration ω 1 , . . . , ω n of its elements. We assume that there is a finite set of times T := {0, . . . , T }, with T ∈ N, at which trading may occur, and fix a filtration (F t ) t∈T where F T ⊆ A and F 0 = {∅, Ω}. The (deterministic) riskless interest rate is r > −1, and we have d risky assets with prices S t = (S 1 t , . . . , S d t ) at time t ∈ T, where S t is assumed to be non-negative and F t -measurable. The price of the riskless asset is denoted by S 0 t := (1 + r) t for t ∈ T, and we denote the market price processes byS := (S 0 , S).
We are interested in trading strategies that consist of integer positions in the risky assets. All trading strategies we consider are self-financing.
For convenience, we will sometimes use the notationφ 0 :=φ 1 . The set of integer trading strategies is denoted by Z.
(ii) Analogously, we define the set R of all (real) trading strategies and the set Q of rational strategies, with values in R × Q d .
We obviously have Z ⊆ Q ⊆ R. For any trading strategyφ ∈ R we denote its value at time t ∈ T by and its discounted value byV t (φ) := V t (φ)/S 0 t . Often it is convenient to work with discounted asset values or discounted gains which are denoted bŷ for t ∈ T resp. t ∈ T \ {0}. The discounted value process then equals Definition 2.2.
(i) An integer arbitrage is a strategyφ ∈ Z which is an arbitrage for the marketS.
(ii) A model satisfies the no-integer-arbitrage condition (NIA), if it admits no integer arbitrage. (iii) Define the set (a Z-module) of discounted net gains realizable by integer strategies, and Assuming (F), we define the condition NIFL (no integer free lunch) as The closure is taken w.r.t. the Euclidean topology, upon identifying L ∞ with R n .
Clearly, NIA is weaker than the classical no-arbitrage property NA or NIFL. It turns out that the classical no-arbitrage property NA and NIFL are equivalent (for finite probability spaces), see Theorem 2.5 below. The following simple properties will be used often: (i) If (F) holds, then in the definition of integer arbitrage the conditionφ ∈ Z can be replaced byφ ∈ Q.
(ii) In the definition of integer arbitrage the condition V 0 (φ) ≤ 0 can be replaced for anyφ ∈ Z (or, under (F), for anyφ ∈ Q).
Proof. (ii) and (iii) are proved precisely as in the classical case. Part (i): Clearly, any arbitrage strategy in Z is also in Q. Now assume that there is an arbitrageφ such that (ϕ 1 t , . . . , ϕ d t ) ∈ Q d for any t ∈ T. Define N := inf{n ∈ N : nϕ ∈ Z d·T }.
Then N ϕ t ∈ Z d for any t ∈ T, and Nφ is an arbitrage.
By (2.1), the implication (2.2) can be written as Although our main results assume (F), we mention that (F) is actually not necessary in parts (i) and (iii) of Lemma 2.3. This follows easily from the fact that arbitrage in a multi-period model implies the existence of a period that allows arbitrage. In the classical setup, this is Proposition 5.11 in [12]; the proof works for integer and rational strategies, too. Under the finiteness condition (F) on Ω, we can show that any real trading strategy can be approximated by an integer trading strategy with a certain rate. The proof is based on Dirichlet's approximation theorem (Theorem A.1).

Lemma 2.4.
(i) If S is bounded, then for any strategyφ ∈ R and any ǫ > 0, we can find a strategyψ ∈ Q such that (ii) Assume (F) and letφ ∈ R and ǫ > 0. Then there is q ∈ N and a strategȳ ψ ∈ Z such that V 0 (φ) = V 0 (ψ)/q and sup t∈T,j=1,...d, l=1,...,n In particular, for any strategyφ ∈ R we can find strategiesψ ∈ Q,η ∈ Z and q ∈ N such that Proof. The first part is trivial as any real number can be approximated by rational numbers. Thus we find a sequence of strategies (ψ (k) ) k∈N in Q such that ψ (k) → ϕ uniformly in ω for k → ∞. This and the boundedness of S imply the convergence of the value at any time t ∈ T if the initial value is being fixed as equal.
To show part (ii), let R t := {ϕ j t (ω l ) : j = 1, . . . , d, l = 1, . . . , n}. For any t ∈ T let a 1 t , . . . , a Kt t be an enumeration of the elements of R t . We have K t ≤ dn for any t ∈ T and thus t∈T K t ≤ nd(1 + T ). By Dirichlet's approximation theorem (Theorem A.1), we find q ∈ N with q −1/(nd(1+T )) < ǫ and Thus ψ t is measurable w.r.t. to the σalgebra generated by ϕ t and, hence, F t−1 -measurable. Therefore, ψ is a predictable Z d -valued process. The uniform distance of ψ and ϕ is less than q −1/(nd(1+T )) by construction.
With the previous lemma at hand we can show that under the finiteness condition classical no-arbitrage is equivalent to NIFL. (iv) The model satisfies NIA.
Proof. The equivalence of (i) and (ii) is the classical FTAP, see [12,Theorem 5.16]. Furthermore, NA is equivalent to the classical no free lunch condition in our setup (see [8,16]), which yields the implication (ii)⇒(iii). Now we assume (iii) and show (ii). Letφ ∈ R such that V 0 (φ) = 0 and V T (φ) ≥ 0. By part (ii) of Lemma 2.4 we find q N ∈ N and strategies ψ (N ) ∈ Z such that W.l.o.g., the sequence q N increases. We get Since Z N ∈ L ∞ , there is a convergent subsequence, and w.l.o.g. Z N itself converges to some Z ∈ cl(C Z ). By (2.4), we have Z ≥ 0. Then, NIFL implies that Z = 0, 3) (recall that q N increases), we conclude that V T (φ) = 0. Now assume that d = 1. (ii)⇒(iv) is obvious. We assume that (ii) does not hold. Proposition 5.11 in [12] yields the existence of a one-period arbitrage, i.e. an arbitrageφ and t 0 ∈ T such that ϕ t = 0 for any t ∈ T \ {t 0 }. Sinceφ is an arbitrage we must have ϕ 1 t0 = 0. Define Then ψ is an arbitrage as well. Moreover, ψ 1 t ∈ {−1, 0, 1} ⊆ Z for any t ∈ T, thus ψ ∈ Z. Consequently, (iv) does not hold.
In practice, all values occurring in the model specification are floating-point numbers. The following result shows that in this case the existence of an arbitrage opportunity is not affected by integrality constraints.
Theorem 2.6. Assume (F), and that the interest rate r and all asset values are rational: r ∈ Q, and S t ∈ Q d for t ∈ T. Then NIA is equivalent to NA.
Proof. NA always implies NIA. Now suppose that we have a real arbitrage opportunity. By part (iii) of Lemma 2.3, there is a predictable process ϕ such that The assertion now follows from Lemma 2.7 below. Note that predictability of the resulting rational process is easy to guarantee, by introducing for all k, j a single variable for the ϕ j k (ω) for which the ωs belong to the same atom of F k−1 .
In the proof of the preceding result, we applied the following simple lemma. Using Ehrhart's theory of lattice points in dilated polytopes [4,21], it is certainly possible to state much more general results along these lines. 1 Therefore, we do not claim originality for Lemma 2.7, but give a short self-contained proof for the reader's convenience.
Lemma 2.7. Let (a ij ) 1≤i≤r,1≤j≤s be a matrix with rational entries a ij ∈ Q. Suppose that there is a real vector (x 1 , . . . , x s ) such that a ij x j ≥ 0, i = 1, . . . , r, with at least one inequality being strict.
Then there is a rational vector satisfying (2.5). 1 We thank Manuel Kauers for pointing this out.
Proof. After possibly reordering the lines of the matrix (a ij ), we may assume that there is u ∈ {1, . . . , r} such that By defining y i := s j=1 a ij x j for 1 ≤ i ≤ u, we get that the vector (x 1 , . . . , x s , y 1 , . . . , y u ) solves the system Equations (2.6) and (2.7) constitute a homogeneous linear system of equations with rational coefficients, which has a basis B ⊂ Q s+u of rational solution vectors, by Gaussian elimination. The vector (x 1 , . . . , x s , y 1 , . . . , y u ) can be written as a linear combination of vectors in B. By approximating the (real) coefficients of this linear combination with rational numbers, we get a vector (x 1 , . . . ,x s ,ỹ 1 , . . . ,ỹ u ) ∈ Q s+u satisfying (2.6)-(2.8). Then (x 1 , . . . ,x s ) is the desired rational vector.
The assertion of Theorem 2.6 does not hold for infinite probability spaces, as the following example illustrates.
where p i , q i ,p i ,q i are natural numbers satisfying Thus, the increments are with at least one inequality being strict. By (2.9), the vector (ϕ 1 1 , ϕ 2 1 ) = (1, −π) satisfies this, and so NA does not hold. By letting i tend to infinity, we see that there is no integer vector satisfying (2.10)-(2.11), which shows that the model satisfies NIA.
Our next goal is to characterise NIA, without restricting the asset prices to rational numbers. As we will see, for d > 1 NIA is not equivalent to the existence of an equivalent martingale measure, but rather to the existence of an absolutely continuous martingale measure with an additional property. We first introduce sets of strategies which do not yield any net profit. Definition 2.9.
(i) Let Q be a probability measure on (Ω, A), and denote the set of trading strategies with zero initial value by R 0 := {φ ∈ R : V 0 (φ) = 0}. We denote the set of all integer-valued (resp. rational-valued, resp. real-valued) trading strategies with zero initial capital and Q-a.s. zero gain by Thenφ is the required trading strategy.
Let Q ′ be a martingale measure with support equal to Ω \ A. Thenφ satisfies the existence statement of (ii) in Definition 2.9. Letψ ∈ Q 0 We need to show that any measure in Q max Z is a martingale measure with support Ω \ A. This, however, follows as soon as we have shown that Q ′ ≪ Q for any Finally, we have to show that Q ′ can be approximated by elements in Q max Z in total variation. Define Q α := αQ ′ +(1−α)Q for any α ∈ [0, 1]. Then Q ′ = Q 1 ← Q α as α → 1. However, Q α is a martingale measure with the same support as Q for α = 1 and, hence, it is in Q max Z by what we have shown so far.
We can now state an FTAP for integer trading.
Theorem 2.11. Assume (F). Then the following statements are equivalent: Thusψ is an integer arbitrage. A contradiction. Consequently, A Ω. We have shown that the marketS is free of arbitrage on Ω \ A. The classical fundamental theorem yields a martingale measure Q on Ω \ A forS. We denote its extension to a probability measure on Ω by Q, i.e.
An immediate consequence is the following sufficient criterion for the construction of markets with no integer arbitrage. Corollary 2.12. Let Q ≪ P be a martingale measure and assume that Z Q 0 = {0}. Then the market satisfies NIA.
The following example is a simple application of the preceding corollary.
Thus, Corollary 2.12 yields that the market does not allow for integer arbitrage. Observe that this holds regardless of the specification of (S 1 1 , S 2 1 )(ω j ) for j ≥ 3. Another immediate consequence is the existence of absolutely continuous martingale measures.
Corollary 2.14. Suppose that a model satisfies NIA and assume (F). Then there is an absolutely continuous martingale measure.
The following example shows that the existence of an absolutely continuous martingale measure alone is insufficient to exclude integer arbitrage.
Finally, we provide a technical statement that will be used in Section 4.
Proof. Let A ∈ F 1 be maximal with Q[A] = 0. Then B := Ω \ A is an atom, and its only strict subset contained in F 1 is the empty set. If A = ∅, then the claim follows trivially. Assume for contradiction that A = ∅. We claim that the model restricted to A still satisfies NIA. To this end letφ ∈ Q, t = 1, . . . T with (Since existence of an arbitrage implies existence of a one period arbitrage, it suffices to consider this kind of strategy.) Thus (S 0 , . . . , S d ) restricted to A satisfies NIA. By Theorem 2.11 there is Q ′ ∈ Q max Z for the model (S 0 , . . . , S d ) restricted to A. We denote its extension to Ω by Q ′ as well, i.e. Q ′ [C] = Q ′ [C ∩ A] for any C ∈ A. DefineQ := Q/2 + Q ′ /2 and observe thatQ ∈ Q Z . However, Q ≈Q because Q ′ has disjoint support with Q.
But Proposition 2.10 implies Q ≈Q, which yields a contradiction. Thus A = ∅ and, hence, F 1 = F 0 .

Claims and integer trading
satisfies NIA. The set of integer arbitrage free prices is denoted by for the infimum of prices of integer superreplication strategies for C.
Analogously to Π Z (C), we write Π(C) for the set of classical arbitrage free prices in models satisfying NA. We recall the classical superhedging theorem (Corollaries 7.15 and 7.18 in [12]): Assume that NA holds, and let C be a claim with sup Π(C) < ∞. Then there is a strategyφ ∈ R with V 0 (φ) = sup Π(C) and V T (φ) ≥ C. Moreover, sup Π(C) is the smallest number with this property.
We find analogue statements to the preceding theorem under the weaker assumption NIA. Proposition 3.8 below states that NIA suffices for the existence of a real cheapest superhedge whose price is the infimum of all rational superhedging prices. Moreover, Theorem 4.3 below implies that either the set of NIA compatible prices for the claim is empty, or its supremum equals the cheapest superhedging price.
There is no need to define the notion of integer completeness, because there would be no interesting models that have this property: Proposition 3.3. The following statements are equivalent: (i) Every claim is replicable by an integer strategy, (ii) The probability space (Ω, A, P ) consists of a single atom.
Proof. If (ii) holds and C is a claim, then there is a constant c ∈ [0, ∞) such that C = c a.s. Then C is replicated by the integer strategyφ = (ϕ 0 , 0) with Now suppose that every claim is integer replicable. In particular, then, each claim is replicable in the classical sense. It is well known that this implies that Ω has a partition into finitely many atoms. (This result is, of course, usually proved in the framework of a model satisfying NA. Assuming NA is not necessary though, as seen from the proof of Theorem 5.37 in [12].) If Ω does not consist of a single atom, then we can fix two distinct atoms A and B. For a random variable X we can find its essential value on A (resp. B) and denote it by δ A (X) (resp. δ B (X)). A self-financing integer trading strategyφ is uniquely defined by specifying its initial wealth V 0 (φ) and the predictable Z d -valued process ϕ = (ϕ t ) t=1,...,T . Thus there is a bijective map Γ : For each ϕ ∈ Z c , the set δ A (V T (Γ(v, ϕ))), δ B (V T (Γ(v, ϕ))) : v ∈ R is a null set for the two-dimensional Lebesgue measure, because it is a one dimensional affine space in R 2 . We conclude that (3.3) has Lebesgue measure zero, and hence (3.2) is a null set, too. This contradicts our assumption.
Recall that in the classical case (assuming (F), so that integrability holds), the set of arbitrage free prices has the representation where Q is the set of equivalent martingale measures. The corresponding result for NIA looks as follows: Proposition 3.4. Assume (F) and that the model satisfies NIA. Let C be a claim.
Proof. Suppose that p ∈ Π Z (C). Then there is an adapted process X such that X 0 = p, X T = C and (S 0 , . . . , S d , X) satisfies NIA. LetQ Z be the set of absolutely continuous martingale measures that satisfy part (iii) of Definition 2.9 for this market. By Theorem 2.11 there is Q ∈Q Z ⊆ Q Z . Then p = E Q [ C (1+r) T ]. The following example shows that the inclusion in Proposition 3 can be strict. In fact, in this example we have Π Z (C) = ∅.
Define the extended model (S 0 , S 1 , S 2 , X), where X 0 := 0, X 1 := C. Since (0, 0, 0, 1) is an integer arbitrage for the extended model, it follows that 0 / ∈ Π Z (C), and so If the model satisfies NA (and not just NIA), then we can compare the sets of classical resp. integer arbitrage free prices. It is well known that Π(C) is an interval, which is open for non-replicable C and consists of a single point if C is replicable. It turns out that under NA the set Π Z (C) is an interval, too, which may differ from Π(C) only at the endpoints. In particular, if NA holds, then Π Z (C) cannot be empty.
Theorem 3.6. Suppose (F), that the model satisfies NA and let C be a claim.
, then either C has a duplication strategy in Q or there is no cheapest classical superhedging strategy that is in Q.
Proof. The first inclusion is trivial. Proposition 2.10 in combination with NA yields that Q max Z is the set of martingale measures which are equivalent to P and that this set is dense in Q Z . Thus, Proposition 3.4 implies To show the second assertion, suppose that s := sup(Π Z (C)) ∈ Π Z (C), and that there is a cheapest classical superhedging strategyφ ∈ Q. This means thatφ has price V 0 (φ) = s and payoff V T (φ) ≥ C. Since s ∈ Π Z (C), there is an integerarbitrage free extension of the model where C trades at price s. Consider the strategy (φ, −1) in the extended model. Its cost is zero, and its payoff is V T (φ)−C ≥ 0. By part (i) of Lemma 2.3, we conclude C = V T (φ), and soφ ∈ Q is a duplication strategy for C.
In the preceding theorem the interval boundaries may or may not be contained in Π Z (C), as the following example shows. The computations needed for parts (ii)-(iv) are similar to (i), and we omit the details.
The equivalent martingale measures are given by , and so the model satisfies NA.
(i) Define the claim Using (3.4), we find the classical arbitrage free prices .
We now check the boundary points for integer arbitrage, using part (iii) of Lemma 2.
3. An integer arbitrage in the market extended by C with price p thus amounts to ϕ ∈ Z 2 such that ϕ(∆S 1 1 , C−p) ≥ 0 and ϕ(∆S 1 1 , C−p) = 0. For p = √ 2, we get the inequalities The solution set {(ϕ 1 , ϕ 1 / √ 2) : ϕ 1 ∈ [0, ∞)} has trivial intersection with Z 2 , and so √ 2 is an integer arbitrage free price for C. Similarly, we obtain that 3 √ 2 ∈ Π Z (C) as well, and we conclude that the interval Π Z (C) contains both endpoints: . We now verify that there is no cheapest classical superhedge in Q, in accordance with the second assertion of Theorem 3.6. (Note that C is not replicable, as |Π(C)| > 1; in particular, there is no replication strategy in Q.) Clearly, ifφ ∈ R 2 is a cheapest superhedge, then ϕ 0 must satisfy ϕ 0 = max ω∈Ω (C(ω) − ϕ 1 S 1 1 (ω)). The cost of this strategy then is Our optimal strategy isφ = (3 √ 2, √ 2) / ∈ Q, because the problem Similarly, we obtain that the most expensive classical subhedge is not in Q, agreeing with the (obvious) subhedging variant of the second assertion of Theorem 3.6.
. The cheapest classical superhedge is in Q, whereas the most expensive classical subhedge is not in Q.
The cheapest classical superhedge is not in Q, whereas the most expensive classical subhedge is in Q.
then Π Z (C) = (0, 1). The cheapest classical superhedge and the most expensive classical subhedge are both in Q.
It might make sense to restrict attention to static trading strategies in the claim, e.g., as a simple approach for modelling the typically reduced liquidity of derivatives compared to their underlyings. This means that the claim can initially be bought or sold, but not traded until maturity. In the classical case, the superhedging theorem (Theorem 3.2) readily yields that the set Π stat (C) of static-arbitrage-free claim prices defined in this way satisfies Π stat (C) = Π(C). Now suppose that our model satisfies only NIA. Analogously to (3.1), definê For p / ∈ [σ Z (C), σ Z (C)], we clearly have p / ∈ Π stat Z (C), because, using appropriate integer sub-resp. superhedges, one can easily construct a static integer arbitrage for the extended model. Therefore, we obtain Π . We now proceed to identify the value of the 'cheapest' superhedge in Q. The only difference to the classical case is that the cheapest superhedge is not necessarily in Q, but can be approximated arbitrarily well with superhedges in Q (even if only NIA holds). For results on the 'cheapest' superhedge in Z, see Section 5.
Proof. Proposition 2.10 yields Then,S restricted to Ω \ A satisfies NA because Q is a martingale measure. By (F), there is a cheapest superhedge for C on this market, which we denote bȳ η ∈ R. It satisfies V T (η) ≥ C Q-a.s. Since Q ∈ Q max Z there isψ ∈ R 0 such that V T (ψ) ≥ 0 and {V T (ψ) > 0} = A. As Ω is finite there is a ∈ R such that V T (aψ +η) = aV T (ψ) + V T (η) ≥ C. By Proposition 2.10 and Theorem 3.2, we find thatφ := aψ +η is a superhedge for C with initial price Now letγ ∈ R be any superhedge for C. Then V T (γ) ≥ C Q-a.s. for any Q ∈ Q max Z and, hence, Consequently,φ is a cheapest superhedge for C.
The second assertion follows easily from part (i) of Lemma 2.4.

The structure of the set of integer-arbitrage-free prices
The main result of this section is that the set Π Z (C) of NIA-compatible claim prices is always dense in an interval (Theorem 4.3). First, we give some sufficient conditions that imply that Π Z (C) equals an interval. Proof. If (ii) holds, then Theorem 2.5 yields that (iii) holds. If we assume (iii), then Theorem 3.6 yields the claim. Now assume that (i) holds. Proposition 3.4 implies that If J is a singleton, then we have equality by assumption and, hence, the claim follows. Thus we may assume that J contains at least two points. The set J is an interval. Let p ∈ int(J) and define X 0 := p, X 1 := C. Assume for contradiction that there is an integer arbitrage (φ, ϕ d+1 ) for the model (S 0 , . . . , S d , X). By part (ii) of Lemma 2.3, we may assume that V 0 (φ, ϕ d+1 ) = 0. We have ϕ d+1 1 = 0, because otherwiseφ is an integer arbitrage for (S 0 , . . . , S d ) with V 0 (φ) = 0.
Throughout the remainder of this section, we will always assume (F), NIA and F T = A. Also, let C be a claim. The following theorem is our main result on the structure of Π Z (C) in the general case.
The theorem will follow from Lemmas 4.8 and 4.9 below. Note that Q max Z can be replaced by Q Z , due to the last assertion of Proposition 2.10. In order to prove Theorem 4.3, we assume that Π Z (C) is non-empty, and choose p * ∈ Π Z (C). By definition, there is an adapted process (X * t ) t∈T such that X * 0 = p, X * T = C, and the model (S 0 , . . . , S d , X * ) satisfies NIA. We also define for any t ∈ T \ {T }. By the same argument as in the proof of Theorem 2.11, there is Definition 4.4. We write Q t for the set of measures Q such that (Ŝ u ) u=t,...T is a Q-martingale, Ω \ A t is the support of Q, and Q[B] > 0 for any non-empty set B ∈ F t . Now we define two sequences of sets: and Lemma 4.9 below yields a countable exception set F such that C 0 = K 0 \ F . Finally, Lemma 4.8 states that C 0 is contained in the set Π Z (C) of NIA-compatible prices, which establishes Theorem 4.3. For technical reasons, we first analyse the sets Q t , and we will need the stochastic convexity of K t given in Lemma 4.7 below.
Lemma 4.5. Let t ∈ T. Then Q t is non-empty. If t = 0, then for any Q ∈ Q t−1 there is Q ′ ∈ Q t such that for any s = t, . . . , T and any random variable X : Ω → R.
Proof. Let I := {t ∈ T : the claim holds for t}. We have 0 ∈ I by Theorem 2.11 (i). Let t ∈ T such that t − 1 ∈ I. We show t ∈ I which implies I = T and, hence, the claim. We directly produce the measure with the given extra property. To this end let Q ∈ Q t−1 . Let B 1 . . . , B m be an enumeration of the minimal non-empty elements of F t and define k := |{B l : l = 1, . . . , m, B l ⊆ A t−1 \ A t }|. We may assume that B 1 , . . . , Define the probability measures P l := P P [B l ] on B l . Since the model (Ŝ u ) u=t,...,T satisfies NIA we get from Theorem 2.11 (i) a martingale measure Q l ≪ P l on B l . Define the probability measure Clearly, the support of Q ′ is Ω\A t and Q ′ [D] > 0 for any D ∈ F t . Also, (Ŝ u ) u=t,...,T is a Q ′ -martingale. Thus, we have Q ′ ∈ Q t . Now, let X : Ω → R be a random variable and s ∈ {t, . . . , T }. We show that E Q ′ [X|F s ] is a version of the F s -conditional expectation of X under Q. To this end, let D ∈ F s and define Thus, t ∈ I.
Proof. Define Obviously, T ∈ I. Let t ∈ I \ {0}. We show that t − 1 ∈ I which implies I = T and, hence, the claim. To this end, let X t−1 ∈ K t−1 . Then there is Q ∈ Q t−1 and X t ∈ K t such that Since R ∈ Q t we have R[B] > 0 for any non-empty set B ∈ F t . Let B ∈ F t−1 ⊆ F t be non-empty. Then R[B|F t ] = 1 B and, hence, However, A t is an R null set, hence R[A t |F t ] = 0 R-a.s. Since F t has no nonempty R null sets, we have Thus, s. for any random variable Y : Ω → R and any s = t, . . . , T . Define Thus, t − 1 ∈ I. Lemma 4.7. For any t ∈ T and any F t -measurable random variable α with values in [0, 1] and any X, Y ∈ K t we have Proof. Lemma 4.6 yields measures Q, R ∈ Q t such that It is clear that Q and Q ′ agree on F t , and one easily verifies Q ′ ∈ Q t . Let B ∈ F t . Then We find Case 1: ∆X 1 / ∈ D 1 or ∆X * 1 / ∈ D 1 . We defineF := {α ∈ [0, 1] : ∆X α 1 ∈ D 1 }. Since D 1 is a vector space, we find thatF contains at most one element. We claim that (0, 1] \ (F 1 ∪F ) does not contain any element of F 0 . To this end, let α ∈ (0, 1] \ (F 1 ∪F ) and assume for contradiction that there is an integer arbitrage in the first period. Hence, there is ξ ∈ Z d+1 such that ξ d+1 = 0 and ξ∆Ŝ 1 +ξ d+1 ∆X α 1 ≥ 0. Since Q is a martingale measure we get ξ∆Ŝ 1 +ξ d+1 ∆X α 1 = 0 Q-a.s., and after solving for ∆X α 1 we find ∆X α 1 ∈ D 1 . A contradiction. Case 2: ∆X 1 , ∆X * 1 ∈ D 1 and there is a set with positive Q-measure on which ∆X 1 = ∆X * 1 . Since D Q 1 restricted to Ω \ A 0 has countably many elements we find that ∆X α 1 ∈ D Q 1 at most countably often. Denote the set of α ∈ [0, 1] where ∆X α 1 ∈ D Q 1 byF . We claim F 0 ⊆ F 1 ∪F . To this end, let α ∈ (0, 1] \ (F 1 ∪F ) and assume for contradiction that there is an integer arbitrage in the first period. Then there is ξ ∈ Z d+1 such that ξ d+1 = 0 and ξ∆Ŝ 1 + ξ d+1 ∆X α 1 ≥ 0. Since Q is a martingale measure we get ξ∆Ŝ 1 + ξ d+1 ∆X α 1 = 0 Q-a.s., and after solving for ∆X α 1 we find ∆X α 1 ∈ D Q 1 . A contradiction. Case 3: ∆X 1 , ∆X * 1 ∈ D 1 and ∆X 1 = ∆X * 1 Q-a.s. Then, we have and any B ∈ F 1 , then Lemma 2.16 yields F 1 = F 0 and, hence, ∆X 1 = 0 = ∆X * 1 which yields that F 0 = F 1 . Thus, we may assume that there is R ∈ Q max Z and B ∈ F 1 with R[B] ∈ (0, 1). By Proposition 2.10 we find that there is B ∈ F 1 such that for any R ∈ Q max Z = Q 0 we have R[B] ∈ (0, 1). In particular, we have Q[B] ∈ (0, 1).

Integer superhedging
In this section we discuss some properties of the integer superhedging price σ Z (C) of a claim, as defined in (3.1). First, we give a simple example where it does not agree with the classical superhedging price sup Π(C).
Example 5.1. In this example, the gap between sup Π(C) and the cheapest integer superhedging price σ Z (C) has size a, for an arbitrary number a > 0. On the probability space Ω = {ω 1 , ω 2 }, consider the one-dimensional model We obtain that the interval of prices of integer superhedges is [2a, ∞). For real ϕ, the minimum in (5.2) is attained at ϕ = 1 2 , yielding the classical superhedging price a = sup Π(C).
As soon as a model is fixed, the gap considered in the preceding example can be bounded for all claims. In Example 5.1, we have equality in (5.3). On R n , we always use the Euclidean norm · = · 2 .
Proposition 5.2. Assume (F) and NA, and let C be a claim. Then Proof. Letψ ∈ R be a cheapest classical superhedge. By Theorem 3.2, it satisfies V 0 (ψ) = sup Π(C). By rounding the risky positions ofψ to the closest integers (with any convention for half-integers), we get a strategy (ψ 0 , ⌊ψ⌉) ∈ Z. Clearly, we get the necessary condition ifφ should be a cheapest integer superhedge. It follows that The following example shows that, contrary to the case of classical superhedging, there need not exist a cheapest integer superhedge.
This model satisfies NA, and it is complete in the classical sense. Indeed, the only martingale measure is given by Q[{ω j }] = 1/2 for j = 1, 2. Consider the claim whose set of (integer) arbitrage free prices is the singleton Then there is no minimizer for the superhedging problem (see (5.4)) Indeed, for ϕ ∈ R 2 , the set of minimizers would be yielding inf ϕ∈R 2 f (ϕ) = 1.
Since f is Lipschitz continuous (with constant L, say), we have Thus, the infimum of the prices of integer superhedges is σ Z (C) = 1, but there is no cheapest integer superhedge.
Financial institutions usually hedge large portfolios of identical (or at least similar) options. The following theorem shows that, when superhedging N copies of C, the integer superhedging price per claim converges to the classical superhedging price: lim N →∞ N −1 σ Z (N C) = sup Π(C). The second part of Theorem 5.4 gives an estimate on superhedging C with rational strategies with controlled denominators.
Theorem 5.4. Assume (F) and NA, and let C be a claim. Then (ii) There is a sequence of rational strategiesψ (N ) ∈ Q such that all denominators occurring inψ (N ) have absolute value at most N , andψ (N ) is a superhedging strategy for C.
Proof. (i) Assumption (F) implies that the classical superhedging price sup Π(C) = sup{E Q [C/(1 + r) T ] : Q ∈ Q} is finite. It is clear that for all N . For the converse estimate, letφ be a classical superhedging strategy for C with price sup Π(C) (see Theorem 3.2). Define We choose an arbitrary map f : N → R satisfying lim N →∞ f (N ) = ∞ and put η (N ),0 1 Then we define η (N ),0 t for t = 2, . . . , T recursively to obtain a self-financing integer strategyη (N ) for each N . By the definition of η (N ) , and sinceφ is a superhedging strategy, we havê for large N . This shows thatη (N ) is an integer superhedging strategy of N C for large N , and hence It is easy to see that a quantity that is O(f (N )) for any f tending to infinity is O(1). Since f was arbitrary, the statement follows.
We define which yields After fixing the initial bank account position ψ (N ),0 1 a strategyψ (N ) ∈ Q is defined for each N . By definition, It remains to show thatψ (N ) is a superhedge for C for large N . This follows from For those finitely many N where the last inequality does not hold, we can simply add a sufficient amount of initial capital to obtain a superhedge; this does not change the convergence rate.
From the proof of (ii), it is clear that log N can be replaced by an arbitrary function tending to infinity.

Variance optimal hedging in one period
We consider a one-period model satisfying (F) and NA. Moreover, we suppose that d ≤ n. Our goal is to approximately hedge a given (non-replicable) claim C. For tractability, the error is measured by the norm of L 2 (P * ), where P * is a fixed EMM; we denote this norm by · throughout this section. In the classical case, this leads to the optimization problem (6.1) inf The problem (6.1) is then solved by projectingC orthogonally to the space {ϕ∆S 1 : ϕ ∈ R d }, which is closed by Theorem 6.4.2 in [9]. For more details on varianceoptimal hedging (in particular, on the multi-period problem), we refer to Chapter 10 of [12] and the references given there. Now we proceed to our setup, and restrict ϕ to Z d . The minimization w.r.t. V 0 is done as in (6.1), and we thus have to compute  w.r.t. the Euclidean norm. This is an instance of the closest vector problem (CVP), a well-known computational problem with applications in cryptography, communications theory and other fields. The survey paper [14] offers an accessible introduction to this subject with many references. By the Pythagorean theorem, the closest lattice point is the lattice point closest to the projection of (6.4) to the subspace generated by the lattice. A cheap method to compute a (hopefully) close lattice point consists of rounding the coefficients of this projected point to the closest integers. It is well-known, though, that the resulting point may be far from optimal. This happens in the following example. We consider a toy example with d = 2, |Ω| = 4, and r = 0, specified in Table 1. The numbers are not calibrated to any market data, but are chosen to illustrate the point that a naive approach at integer approximate hedging (as mentioned above) can lead to significant errors. A detailed investigation of integer variance-optimal hedging over several periods in realistic models is left to future work.  Table 1. Model parameters, risk neutral measure, and a claim.
We wish to approximately hedge N copies of the claim, i.e., the claim N C, for N ∈ N. First, we computed the classical variance-optimal hedge ϕ (N ) = N ϕ (1) ∈ R 2 by projection (see (6.1)). The relative L 2 (P * )-error which of course does not depend on N , is displayed in the second line of Table 2. The third line of Table 2 contains the maximal position size max i=1,2 |ϕ (N ),i | = N max i=1,2 |ϕ (1),i | in the underlying assets. Then, we solved the integer varianceoptimal hedging problem (6.2) exactly, using the algorithm CLOSESTPOINT described in [1], which is based on the Schnorr-Euchner algorithm [20]. CVP is known as a computationally hard problem, with the fastest algorithms having exponential complexity in the dimension. Since our dimension is only |Ω| = 4, this was not an issue in this toy example. In more sophisticated examples, a preprocessing using the LLL-algorithm [18] might faciliate the task of computing a closest vector. Table 2 shows the relative L 2 (P * )-error CVP ∆S 1 NC and the maximum position size. Finally, we used a poor man's approach at solving (6.2) approximately, by simpling rounding the positions of the classical hedge ϕ (N ) to the closest integers. From Table 2, we see that this works fine for large N , but gives significantly worse results than solving CVP for small N . Note that, in this example, the position sizes of the integer hedge are much smaller than that of the classical hedge. Finally, we mention that computing the so-called covering radius [14] of the lattice (6.3) yields an upper bound for the hedging error for any claim.  Table 2. Errors and position sizes for variance optimal hedging.
Appendix A. Tools from number theory In this appendix we collect the classical number theoretic theorems we have used in this paper. The theorems of Dirichlet and Kronecker are fundamental results in Diophantine approximation (i.e., the approximation of real numbers by rational numbers).
We also mention here the following classical theorem [13]: Theorem A.3 (Minkowski). Let K ⊂ R d be closed, convex, zero-symmetric, and bounded. If the volume of K satisfies vol(K) ≥ 2 d , then K contains a non-zero point with integral coordinates.
We did not apply Theorem A.3 in the rest of the paper, but hint at a possible application. Consider the following one-period portfolio optimization problem with maximum loss constraint, where c > 0 and U is some utility function: