Dynamic trading under integer constraints
Stefan Gerhold Paul Krühner









[2010]91G10,91G20,11K60



TU Wien


sgerhold@fam.tuwien.ac.at, paulkrue@fam.tuwien.ac.at



  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.


Dynamic trading under integer constraints
    Stefan Gerhold, Paul Krühner
    December 30, 2023
=========================================











§ INTRODUCTION



Classical, frictionless no-arbitrage theory <cit.> 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 <ref>).
 
To the best of our knowledge, the existing literature on arbitrage, pricing and hedging under trading
constraints <cit.> 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 <cit.>,
which employ mixed-integer nonlinear programming to solve the Markowitz portfolio selection problem.
In the literature, other keywords 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. <cit.>,
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 <ref>). 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 <ref>, we introduce the notions of no integer arbitrage (NIA)
and no integer free lunch (NIFL) in a straightforward way.
It turns out (Theorem <ref>) 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 <ref>)
 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 <ref>,
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 <ref>), 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 <ref>.
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 <ref>, Example <ref>, and
Theorem <ref>)
we invoke non-trivial results from number theory, collected in Appendix <ref>.

Readers who are mainly interested in the practical consequences of integer restrictions are invited
to read (besides the basic definitions) Theorem <ref>,
Theorem <ref>, and Section <ref>.
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 <ref>). 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 <ref> 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_t^1,…,S_t^d) 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_t^0 := (1+r)^t for t∈𝕋, and we denote the market price processes by S̅ := (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.


    
  
  (i)
     An integer (trading) strategy is a predictable process (ϕ̅_t)_t∈𝕋∖{0} 
     with values in ℝ×ℤ^d and ϕ̅_t S̅_t=ϕ̅_t+1S̅_t for t∈𝕋∖{0,T}. For convenience, we will sometimes use the notation ϕ̅_0:=ϕ̅_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 strategy ϕ̅∈ℛ we denote its value at time t∈𝕋 by

    V_t(ϕ̅) := ϕ̅_t S̅_t=  ∑_j=0^d ϕ_t^jS_t^j,

and its discounted value by V̂_t(ϕ̅):=V_t(ϕ̅)/S_t^0.
Often it is convenient to work with discounted asset values or discounted gains which are denoted by
 
    Ŝ_t    := (S_t^1,…,S_t^d)/S_t^0, 
    ΔŜ_t    := Ŝ_t-Ŝ_t-1

for t∈𝕋 resp. t∈𝕋∖{0}.
The discounted value process then equals

    V̂_t(ϕ̅) = V_0(ϕ̅) +∑_k=1^t ϕ_k ΔŜ_k,   t∈𝕋.



    
  
  (i)
    An integer arbitrage is a strategy ϕ̅∈𝒵 which is an arbitrage for the market S̅.
  
  (ii)
    A model satisfies the no-integer-arbitrage condition (NIA), if it admits no integer arbitrage.
    
  (iii) Define the set (a ℤ-module)
    
    𝒦_ℤ := {∑_k=1^T ϕ_k ΔŜ_k
          : ϕ̅∈𝒵}

    of discounted net gains realizable by integer strategies,
    and 
    𝒞_ℤ:= (𝒦_ℤ-L_+^0)∩ L^∞.

    Assuming (F), we define the condition NIFL (no integer free lunch) as
    
    cl(𝒞_ℤ) ∩ L_+^0  ={0}.

    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 <ref> below.
The following simple properties will be used often:


  
   
  
  (i)
    If (F) holds, then in the definition of integer arbitrage the condition ϕ̅∈𝒵 can be replaced by ϕ̅∈𝒬.
  
  (ii)
    In the definition of integer arbitrage  the condition
    V_0(ϕ̅) ≤ 0 can be replaced by
    V_0(ϕ̅) = 0.
  
  (iii) NIA is equivalent to
  
    V̂_T(ϕ̅) - V̂_0(ϕ̅) ≥ 0 ⇒ V̂_T(ϕ̅) = V̂_0(ϕ̅)

   for any ϕ̅∈𝒵 (or, under (F), for any ϕ̅∈𝒬).





  



  (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 arbitrage ϕ̅ such that (ϕ_t^1,…,ϕ_t^d)∈ℚ^d for any t∈𝕋. Define 
   
    N := inf{n ∈ℕ: nϕ∈ℤ^d· T}.

  Then Nϕ_t ∈ℤ^d for any t∈𝕋, and Nϕ̅ is an arbitrage.


By (<ref>), the implication (<ref>) can be written as

    ∑_k=1^T ϕ_k ΔŜ_k ≥ 0  ⇒ ∑_k=1^T ϕ_k ΔŜ_k = 0.

Although our main results assume (F), we mention that (F)
is actually not necessary in parts (i) and (iii) of Lemma <ref>.
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 <cit.>; 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 <ref>).

  
  
  (i)
  If S is bounded, then for any strategy ϕ̅∈ℛ and any ϵ>0,
  we can find a  strategy ψ̅∈𝒬 such that
 
    sup_t∈𝕋 ess  sup |V_t(ϕ̅)-V_t(ψ̅)| < ϵ

and V_0(ϕ̅)=V_0(ψ̅).
   
  (ii)
  Assume (F) and let ϕ̅∈ℛ and ϵ>0. Then there is q∈ℕ and a strategy ψ̅∈𝒵 such that V_0(ϕ̅)=V_0(ψ̅)/q and
    
    sup_t∈𝕋,j=1,… d,
    l=1,…,n |ψ_t^j(ω_l) - qϕ_t^j(ω_l)| < q^-1/(nd(T+1)) <ϵ.

  In particular, for any strategy ϕ̅∈ℛ we can find strategies ψ̅∈𝒬, η̅∈𝒵 and q∈ℕ such that
  
    sup_t∈𝕋 ess  sup |V_t(ϕ̅)-V_t(ψ̅)|    < ϵ, 
    sup_t∈𝕋 ess  sup |qV_t(ϕ̅)-V_t(η̅)|    < ϵ

   and V_0(ϕ̅) = V_0(ψ̅) = V_0(η̅)/q.
   


  The first part is trivial as any real number can be approximated by rational numbers. Thus we find a sequence of strategies (ψ̅^(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_t^j(ω_l): j=1,…, d,l=1,…,n}. For any t∈𝕋 let a_t^1,…,a_t^K_t be an enumeration of the elements of R_t. We have K_t≤ dn for any t∈𝕋 and thus ∑_t∈𝕋K_t ≤ nd(1+T). By Dirichlet's approximation
  theorem (Theorem <ref>), we find q∈ℕ with q^-1/(nd(1+T)) < ϵ and p_t^k ∈ℤ with |p_t^k-qa_t^k| < q^-1/(nd(1+T)) for any t∈𝕋, k=1,…,K_t. For t∈𝕋 we define
   
    ψ_t^j(ω_l) := p_t^k

 where k∈{1,…,K_t} is such that ϕ_t^j(ω_l) = a_t^k. Then 
    {ψ_t^j = p_t^k}⊆⋃_m ∈ A_k{ϕ_t^j = a_t^m}

  where A_k = {m=1,…, K_t: p_t^m = p_t^k}. Thus ψ_t is measurable w.r.t. to the
  σ-algebra 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.
 
 Assume (F). Then the following statements are equivalent:
  
     
  (i) There is an equivalent martingale measure Q≈ P,
     
  (ii) The model satisfies  the classical no-arbitrage property NA and
     
  (iii) The model satisfies NIFL.
  
 Moreover, if the number of risky assets is d=1, then the following statement is equivalent as well:
  
    
  (iv) The model satisfies NIA.
  


  The equivalence of (i) and (ii) is the classical FTAP, see <cit.>. Furthermore,  NA is equivalent to the classical no free lunch
  condition in our setup (see <cit.>), which yields the implication (ii)⇒(iii).
  
Now we assume (iii) and show (ii). Let ϕ̅∈ℛ such that V_0(ϕ̅) = 0 and V_T(ϕ̅) ≥ 0.

By part (ii)
of Lemma <ref> we find q_N∈ℕ and strategies
ψ^(N)∈𝒵 such that

    ess  sup  |q_N V_T(ϕ̅)-V_T(ψ̅^(N))| ≤1/N.

W.l.o.g., the sequence q_N increases.
We get
 
    V_T(ψ̅^(N)) ≥ q_NV_T(ϕ̅) - 1/N≥ -1/N.

 Define
 
    Z_N    :=
       
         1     V_T(ψ̅^(N))  > 1, 
    
         V_T(ψ̅^(N))     V_T(ψ̅^(N)) ≤ 1
       
       = V_T(ψ̅^(N)) -(V_T(ψ̅^(N)) -1)  1_{V_T(ψ̅^(N)) >1}∈𝒞_ℤ,    N∈ℕ.

 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 (<ref>), we have Z≥0. Then, NIFL implies that
 Z=0, and thus V_T(ψ̅^(N))  →0. Since q_N^-1V_T(ψ̅^(N)) → V_T(ϕ̅) by (<ref>) (recall that q_N increases), we conclude
 that V_T(ϕ̅) =0.

Now assume that d=1. (ii)⇒(iv) is obvious. We assume that (ii) does not hold.
Proposition 5.11 in <cit.> yields the existence of a one-period arbitrage, i.e. an arbitrage ϕ̅ and t_0∈𝕋 such that ϕ_t=0 for any t∈𝕋∖{t_0}. Since ϕ̅ is an arbitrage we must have ϕ^1_t_0≠ 0. Define
 
    ψ^j_t := ϕ^t_j / |ϕ^1_t_0|,   t∈𝕋,j=0,1.

Then ψ is an arbitrage as well. Moreover, ψ_t^1∈{-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. 


  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.


   NA always implies NIA. Now suppose that we have a real arbitrage opportunity.
   By part (iii) of Lemma <ref>, there is a predictable process ϕ such that
   
    ∀ ω∈Ω: ∑_k=1^T ϕ_k(ω) ΔŜ_k(ω)   ≥ 0, 
    ∃ ω∈Ω: ∑_k=1^T ϕ_k(ω) ΔŜ_k(ω)   > 0.

   The assertion now follows from Lemma <ref> 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_k^j(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 <cit.>, it is certainly
possible to state much more general results along these lines.[We thank Manuel Kauers
for pointing this out.]
Therefore, we do not claim originality for Lemma <ref>, but give a short
self-contained proof for the reader's convenience.


  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
  
    ∑_j=1^s a_ij x_j ≥ 0,    i=1,…,r,  with at least one inequality being strict.

  Then there is a rational vector satisfying (<ref>).


  After possibly reordering the lines of the matrix (a_ij), we may assume that there
  is u∈{1,…,r} such that
  
    ∑_j=1^s a_ij x_j > 0,    1≤ i≤ u, 
    ∑_j=1^s a_ij x_j = 0,    u<i≤ r.

  By defining y_i:=∑_j=1^s a_ij x_j for 1≤ i≤ u, we get that
  the vector (x_1,…,x_s,y_1,…,y_u) solves the system
  
    ∑_j=1^s a_ij x_j - y_i    = 0,    1≤ i≤ u, 
    ∑_j=1^s a_ij x_j    = 0,    u<i≤ r, 
    
        y_1,…,y_u   >0.

  Equations (<ref>) and (<ref>) 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
  (x̃_1,…,x̃_s,ỹ_1,…,ỹ_u)∈ℚ^s+u
  satisfying (<ref>)–(<ref>). Then
  (x̃_1,…,x̃_s) is the desired rational vector.


The assertion of Theorem <ref> does not hold for infinite probability spaces,
as the following example illustrates.


  Let Ω={ω_1,ω_2,…} be countable, 𝒜=2^Ω, and fix an
  arbitrary probability measure P with P[{ω_i}]>0 for all i∈ℕ.
  We choose d=2, T=1, and r=0. The asset prices are defined by S_0=(1,1)
  and
  
    S_1(ω_i)=
        
           (1+p_i,1+q_i)     i even, 
    
           (1-p̂_i,1-q̂_i)     i odd,

  where p_i,q_i,p̂_i,q̂_i are natural numbers satisfying
  
    p_i/q_i↘π  and  p̂_i/q̂_i↗π,  
        i→∞.

  Thus, the increments are
  
    Δ S_1(ω_i)=
        
           (p_i,q_i)     i even, 
    
           (-p̂_i,-q̂_i)     i odd.

  A vector (ϕ_1^1,ϕ_1^2)∈ℝ^2 yields an arbitrage
  if and only if
  
    p_i/q_iϕ_1^1+ϕ_1^2    ≥ 0,   i even, 
    p̂_i/q̂_iϕ_1^1+ϕ_1^2    ≤ 0,   i odd,

  with at least one inequality being strict.
  By (<ref>), the vector (ϕ_1^1,ϕ_1^2)=(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 (<ref>)-(<ref>),
  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.


  
  
  (i)
  Let Q be a probability measure on (Ω,𝒜), and denote the set of trading strategies with zero initial value by ℛ_0:={ϕ̅∈ℛ:V_0(ϕ̅)=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
    
    𝒵^0_Q    := {ϕ̅∈ℛ_0 ∩𝒵 : V_T(ϕ̅)=0  Q-a.s.}, 
    𝒬^0_Q    := {ϕ̅∈ℛ_0 ∩𝒬: V_T(ϕ̅)=0  Q-a.s.}, 
    ℛ^0_Q    := {ϕ̅∈ℛ_0: V_T(ϕ̅)=0  Q-a.s.}.


  
  (ii) If we assume[This ensures that the sets {ω} occurring
  in part (ii) of Definition <ref> are measurable.] (F) then we write ^max for the set of martingale measures Q≪ P such that
        
    ∀ϕ̅∈𝒬^0_Q:(  V_T(ϕ̅)≥ 0 ⇒  V_T(ϕ̅) = 0)  and
       ∃ϕ̅∈ℛ^0_Q: V_T(ϕ̅)≥ 0 and {V_T(ϕ̅) > 0} = {ω∈Ω: Q[{ω}] = 0 }.

  
  (iii)  denotes the set of martingale measures Q≪ P such that
  
    ∀ϕ̅∈𝒵^0_Q:(  V_T(ϕ̅)≥ 0 ⇒  V_T(ϕ̅) = 0).

  


Obviously, we have 𝔔⊆^max⊆, where 𝔔
denotes the set of equivalent martingale measures. (As for the first inclusion, ϕ̅=0
satisfies the existence statement in (ii).) Before giving an FTAP for integer trading we show further properties of the measures in ^max.

 Assume (F) and that ^max≠∅. Then there is a set A⊊Ω such that ^max is the set of martingale measures whose support is Ω∖ A. Also, there is ϕ̅∈ℛ_0 with V_T(ϕ̅)≥ 0 and {V_T(ϕ̅)>0} =A.
 
 Now, let Q∈^max and let Q'∈. Then Q'≪ Q. Moreover, ^max is dense in  with respect to the total variation distance. 


  Choose Q∈^max and let ϕ̅∈ℛ_Q^0 satisfy the existence statement in (ii) of Definition <ref>. Define A:={V_T(ϕ̅)>0}. Then ϕ̅ is the required trading strategy.

  Let Q' be a martingale measure with support equal to Ω∖ A. Then ϕ̅ satisfies the existence statement of (ii) in Definition <ref>. Let ψ̅∈𝒬_Q'^0 with V_T(ψ̅)≥ 0. Then V_T(ψ̅) = 0 Q'-a.s., i.e. V_T(ψ̅) = 0 on Ω∖ A. Consequently, ψ̅∈𝒬_Q^0. (ii) of Definition <ref> yields that V_T(ψ̅) = 0. Thus, Q'∈^max.
  
 We need to show that any measure in ^max is a martingale measure with support Ω∖ A. This, however, follows as soon as we have shown that Q'≪ Q for any Q'∈. Let Q'∈. Observe that V_T(ϕ̅) = 0 Q'-a.s. because Q' is a martingale measure. Thus A = {V_T(ϕ̅) > 0} is a Q'-null set. We find Q'≪ Q.
 
 Finally, we have to show that Q' can be approximated by elements in ^max in total variation. Define Q_α := α Q'+(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 ^max by what we have shown so far.




We can now state an FTAP
for integer trading.


   Assume (F). Then the following statements are equivalent:
    
      
  (i) ^max≠∅
      
  (ii) ≠∅
      
  (iii) The market satisfies NIA.
    
  The implication (ii)⇒(iii) does not need assumption (F).


  (i)⇒(ii) is trivial.
  
  (ii)⇒(iii): We fix a measure Q∈. Let ϕ̅∈ℛ_0∩𝒵 with V_T(ϕ̅)≥ 0. Since Q is a martingale measure we have V̂_T(ϕ̅) = 0 Q-a.s. and, hence, V_T(ϕ̅) = 0 Q-a.s. Thus ϕ̅∈𝒵^0_Q. By part (iii) of
  Definition <ref> we have V_T(ϕ̅) = 0. Hence, we have NIA.
  
  (iii)⇒(i): Let
  
    A:={ω∈Ω: ∃ϕ̅∈ℛ_0: V_T(ϕ̅) ≥ 0∧ V_T(ϕ̅)(ω) > 0}.

    For every ω∈ A choose an according strategy ϕ̅^(ω)∈ℛ_0 with V_T(ϕ̅^(ω)) ≥ 0 and V_T(ϕ̅^(ω))(ω) > 0. Define
    
    ϕ̅:= ∑_ω∈ Aϕ̅^(ω).

    Then ϕ̅∈ℛ_0, V_T(ϕ̅)≥ 0 and {V_T(ϕ̅) > 0} = A.
  
  We claim that for any ψ̅∈ℛ_0 with V_T(ψ̅)1_Ω∖ A≥ 0 we have V_T(ψ̅)1_Ω∖ A = 0. Let ψ̅∈ℛ_0 with V_T(ψ̅)1_Ω∖ A≥ 0. If V_T(ψ̅) ≥ 0 on A, then V_T(ψ̅) ≥ 0 and, hence, V_T(ψ̅)1_Ω∖ A = 0 by construction of A. Thus, we may assume that V_T(ψ̅)(ω) < 0 for some ω∈ A. Then
  
    c:= -min{V_T(ψ̅)(ω):ω∈ A}/min{V_T(ϕ̅)(ω):ω∈ A}> 0.

  The strategy ψ̅+cϕ̅ is in ℛ_0 and satisfies V_T(ψ̅+cϕ̅)≥ 0. Thus, V_T(ψ̅+cϕ̅) = 0 outside A. Hence, V_T(ψ̅) = 0 outside A, i.e. V_T(ψ̅)1_Ω∖ A = 0.
  
   Assume for contradiction that A=Ω. Then V_T(ϕ̅)>0. Define 
   
    e := min{ V_T(ϕ̅)(ω): ω∈Ω} > 0.

   Lemma <ref> yields q∈ℕ and ψ̅∈𝒵 such that |V_T(ψ̅) - V_T(qϕ̅)| < e. Thus, V_T(ψ̅) > qV_T(ϕ̅)-e ≥ 0. Thus ψ̅ is an integer arbitrage. A contradiction.
   
   Consequently, A⊊Ω. We have shown that the market S̅ is free of arbitrage on Ω∖ A. The classical fundamental theorem yields a martingale measure Q on Ω∖ A for S̅. We denote its extension to a probability measure on Ω by Q, i.e. Q[M] = Q[M∖ A] for any M⊆Ω. Then Q≪ P and Q is a martingale measure with {ω∈Ω: Q[{ω}] = 0} = A. Since {V_T(ϕ̅)>0} = A we have the existence statement in
   part (ii) of Definition <ref>.
   Now let ψ̅∈𝒬_Q^0 with V̂_T(ψ̅)≥ 0. Let q be a common denominator for {ψ_t^j(ω_l): t∈𝕋,  j=1,…,d,  l=1,…,n}. Then qψ̅∈𝒵_Q^0. Since we have NIA we get V_T(ψ̅) = 1/qV_T(qψ̅) = 0 as claimed.




An immediate consequence is the following sufficient criterion for the construction of markets with no integer arbitrage.

  Let Q≪ P be a martingale measure and assume that 𝒵_0^Q = {0}. Then the market satisfies NIA.


The following example is a simple application of the preceding corollary.

  Assume that d=2, n≥ 2, T=1, r=0 and choose  (S^1_0,S^2_0)=(1,π) and
  
    (S_1^1,S_1^2)(ω_j):=
        
          (3/2,3π/2)     j=1, 
    
          (1/2,π/2)     j=2.

  Define Q[{ω_j}] = 1_{j=1,2}/2 for j=1,…,n. Then Q≪ P is a martingale measure, and 
  ℛ^0_Q ={(0,-πϕ^2,ϕ^2):ϕ^2∈ℝ}. Consequently, we have 𝒵^0_Q={0}. Thus, Corollary <ref> yields that the market does not allow for integer arbitrage.  
  Observe that this holds regardless of the specification of (S_1^1,S_1^2)(ω_j) for j≥ 3.



Another immediate consequence is the existence of absolutely continuous martingale measures.

  Suppose that a model satisfies NIA and assume (F). Then there is an absolutely continuous martingale measure.



  
  
 Immediate from Theorem <ref> (iii)⇒(i).




The following example shows that the existence of an absolutely continuous martingale measure alone is insufficient to exclude integer arbitrage.

  Let Ω = {ω_1,ω_2}, S^0_0 = 1 = S^0_1, S^1_0= 1 and S_1^1(ω_i) = i for i=1,2 (i.e. T=1, d=1, n=2). Then Q := δ_ω_1 is a martingale measure which is absolutely continuous with respect to P := (δ_ω_1 + δ_ω_2)/2, where δ_ω_j denotes the Dirac-measure on ω_j. The strategy ϕ̅_1 := (-1,1) is an integer arbitrage.



Finally, we provide a technical statement that will be used in Section <ref>.

  Assume (F), let Q∈ and assume that for any B∈ℱ_1 we have Q[B]∈{0,1}. Then ℱ_1=ℱ_0.


  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 let ϕ̅∈𝒬, t=1,… T with ϕ̅_1=…ϕ̅_t-1=0 and V̂_t(ϕ̅)=… =V̂_T(ϕ̅)≥ 0 on A.
  (Since existence of an arbitrage implies existence of a one period arbitrage,
  it suffices to consider this kind of strategy.)
  
  Case 1: t=1. Since Ŝ_0 = E_Q[Ŝ_1] = Ŝ_1(B), we find that V̂_1(ϕ̅)(B) = V_0(ϕ̅) = 0 and, hence, V_1(ϕ̅)≥ 0 everywhere. Since (S^0,…,S^d) satisfies NIA, we obtain that V_1(ϕ̅) = 0 and, hence, V_s(ϕ̅)=0 for any s∈𝕋.
  
  Case 2: t≥ 2. Define ψ̅:= 1_Aϕ̅. Since A∈ℱ_1 and ϕ̅_1=0 we find that ψ̅∈𝒬 with ψ̅_0=…ψ̅_t-1=0 and V̂_t(ψ̅)=… =V̂_T(ψ̅)≥ 0. Since the model (S^0,…,S^d) on Ω satisfies NIA by assumption we find that 0 = V_t(ψ̅) = 1_A V_t(ϕ̅) and, hence, V_t(ϕ̅) = 0 on A. 
  
  Thus (S^0,…,S^d) restricted to A satisfies NIA. By Theorem <ref> there is Q'∈ 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. Define Q̃:=Q/2+Q'/2 and observe that Q̃∈. However, Q ≉Q̃ because Q' has disjoint support with Q.
  But Proposition <ref> implies Q≈Q̃, which yields
  a contradiction. Thus A=∅ and, hence, ℱ_1=ℱ_0.








§ CLAIMS AND INTEGER TRADING


  Fix a model that satisfies NIA.
  
  
  (i)A claim is a random variable C≥0. A real number p≥0 is an
  integer arbitrage free price of C, if there is an adapted non-negative stochastic process (X_t)_t∈𝕋 with X_0=p, X_T=C such that the market 
    
    (S^0,…,S^d,X)

  satisfies  NIA. The set of integer arbitrage free prices is denoted
  by Π_ℤ(C).
  
  (ii)
  An integer superhedge for C is a trading strategy ϕ̅∈𝒵 such that V_T(ϕ̅) ≥ C,
  and it is an integer replication strategy if it satisfies V_T(ϕ̅) = C.
  We write
  
    σ_ℤ(C) = inf{V_0(ϕ̅): ϕ̅∈𝒵, V_T(ϕ̅)≥ C}

  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 <cit.>):

  Assume that NA holds, and let C be a claim with supΠ(C)<∞.
  Then there is a strategy ϕ̅∈ℛ with V_0(ϕ̅)=supΠ(C)
  and V_T(ϕ̅)≥ C. Moreover, supΠ(C) is the smallest number with this property.


We find analogue statements to the preceding theorem under the weaker assumption NIA. Proposition <ref> 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 <ref> 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:



  
  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.
  


  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 strategy
  ϕ̅=(ϕ^0,0) with ϕ^0_t=c/(1+r)^T-t, t∈𝕋.
  
  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 <cit.>.) 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 strategy ϕ̅ is uniquely defined by specifying its initial wealth V_0(ϕ̅) and the predictable ℤ^d-valued process ϕ=(ϕ_t)_t=1,…,T. Thus there is a bijective map
   
    Γ: ℝ×𝒵_c→𝒵

 where
  𝒵_c := { (ϕ^1,…,ϕ^d): ϕ̅∈𝒵} is countable with V_0(Γ(v,ϕ)) = v for any v∈ℝ. In particular, v↦ V_T(Γ(v,ϕ)) is affine.  
  We have
   
    { (a,b)∈[0,∞)^2 : a1_A+b1_B can be integer replicated}
               ⊆{(δ_A(V_T(ϕ̅)),δ_B(V_T(ϕ̅))): ϕ̅∈𝒵}
                =⋃_ϕ∈𝒵_c{(δ_A(V_T(Γ(v,ϕ))),δ_B(V_T(Γ(v,ϕ)))): v∈ℝ}.

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 (<ref>) has Lebesgue measure zero, and hence (<ref>) 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

    Π(C) = { E_Q[C/(1+r)^T]: Q∈𝔔},

where 𝔔 is the set of equivalent martingale measures. The corresponding
result for NIA looks as follows:


  Assume (F) and that the model satisfies NIA. Let C be a claim. Then
  
    Π_ℤ(C) ⊆{ E_Q[C/(1+r)^T] : Q ∈}.



  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. Let  be the set of absolutely continuous
martingale measures that satisfy part (iii) of Definition <ref> for this market.
By Theorem <ref> there is Q∈⊆. Then p = E_Q[C/(1+r)^T].



The following example shows that the inclusion in Proposition <ref> can be strict. In fact, in this example we have Π_ℤ(C)=∅.

  Let Ω = {ω_1,ω_2,ω_3}, r=0, d=2, T=1 and P[{ω_j}]=1/3 for any j=1,2,3. Then the riskless asset is constant 1, i.e. S_t^0 = 1 for t∈{0,1}. We choose (S_0^1,S_0^2) = (π,1) and 
   
    (S_1^1,S_1^2)(ω_j) = (2π,2)   j=1,
    (π/2,1/2)   j=2,
    (π,2)   j=3.

  A short calculation reveals that Q[{ω_j}] := 1_{j≠ 3}j/3 is the only martingale measure. Obviously, we have Q≪ P, and the only integer strategy with zero initial wealth and Q-a.s. zero final wealth is identically zero. Thus Q satisfies (ii) in Theorem <ref> and, hence, we have NIA. Since Q is the only martingale measure we have ={Q}=. 
  
  Now, we consider the claim C:=1_{ω_3}.
   
 Proposition <ref> yields that 
  
    Π_ℤ(C) ⊆{ E_Q[C] } = {0}.

  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
 
    Π_ℤ(C) = ∅⊊{0} = { E_R[C/(1+r)^T] : R ∈}.




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.



  Suppose (F), that the model satisfies NA and let C be a claim. Then
   
    Π(C) ⊆Π_ℤ(C) ⊆cl(Π(C)).

   Moreover, if sup(Π_ℤ(C)) ∈Π_ℤ(C), then either C has a
   duplication strategy in 𝒬
   or there is no cheapest classical superhedging strategy that is in 𝒬.


  The first inclusion is trivial. Proposition <ref> in combination with NA yields that ^max is the set of martingale measures which are equivalent to P and that this set is dense in . Thus, Proposition <ref> implies
  
    Π_ℤ(C)    ⊆{ E_Q[C/(1+r)^T]:Q∈}
       ⊆cl{ E_Q[C/(1+r)^T]:Q∈^max}
       = cl(Π(C)).

  To show the second assertion, suppose that s:=sup(Π_ℤ(C)) ∈Π_ℤ(C),
  and that there is a cheapest classical superhedging strategy ϕ̅∈𝒬. This means that ϕ̅ has price V_0(ϕ̅)=s
  and payoff V_T(ϕ̅) ≥ C. Since s ∈Π_ℤ(C),
  there is an integer-arbitrage free extension of the model where C
  trades at price s.  Consider the strategy (ϕ̅,-1) in the extended model.
  Its cost is zero, and its payoff is V_T(ϕ̅)-C≥0.
  By part (i) of Lemma <ref>, we conclude C=V_T(ϕ̅),
  and so ϕ̅∈𝒬 is a duplication strategy for C.


Alternatively, the inclusion Π_ℤ(C) ⊆cl(Π(C)) can be proved
using Lemma <ref> (i), Lemma <ref> (i), and Theorem <ref>.

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.

  Let Ω={ω_1,ω_2,ω_3}, r=0, T=1 and assume that the number of risky assets is d=1. Let S_0^1 = 2 and
   
    S_1^1(ω_j) = 1     j=1, 
     3     j=2,
     3   j=3.

   The equivalent martingale measures are given by
   
    Q_α := 12 δ_ω_1 + αδ_ω_2
          + (12 -α)δ_ω_3,   α∈(0,12),

   and so the model satisfies NA.

  
  (i) Define the claim
  
    C(ω_j) = 2√(2)    j=1, 
     0     j=2,
     4√(2)   j=3.

  Using (<ref>), we find the classical arbitrage free prices
  
    Π(C)={ E_Q_α[C]: α∈(0,12)}=(√(2),3√(2)).

  We now check the boundary points for integer arbitrage, using part (iii) of Lemma <ref>.
  An integer arbitrage in the market extended by C with price p thus amounts to
  ϕ∈ℤ^2 such that ϕ(Δ S_1^1,C-p)≥ 0
  and ϕ(Δ S_1^1,C-p)≠ 0. For p=√(2), we get the inequalities
  
    (
        
          -1    √(2)
    
          1     -√(2)
    
          1     3√(2))
        (
        ϕ^1 
    ϕ^2
        )
        ≥0.

  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: Π_ℤ(C) = [√(2),3√(2)]. 
  
  We now verify that there is no cheapest classical superhedge in 𝒬,
  in accordance with the second assertion of Theorem <ref>.
  (Note that C is not replicable, as |Π(C)|>1; in particular, there is no replication strategy
  in 𝒬.)
  Clearly, if ϕ̅∈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
  
    V_0(ϕ̅)    = max_ω∈Ω(C(ω)-ϕ^1 S_1^1(ω)) + ϕ^1 S_0^1
       = max_ω∈Ω(C(ω)-ϕ^1 Δ S_1^1(ω)).

  Our optimal strategy is ϕ̅=(3√(2),√(2))∉𝒬, because
  the problem
  
    inf_ϕ^1 ∈ℝmax_ω∈Ω(C(ω)-ϕ^1 Δ S_1^1(ω))
        =inf_ϕ^1 ∈ℝmax{2√(2)+ϕ^1,-ϕ^1,4√(2)-ϕ^1 }

  has the unique solution ϕ^1=√(2). 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 <ref>.
  
  
  (ii) If 
    C(ω_j) = 2√(2)    j=1, 
     0     j=2,
     2√(2)   j=3,

  then Π_ℤ(C) = [√(2),2√(2)). The cheapest classical superhedge
  is in 𝒬, whereas the most expensive classical subhedge is not in 𝒬.
  
  (iii) If 
    C(ω_j) = 0     j=1, 
     0     j=2,
     2√(2)   j=3,

  then Π_ℤ(C) = (0,√(2)].
   The cheapest classical superhedge
  is not in 𝒬, whereas the most expensive classical subhedge is in 𝒬.
  
  (iv) If 
    C(ω_j) = 0     j=1, 
     0     j=2,
     2   j=3,

  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 <ref>) 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 (<ref>), define

    σ̂_ℤ(C) = sup{V_0(ϕ̅): ϕ̅∈𝒵, V_T(ϕ̅)≤ C}.

For p∉ [ σ̂_ℤ(C), σ_ℤ(C)], we clearly have
p∉Π_ℤ^stat(C), because, using appropriate integer sub- resp. superhedges,
one can easily construct a static integer arbitrage for the extended model.
Therefore, we obtain Π_ℤ(C)
⊆Π_ℤ^stat(C)⊆[ σ̂_ℤ(C), σ_ℤ(C)].




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 <ref>.

  Assume (F), that the model satisfies NIA and let C be a claim. Then there is a cheapest superhedge ϕ̅∈ℛ which satisfies

    V_0(ϕ̅)    = sup{E_Q[C/(1+r)^T]:Q∈}
       = sup{E_Q[C/(1+r)^T]:Q∈^max}.

   Moreover, for any ϵ>0 there is a superhedge ξ̅∈𝒬 for C such that V_0(ξ̅) ≤ V_0(ϕ̅) + ϵ.


   Proposition <ref> yields 
   
    sup{E_Q[C/(1+r)^T]:Q∈} = sup{E_Q[C/(1+r)^T]:Q∈^max}.


  Let Q∈^max and define
  
    A:={ω∈Ω: Q[{ω}]=0}.

   Then, S̅ 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 by η̅∈ℛ. It satisfies V_T(η̅) ≥ C Q-a.s. Since Q∈^max there is ψ̅∈ℛ_0 such that V_T(ψ̅)≥ 0 and {V_T(ψ̅)>0} = A. As Ω is finite there is a∈ℝ such that V_T(aψ̅+η̅) = aV_T(ψ̅)+V_T(η̅) ≥ C. By Proposition <ref> and
   Theorem <ref>,
   we find that ϕ̅:= aψ̅+η̅ is a superhedge for C with initial price 
   
   
    V_0(ϕ̅) = V_0(η̅) = sup{E_Q[C/(1+r)^T]:Q∈^max}.

  Now let γ̅∈ℛ be any superhedge for C. Then V_T(γ̅) ≥ C Q-a.s. for any Q∈^max and, hence, 
  
    V_0(γ̅) =  E_Q[V_T(γ̅)/(1+r)^T] ≥sup{E_Q[C/(1+r)^T]:Q∈^max}.

  Consequently, ϕ̅ is a cheapest superhedge for C.
  
  The second assertion follows easily from part (i) of Lemma <ref>.











§ 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 <ref>).
First, we give some sufficient conditions that imply that Π_ℤ(C) equals an interval.


  Assume (F) and that the model satisfies NIA.
  Then Π_ℤ(C) is an interval if any of the following statements holds:
   
     
  (i) there is only one trading period (T=1), and Π_ℤ(C) is not empty,
     
  (ii) there is only one risky asset (d=1), or
     
  (iii) the model satisfies NA.
   





  If (ii) holds, then Theorem <ref> yields that (iii) holds.  
  If we assume (iii), then Theorem <ref> yields the claim.  
  Now assume that (i) holds. Proposition <ref> implies that
  
    Π_ℤ(C) ⊆{E_Q[C/(1+r)^T] : Q∈} :=J.

   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∈(J) and define X_0:=p, X_1:=C. Assume for contradiction that there is an integer arbitrage (ϕ̅,ϕ^d+1) for the model (S^0,…,S^d,X).
   By part (ii) of Lemma <ref>, we may assume that V_0(ϕ̅,ϕ^d+1)=0.   
   We have ϕ_1^d+1≠ 0, because otherwise ϕ̅ is an integer arbitrage for (S^0,…,S^d) with V_0(ϕ̅) = 0. 
   
   Case 1: ϕ_1^d+1<0. Then C ≤ -(1/ϕ_1^d+1)∑_j=0^d ϕ_1^j S_1^j. Thus, 
    
    ψ_1^j := -ϕ^j/ϕ^d+1,   j=0,…,d,

 is a superhedge for C. We have V_0(ψ̅) = p. Proposition <ref> yields that p = V_0(ψ̅) ≥sup(J) > p. A contradiction.
  
   Case 2: ϕ_1^d+1>0; analogous.
 
  Thus, p∈Π_ℤ(C) which yields that (J) ⊆Π_ℤ(C) ⊆ J and, hence, Π_ℤ(C) is an interval.




 We now give an example in which the set of integer arbitrage compatible prices is not an interval. More precisely, we exhibit a model satisfying NIA and a claim C where the set of NIA-consistent prices is given by Π_ℤ(C) = [0,1/2]∖ (ℚ + ℚπ).

  Let Ω := {ω_1,ω_2,ω_3,ω_4}, d=2, T=2 and r=0. We use the filtration ℱ_0 := {∅,Ω}, ℱ_1:=σ({ω_1},{ω_2,ω_3},{ω_4}) and ℱ_2:=2^Ω. We choose the market model given by (S^1_0,S^2_0) := (1,π) and
  
    S_2(ω_j) := S_1(ω_j) :=  (3/2,3π/2)     j=1,
     (1/2,π/2)     j=2,3, 
     (1,1+π)     j=4.

  This market allows for the static real arbitrage η̅_t := (0,-π,1) which is self-financing, satisfies V_0(η̅) = 0 and V_2(η̅) = 1_{ω_4}. Thus, any martingale measure Q must satisfy Q[{ω_4}] = 0. Define the measure Q_α by
   
    Q_α[{ω_j}] :=  1/2   j=1,
    α/2    j=2,
     (1-α)/2    j=3,
     0     j=4.

   for any α∈ [0,1]. Then Q_α is a martingale measure and 𝒵_Q_α^0 = {0}. In particular, Theorem <ref> yields that NIA holds. Moreover, = {Q_α: α∈[0,1]}.
   
   Now we choose the claim C := 1_{ω_2}. Proposition <ref> yields
    
    Π_ℤ(C) ⊆{ E_Q[C] : Q∈} = [0,1/2].




 Let p∈ [0,1/2]∩ (ℚ+ ℚπ), p≠ 0 and assume for contradiction that p∈Π_ℤ(C). Then there is an adapted process (X_0,X_1,X_2) such that X_0=p, X_2=C and the model (S^0,S^1,S^2,X)satisfies NIA. Define α := 2p, and let u,v∈ℚ
 be such that α=u+vπ.
 
 Define the strategy (ϕ̅_t,ϕ^3_t)_t=1,2∈𝒬 by
 
    ϕ^3_1:= sgn((2-π)v-u) ∈{-1,1}

 and
 
    (ϕ̅_1,ϕ_1^3)    := (-32 αϕ_1^3,u ϕ_1^3,v ϕ_1^3,ϕ^3_1), 
    
       (ϕ̅_2,ϕ_2^3)(ω_j)    :=
       
         0     j=1,2,3, 
    (12|(2-π)v-u|,0,0,0)     j=4.

 This strategy satisfies V_0(ϕ̅,ϕ^3) = 0 and
 
    V_1(ϕ̅,ϕ^3)=V_2(ϕ̅,ϕ^3) = 12|(2-π)v-u|· 1_{ω_4}.

   Thus, (ϕ̅,ϕ^3) is a rational arbitrage. A contradiction.
 
Let p=0 and assume for contradiction that p∈Π_ℤ(C). Then there is an adapted process (X_0,X_1,X_2) such that X_0=0, X_2=C and the model (S^0,S^1,S^2,X) satisfies NIA. Define the static strategy (ϕ̅,ϕ^3) := (0,0,0,1). We have V_0(ϕ̅,ϕ^3) = 0 and V_2(ϕ̅,ϕ^3) = 1_{ω_2}. Thus, (ϕ̅,ϕ^3) is an integer arbitrage. A contradiction.
 
 We have shown so far that Π_ℤ(C)⊆ [0,1/2]∖ (ℚ + ℚπ).
 Conversely, let now p∈ [0,1/2]∖ (ℚ + ℚπ). We show that p∈Π_ℤ(C). Define α := 2p and  X^α_0 := α/2, X^α_1 := α 1_{ω_2,ω_3}, X^α_2 := C. To see that the model (S^0,S^1,S^2,X^α) satisfies NIA,
assume for contradiction that there is an integer arbitrage. Then there is a one period arbitrage (ϕ̅,ϕ^3). Obviously, there is no arbitrage possibility in the second period, and so we may assume (ϕ_2,ϕ^3_2)=0 (i.e., no risky position in the second period). Thus, V_1(ϕ̅,ϕ^3) = V_2(ϕ̅,ϕ^3) ≥ 0. Since Q_α is a martingale measure for the extended model, we get V_1(ϕ̅,ϕ^3) = 0 Q_α-a.s.
In particular, V_1(ϕ̅,ϕ^3)(ω_2)=0, and together with V_0(ϕ̅,ϕ^3)=0
(see Lemma <ref> (ii)) this implies

    ϕ^1_1 + πϕ^2_1 - αϕ^3_1=0.

As the original model satisfies NIA, we must have ϕ^3_1≠ 0, which leads to the
contradiction

    2p=α = ϕ^1_1 + πϕ^2_1/ϕ^3_1∈ℚ + ℚπ.

 Thus, there is no integer arbitrage, i.e. the model satisfies NIA and, hence, p∈Π_ℤ(C).





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 set Π_ℤ(C) is either empty or dense in
   
    [inf{ E_Q[C/(1+r)^T]: Q∈},sup{ E_Q[C/(1+r)^T]: Q∈}].



The theorem will follow from Lemmas <ref> and <ref> below.
Note that  can be replaced by , due to the last assertion of
Proposition <ref>.
In order to prove Theorem <ref>, 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

    A_t := {ω∈Ω: ∃ϕ̅∈ℛ_0:V_T(ϕ̅)≥ 0,V_T(ϕ̅)(ω)>0, ϕ̅_1,…,ϕ̅_t=0}

for any t∈𝕋∖{T}. By the same argument as in the proof of Theorem <ref>, there is ϕ̅∈ℛ_0 with ϕ̅_1,…,ϕ̅_t=0, V_T(ϕ̅)≥ 0 and {V_T(ϕ̅)> 0} = A_t. From the definition we see that A_t⊆ A_t-1 and A_t-1∖ A_t∈ℱ_t for any t∈𝕋∖{0}.


We write  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:

    𝒦_T    := {C}, 
    𝒦_t   := {E_Q[D/1+r|ℱ_t]: Q∈,D∈𝒦_t+1}, 
    𝒞_T    := {C}, 
    𝒞_t   := {E_Q[D/1+r|ℱ_t]: Q∈,D∈𝒞_t+1, ∀ B∈ℱ_t ∀ξ∈ℚ^d ∀ s∈{-1,1}:
            1_B ξΔŜ_t+1≥ s1_B(D/(1+r)^t+1-E_Q[D/(1+r)^t+1|ℱ_t]) 
           ⇒ 1_B ξΔŜ_t+1 = s1_B(D/(1+r)^t+1-E_Q[D/(1+r)^t+1|ℱ_t])  }

for any t∈𝕋∖{T}.


Lemma <ref> below together with the convexity of =𝔔_0 implies that 
 
    𝒦_0 = [inf{ E_Q[C/(1+r)^T]: Q∈},sup{ E_Q[C/(1+r)^T]: Q∈}],

and Lemma <ref> below yields a countable exception set F such that 𝒞_0 = 𝒦_0∖ F. Finally, Lemma <ref> states that 𝒞_0 is contained in the set Π_ℤ(C) of NIA-compatible prices, which establishes Theorem <ref>. For technical reasons, we first analyse the sets , and we will need the stochastic convexity of 𝒦_t given in Lemma <ref> below.




  Let t∈𝕋. Then  is non-empty. If t≠ 0, then for any Q∈𝔔_t-1 there is Q'∈ such that
   
    E_Q'[X|ℱ_s] = E_Q[X|ℱ_s]   Q-a.s.

 for any s=t,…,T and any random variable X:Ω→ℝ.


  Let I:={t∈𝕋:  the claim holds for t}. We have 0∈ I by
  Theorem <ref> (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,…,B_k⊆ A_t-1∖ A_t. Since Ω∖ A_t-1 is the support of Q we have Q[B_l] > 0 for any l=k+1,…,m. Since A_t∖ A_t-1 is ℱ_t-measurable we have A_t-1∖ A_t = ⋃_l=1^kB_l. 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 <ref> (i) a martingale measure Q_l≪ P_l on B_l. Define the probability measure
   
    Q'[D] := (Q[D] + ∑_l=1^k Q_l[D∩ B_l] )/(1+k),   D∈𝒜.

 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'∈. 
 
 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 D':=D∖ A_t-1. Then D' is Q'-essentially ℱ_s-measurable, because A_t is a Q'-null set and A_t-1∖ A_t is ℱ_t⊆ℱ_s-measurable. We have
 
    E_Q[E_Q'[X|ℱ_s]1_D]    = E_Q[E_Q'[X|ℱ_s]1_D'] 
       = E_Q[E_Q'[X1_D'|ℱ_s]] 
       = E_Q[E_Q[X1_D'|ℱ_s]] 
       = E_Q[X1_D'] 
       = E_Q[X1_D].

 Thus, t∈ I.




  For any t∈𝕋 we have 
     𝒦_t = { E_Q[C/(1+r)^T-t|ℱ_t]: Q∈}.


  Define
   
    I:={ t∈𝕋: 𝒦_t = { E_Q[C/(1+r)^T-t|ℱ_t]: Q∈}}.

 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 X_t-1 = E_Q[X_t/1+r|ℱ_t-1]. Since X_t∈𝒦_t there is R∈ such that X_t = E_R[C/(1+r)^T-t|ℱ_t]. Define the measure
  
    Q'[B] := E_Q[R[B|ℱ_t]],   B∈𝒜.

Since R∈ 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, Q'[B]=Q[B]>0. Also, (Ŝ_u)_u=t-1,…,T is a Q'-martingale. Since A_t⊆ A_t-1 and A_t-1∖ A_t∈ℱ_t we get
 
    R[A_t-1|ℱ_t] = 1_A_t-1∖ A_t + R[A_t|ℱ_t].

However, A_t is an R null set, hence R[A_t|ℱ_t] = 0 R-a.s. Since ℱ_t has no non-empty R null sets, we have R[A_t|ℱ_t] = 0. We get R[A_t-1|ℱ_t] = 1_A_t-1∖ A_t, which yields Q'[A_t-1] = Q[A_t-1∖ A_t] = 0. Let ω∈Ω∖ A_t-1. Then R[{ω}|ℱ_t]≥ 0 and R[{ω}|ℱ_t](ω) > 0. Since Q[{ω}]>0 we get Q'[{ω}] >0. Thus, the support of Q' is Ω∖ A_t-1, which yields Q'∈𝔔_t-1. We have
 
    E_Q'[C/(1+r)^T-(t-1)|ℱ_t-1]    = E_Q[E_R[C/(1+r)^T-t|ℱ_t]/(1+r)|ℱ_t-1] 
       = E_Q[X_t/1+r|ℱ_t-1] 
       = X_t-1.

 Thus, X_t-1∈{ E_Q[C/(1+r)^T-t|ℱ_t-1]: Q∈𝒬_t-1}.
 
 Now, let X_t-1∈{ E_Q[C/(1+r)^T-(t-1)|ℱ_t-1]: Q∈𝔔_t-1}; we have to show that X_t-1∈𝒦_t-1. There is Q∈𝔔_t-1 such that X_t-1 = E_Q[C/(1+r)^T-(t-1)|ℱ_t-1]. By Lemma <ref> we find Q'∈ such that E_Q[Y|ℱ_s] = E_Q'[Y|ℱ_s] Q-a.s. for any random variable Y:Ω→ℝ and any s=t,…, T. Define X_t := E_Q'[C/(1+r)^T-t|ℱ_t] ∈𝒦_t because t∈ I. We find

    𝒦_t-1   ∋ E_Q[X_t/1+r|ℱ_t-1] 
       = E_Q[E_Q'[C/(1+r)^T-t+1|ℱ_t]|ℱ_t-1] 
       = E_Q[C/(1+r)^T-(t-1)|ℱ_t-1] 
       = X_t-1.
 
Thus, t-1∈ I.





  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
   
    α X + (1-α) Y ∈𝒦_t.



 Lemma <ref> yields measures Q,R∈ such that X=E_Q[C/(1+r)^T-t|ℱ_t], Y=E_R[C/(1+r)^T-t|ℱ_t]. Define the measure
  
    Q'[B] := E_Q[ α Q[B|ℱ_t] + (1-α) R[B|ℱ_t]].

  It is clear that Q and Q' agree on ℱ_t, and
one easily verifies Q'∈. Let B∈ℱ_t. Then

    E_Q'[ 1_B C/(1+r)^T-t]    =
        E_Q[ α E_Q[ 1_B C/(1+r)^T-t| ℱ_t]
        + (1-α) E_R[ 1_B C/(1+r)^T-t| ℱ_t]] 
       = E_Q[α 1_B X +(1-α)1_B Y] 
       = E_Q'[α 1_B X +(1-α)1_B Y].

 We find
 
    𝒦_t    ∋ E_Q'[C/(1+r)^T-t|ℱ_t] 
       = α X + (1-α)Y.





 We have 𝒞_0 ⊆Π_ℤ(C).


  Let p∈𝒞_0. Define X_0:=p. We can find recursively X_t+1∈𝒞_t+1 and Q_t∈ such that X_t = E_Q_t[X_t+1/(1+r)|ℱ_t] for t∈𝕋∖{T}. Since X_T∈𝒞_T = {C} we have X_T=C. Assume for contradiction that there is an integer arbitrage for the model (S^0,…,S^d,X). Then there is a one period integer arbitrage (ϕ̅,ϕ^d+1), i.e. there is t_0∈𝕋 such that (ϕ_t,ϕ^d+1_t)=0 for any t∈𝕋∖{0,t_0}. Then there is a minimal set B∈ℱ_t_0-1∖{∅} such that 1_B  (ϕ̅,ϕ^d+1) is still an arbitrage. Define (η,η^d+1):=(ϕ_t_0,ϕ^d+1_t_0)(ω)∈ℤ^d+1 for some ω∈ B. Then
  
    Y := 1_B(ηΔŜ_t_0 + η^d+1(X_t_0/(1+r)^t_0-X_t_0-1/(1+r)^t_0-1)) ≥ 0

and P[Y>0]>0. Since the model (S^0,…,S^d) satisfies NIA, we have
η^d+1≠0 and can define ξ^j := η^j/η^d+1. We get
 
    Y/η^d+1 = 1_B( ξΔŜ_t_0 + (X_t_0/(1+r)^t_0-X_t_0-1/(1+r)^t_0-1)).

 Thus, X_t_0-1∉𝒞_t_0-1. A contradiction.


It is not hard to see that actually 𝒞_0 = Π_ℤ(C), but we will not use this fact.



  Let t∈𝕋. Let X_t∈𝒦_t and define X^α_t := α X_t + (1-α)X_t^* for any α∈[0,1]. Then there is a countable set F⊆ (0,1] such that X_t^α∈𝒞_t for any α∈[0,1]∖ F. In particular, 𝒞_t is dense in 𝒦_t.


  Define
   
    I:={ t∈𝕋:  the claim holds for this t}.

 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 are X_t∈𝒦_t and Q∈𝔔_t-1 such that X_t-1 = E_Q[X_t/(1+r)|ℱ_t-1]. We define
 
    X_t-1^α   := α X_t-1 + (1-α) X^*_t-1, 
    
        X_t^α   := α X_t + (1-α) X^*_t

for any α∈[0,1]. There is F_t⊆ (0,1] countable such that X_t^α∈𝒞_t for any α∈ [0,1]∖ F_t. For any α∈[0,1]∖ F_t we find recursively X^α_s+1∈𝒞_s+1 and Q^α_s∈𝔔_s such that X^α_s = E_Q^α_s[X^α_s+1/(1+r)|ℱ_s], for s≥ t.
 
 We will show that (S_u^0,…,S_u^d,X^α_u)_u=t-1,…,T satisfies NIA for all but countably many choices for α∈[0,1]. Since existence of an integer arbitrage implies existence of a one-period arbitrage and the market (S_u^0,…,S_u^d,X^α_u)_u=t,…,T does not allow for arbitrage, we know that this arbitrage must be in the period from t-1 to t. Since ℱ_t-1 is generated by finitely many atoms it is sufficient to condition on one of the atoms. Thus, we may simply assume that t=1.
We define
  
    F_0 := {α∈(0,1] : α∈ F_1 or  X^α_0∉𝒞_0},

 and we will show that F_0 is countable. To this end, we define the sets
  
    𝒟_1    := { Y∈ L^0((Ω,ℱ_1,P),ℝ): Y=ξΔŜ_1 Q-a.s., ξ∈ℝ^d}, 
    𝒟^ℚ_1    := { Y∈ L^0((Ω,ℱ_1,P),ℝ): Y=ξΔŜ_1 Q-a.s., ξ∈ℚ^d}

and ΔX̂^α_1 := X^α_1/1+r-X^α_0 for α∈[0,1].

 Case 1: ΔX̂_1∉𝒟_1 or ΔX̂^*_1∉𝒟_1.  We define F̅:={α∈ [0,1]:Δ X_1^α∈𝒟_1}. Since 𝒟_1 is a vector space, we find that F̅ contains at most one element. We claim that (0,1]∖ (F_1∪F̅) does not contain any element of F_0. To this end, let α∈ (0,1]∖ (F_1∪F̅) 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 ξΔŜ_1 + ξ^d+1ΔX̂^α_1≥ 0. Since Q is a martingale measure we get ξΔŜ_1 + ξ^d+1ΔX̂^α_1 = 0 Q-a.s., and after solving for ΔX̂^α_1 we find ΔX̂^α_1∈𝒟_1. A contradiction.
 
 Case 2: ΔX̂_1,ΔX̂^*_1 ∈𝒟_1 and there is a set with positive Q-measure on which ΔX̂_1 ≠ΔX̂^*_1. Since 𝒟^ℚ_1 restricted to Ω∖ A_0 has countably many elements we find that ΔX̂^α_1∈𝒟^ℚ_1 at most countably often. Denote the set of α∈[0,1] where ΔX̂^α_1∈𝒟^ℚ_1 by F̅. We claim F_0⊆ F_1∪F̅. To this end, let α∈ (0,1]∖ (F_1∪F̅) 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 ξΔŜ_1 + ξ^d+1ΔX̂^α_1≥ 0. Since Q is a martingale measure we get ξΔŜ_1 + ξ^d+1ΔX̂^α_1 = 0 Q-a.s., and after solving for ΔX̂^α_1 we find ΔX̂^α_1∈𝒟_1^ℚ. A contradiction.
 
 Case 3: ΔX̂_1,ΔX̂^*_1 ∈𝒟_1 and ΔX̂_1 = ΔX̂^*_1 Q-a.s. Then, we have
 
    X_1    = (1+r)(ΔX̂_1+X_0) 
       = (1+r)(ΔX̂^*_1+X_0) 
       = X_1^* + (1+r)(X_0-X_0^*),   Q-a.s.

 If X_0 = X_0^*, then X_0=X_0^*∈𝒞_0. Thus we may assume that X_0≠ X_0^*. If R[B] ∈{0,1} for any R∈ and any B∈ℱ_1, then Lemma <ref> yields ℱ_1=ℱ_0 and, hence, ΔX̂_1 = 0 = ΔX̂^*_1 which yields that F_0 = F_1. Thus, we may assume that there is R∈ and B∈ℱ_1 with R[B]∈ (0,1). By Proposition <ref> we find that there is B∈ℱ_1 such that for any R∈=𝔔_0 we have R[B]∈ (0,1). In particular, we have Q[B]∈ (0,1). 
 
For n∈ℕ we define
  
    Y_1^n    := X_11_B^c + ((1-1/n)X_1+X_1^*/n)1_B
       = X_1 + 1_B(X_1^*-X_1)/n, 
    
         Y_0^n    := X_0Q[B^c] + ((1-1/n)X_0+X_0^*/n)Q[B] 
       = X_0 + Q[B](X_0^*-X_0)/n.

 Lemma <ref> yields that Y_1^n∈𝒦_1 for any n∈ℕ. The measure Q_n[D] := E_Q[Q_n'[D|ℱ_1]], where Q_n'∈𝔔_1 is such that Y_1^n = E_Q_n'[C/(1+r)^T-1|ℱ_1], satisfies
 
    E_Q_n[Y_1^n/1+r] = E_Q[Y_1^n/1+r] = Y_0^n,

  where the last equality follows from the definition of Q and (<ref>).
  Thus, Y_0^n∈𝒦_0 for any n∈ℕ. Observe that

    ΔŶ_1^n := Y_1^n/1+r-Y_0^n = ΔX̂_1 + 1/n(1_BX_1^*-X_1/1+r-Q[B](X_0^*-X_0)).

 We find that ΔŶ_1^n ≠ΔX̂_1^* with positive Q-probability. By appealing to case 1 resp. case 2 we find F^n⊆[0,1] countable such that α Y_0^n+(1-α)X_0^*∈𝒞_0 for any α∈ [0,1]∖ F^n. Define  the countable set
  
    F̅:= {α(1-Q[B]/n): n∈ℕ, α∈ F^n}.

We claim that F_0∖{1}⊆ F_1∪F̅. To this end let α∈ (0,1) ∖ (F_1∪F̅). Choose n∈ℕ such that n > Q[B]/(1-α).
Then there is α'∈ [0,1] such that α = α'(1-Q[B]/n). We find α'∉ F^n because α∉F̅. Thus, we have 

    𝒞_0   ∋α'Y_0^n+(1-α')X_0^*
        = X_0^*+α/1-Q[B]/n(Y_0^n-X_0^*) 
       = X_0^* + α/1-Q[B]/n(X_0-X_0^*)(1-Q[B]/n) 
       = X_0^* + α(X_0-X_0^*) 
       = X_0^α.
 
 Consequently, α∉ F_0, and we have shown that F_0 is countable. 

 








 
 
 
 












§ INTEGER SUPERHEDGING


In this section we discuss some properties of the integer superhedging price
σ_ℤ(C) of a claim, as defined in (<ref>).
First, we give a simple example where it does not agree with the classical
superhedging price supΠ(C).


  In this example, the gap between supΠ(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
  
    S^1_0 = 1,    S^1_1(ω_1) = 1-2a,     S^1_1(ω_2)=1+2a

  with r=0. The unique equivalent martingale measure is
  (δ_ω_1+δ_ω_2)/2, and so the unique arbitrage free price
  of the claim
  
    C(ω_1)=0,    C(ω_2)=2a

  is given by Π(C)={a}. By Theorem <ref>,
  we have Π_ℤ(C)=Π(C)={a}. The  integer superhedging
  price is found by computing
  
    σ_ℤ(C)   =
          inf_ϕ∈ℤmax_ω∈Ω(C(ω)
            -ϕΔ S_1^1(ω))  
       =min_ϕ∈ℤmax{2aϕ,2a-2aϕ}=2a.

  We obtain that the interval of prices of integer superhedges
  is [2a,∞).
  For real ϕ, the minimum in (<ref>)
  is attained at ϕ=12, yielding the classical superhedging
  price a=supΠ(C).








As soon as a model is fixed, the gap considered in the preceding example
can be bounded for all claims. In Example <ref>,
we have equality in (<ref>). On ℝ^n, we always use the
Euclidean norm ·=·_2.



  Assume (F) and NA, and let C be a claim. Then
  
    σ_ℤ(C) - supΠ(C) ≤1/2√(d) max_ω∈Ω∑_k=1^TΔŜ_k (ω).



  Let ψ̅∈ℛ be a cheapest classical superhedge. By Theorem <ref>,
  it satisfies V_0(ψ̅)=supΠ(C).
  By rounding the risky positions of ψ̅
  to the closest integers (with any convention for half-integers),
  we get a strategy (ψ^0,⌊ψ⌉)∈𝒵. Clearly,
  
    ψ_t-⌊ψ⌉_t≤(12,…,12)=12 √(d),
           t∈𝕋.

  Define Ĉ=C/(1+r)^T, and let ϕ̅∈𝒵.
  Since
  
    V̂_T(ϕ̅) = V_0(ϕ̅) + ∑_k=1^T ϕ_k ΔŜ_k,

  we get the necessary condition
  
    V_0(ϕ̅)=max_ω∈Ω(
        Ĉ(ω)- ∑_k=1^T ϕ_k(ω) ΔŜ_k(ω) ),

  if ϕ̅ should be a cheapest integer superhedge. It follows that
  
    σ_ℤ(C)    = inf_ϕ predictable,
    ℤ^d-valued max_ω∈Ω(
          Ĉ(ω)- ∑_k=1^T ϕ_k(ω) ΔŜ_k(ω) ) 
       ≤max_ω∈Ω(
          Ĉ(ω)- ∑_k=1^T ⌊ψ⌉_k(ω) ΔŜ_k(ω) ) 
       ≤max_ω∈Ω(
          Ĉ(ω)- ∑_k=1^T ψ_k(ω) ΔŜ_k(ω) )
            + max_ω∈Ω∑_k=1^T(ψ_k(ω)-⌊ψ⌉_k(ω))
               ΔŜ_k(ω) 
       ≤supΠ(C) + max_ω∈Ω∑_k=1^T
           ψ_k(ω)-⌊ψ⌉_k(ω)·ΔŜ_k(ω)
       ≤supΠ(C) +1/2√(d) max_ω∈Ω∑_k=1^TΔŜ_k (ω).



The following example shows that, contrary to the case of classical superhedging,
there need not exist a cheapest integer superhedge.


  Let Ω={ω_1,ω_2}, r=0, T=1 and d=2. We choose the model with S_0 :=(2,2) and
   
    S_1(ω_j) =  (3,2-√(2))    j=1,
     (1,2+√(2))    j=2.

  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
  
    C(ω_1) = 1-12√(2),    C(ω_2) =1+ 12√(2),

  whose set of (integer) arbitrage free prices is the singleton Π_ℤ(C)=Π(C)={1}.
  Then there is no minimizer for the superhedging problem (see (<ref>))
  
    inf_ϕ∈ℤ^2max_i=1,2
           (C(ω_i)-ϕΔ S_1(ω_i)) =:  inf_ϕ∈ℤ^2 f(ϕ).

  Indeed, for ϕ∈ℝ^2, the set of minimizers would be
  
    ϕ∈{(x,12 + x/√(2)): x∈ℝ},

  yielding inf_ϕ∈ℝ^2 f(ϕ)=1.
  Obviously, this set contains no integer strategies. By Kronecker's approximation
  theorem (Theorem <ref>), the sequence (12+m/√(2)) 1, m∈ℕ,
  is dense in [0,1]. Thus, there is a sequence m_k∈ℕ
  such that
  
    0≤ (12+m_k/√(2)) 1 ≤1/k,    k∈ℕ.

  Define
  
    ϕ^(k):=(m_k,⌊12+m_k/√(2)⌋) ∈ℤ^2,    k∈ℕ.

  Since f is Lipschitz continuous (with constant L, say), we have
  
    |f(ϕ^(k))-1|    = |f(ϕ^(k))- f(m_k,12+m_k/√(2))| 
       ≤ L (0,(12+m_k/√(2)) 1) → 0,    k→∞.

  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^-1σ_ℤ(NC) = supΠ(C). The second part of Theorem <ref> gives an estimate
on superhedging C with rational strategies with controlled denominators.


  Assume (F) and NA, and let C be a claim. Then
  
  
  (i)
  
    σ_ℤ(NC)/N = supΠ(C) + O( 1/N),
           N→∞.

  
  (ii)
  There is a sequence of rational strategies ψ̅^(N)∈𝒬
  such that all denominators occurring in ψ̅^(N) have absolute value at most N,
  
    V_0(ψ̅^(N)) = supΠ(C) + O(N^-1/(nd(T+1))log N),

  and ψ̅^(N) is a superhedging strategy for C.
  



  (i) Assumption (F) implies that the classical superhedging price
  supΠ(C)=sup{E_Q[C/(1+r)^T]:Q∈𝔔} is finite.
  It is clear that
  
    N^-1σ_ℤ(NC) ≥ N^-1supΠ(NC) = supΠ(C)

  for all N. For the converse estimate, let ϕ̅ be a classical superhedging
  strategy for C with price supΠ(C) (see Theorem <ref>).
  Define
  
    η^(N),j_t(ω) := ⌊ N ϕ_t^j(ω) ⌋ = N ϕ_t^j(ω)+O(1),
          ω∈Ω,t∈𝕋,1≤ j≤ d,N∈ℕ.

  We choose an arbitrary map f:ℕ→ℝ satisfying
  lim_N→∞f(N)=∞ and put
  
    η_1^(N),0 := Nϕ_1^0+f(N),    N∈ℕ.

  Then we  define   η_t^(N),0 for t=2,…,T recursively to obtain a self-financing
  integer strategy η̅^(N) for each N. By the definition of η^(N),
  and since ϕ̅ is a superhedging strategy, we have
  
    V̂_T(η̅^(N))/N   = V_0(η̅^(N))/N
            +∑_k=1^T-1η^(N)_k/NΔŜ_k
       =   ϕ_1^0+ f(N)/N + η_1^(N) S_0/N
            +∑_k=1^T-1ϕ_k ΔŜ_k + O(1/N) 
       = f(N)/N +V_0(ϕ̅)
            +∑_k=1^T-1ϕ_k ΔŜ_k + O(1/N) 
       ≥C/(1+r)^T + f(N)/N +O(1/N)
          ≥C/(1+r)^T

  for large N. This shows that η̅^(N) is an integer superhedging strategy
  of NC for large N, and hence
  
    σ_ℤ(NC)    ≤ V_0(η̅^(N)) 
       = N V_0(ϕ̅) + O(f(N)) 
       = N supΠ(C) + O(f(N)),  
         N→∞.

  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.
  
  (ii)   Again, let ϕ̅ be a classical superhedging
  strategy for C with price supΠ(C).
  By Dirichlet's approximation theorem (Theorem <ref>), there are 1≤ q(N)≤ N
  and p(N,t,j,l)∈ℤ such that
  
    |ϕ_t^j(ω_l)q(N)-p(N,t,j,l)| < N^-1/(nd(1+T)),   
        1≤ l≤ n,t∈𝕋,1≤ j≤ d,N∈ℕ.

  We define
  
    ψ_t^(N),j(ω_l) := p(N,t,j,l)/q(N),   
        1≤ l≤ n,t∈𝕋,1≤ j≤ d,N∈ℕ,

  which yields
  
    |ϕ_t^j(ω_l)-ψ_t^(N),j(ω_l)|<N^-1/(nd(1+T)),   
        1≤ l≤ n,t∈𝕋,1≤ j≤ d,N∈ℕ.

  After fixing the initial bank account position
  
    ψ_1^(N),0:= ϕ_1^0 + N^-1/(nd(1+T))log N,    N∈ℕ,

  a strategy ψ̅^(N)∈𝒬 is defined for each N.
  By definition,
  
    V_0(ψ̅^(N))    = ϕ_1^0 + N^-1/(nd(1+T))log N + ψ_1^(N)S_0 
       = ϕ_1^0 + N^-1/(nd(1+T))log N + ϕ_1 S_0 + O(N^-1/(nd(1+T))) 
       = supΠ(C) + O(N^-1/(nd(1+T))log N),    N→∞.

  It remains to show that ψ̅^(N) is a superhedge for C for large N.
  This follows from
  
    V̂_T(ψ̅^(N))    = V_0(ψ̅^(N)) + ∑_k=1^T ψ_k^(N)ΔŜ_k 
       = V_0(ϕ̅) + N^-1/(nd(1+T))log N
          + ∑_k=1^T ϕ_k ΔŜ_k + O(N^-1/(nd(1+T))) 
       ≥C/(1+r)^T + N^-1/(nd(1+T))log N + O(N^-1/(nd(1+T))) 
       ≥C/(1+r)^T,    N large.

 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

    inf_ϕ∈ℝ^dinf_V_0 ∈ℝ C/(1+r)-V_0-ϕΔ S_1 
      = inf_ϕ∈ℝ^dC̃ - ϕΔ S_1 ,

where C̃:=(C-E^*[C])/(1+r). Note that inf_V_0 ∈ℝ is attained
at

    V_0=E^*[C/(1+r)-ϕΔ S_1]=E^*[C]/(1+r).

The problem (<ref>) is then solved
by projecting C̃ orthogonally to the space
{ϕΔ S_1 : ϕ∈ℝ^d}, which is closed
by Theorem 6.4.2 in <cit.>.
For more details on variance-optimal hedging (in particular, on the multi-period problem),
we refer to Chapter 10 of <cit.> 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 (<ref>), and we thus have to compute

    inf_ϕ∈ℤ^dC̃ - ϕΔ S_1 .

We have

    C̃ - ϕΔ S_1 ^2    = ∑_l=1^n P^*[ω_l](C̃(ω_l)-ϕΔ S_1(ω_l))^2
       = ∑_l=1^n (C̃(ω_l)P^*[ω_l]^1/2-ϕΔ S_1(ω_l)P^*[ω_l]^1/2)^2.

The problem (<ref>) thus amounts to computing the element of the lattice

    {ϕ^1
      (
      Δ S_1^1(ω_1)P^*[ω_1]^1/2
    ⋮
    Δ S_1^1(ω_n)P^*[ω_n]^1/2)
       +… + ϕ^d
       (
      Δ S_1^d(ω_1)P^*[ω_1]^1/2
    ⋮
    Δ S_1^d(ω_n)P^*[ω_n]^1/2)
       : ϕ∈ℤ^d }⊂ℝ^n

closest to the vector

    (
      C̃(ω_1)P^*[ω_1]^1/2
    ⋮
    C̃(ω_n)P^*[ω_n]^1/2) ∈ℝ^n

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 <cit.> 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 (<ref>) 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 <ref>.
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.



We wish to approximately hedge N copies of the claim, i.e., the claim NC, for N∈ℕ.
First, we computed the classical variance-optimal hedge ϕ^(N)=N ϕ^(1)∈ℝ^2 by
projection (see (<ref>)). The relative L^2(P^*)-error

    NC̃- ϕ^(N)Δ S_1 /NC̃
        = C̃- ϕ^(1)Δ S_1 /C̃,

which of course does not depend on N, is displayed in the second line of Table <ref>.
The third line of Table <ref> 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 variance-optimal hedging
problem (<ref>) exactly, using the algorithm CLOSESTPOINT described
in <cit.>, which is based on the Schnorr-Euchner algorithm <cit.>.
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
<cit.> might faciliate the task of computing a closest vector.
Table <ref> shows the relative L^2(P^*)-error

    NC̃- ϕ_CVP^(N)Δ S_1 /NC̃

and the maximum position size.
Finally, we used a poor man's approach at
solving (<ref>) approximately, by simpling rounding the positions
of the classical hedge ϕ^(N) to the closest integers. From Table <ref>,
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 <cit.> of the
lattice (<ref>) yields an upper bound for the hedging error for any claim.








§ 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).


  Given α_1,…,α_n∈ℝ and an integer N>1, there are
  q,x_1,…,x_n∈ℤ with 1≤ q≤ N and
  
    |α_i q-x_i| < N^-1/n,    1≤ i≤ n.




  If θ∈ℝ is an irrational number, then the sequence (n θ 1)_n∈ℕ is dense in [0,1].


We also mention here the following classical theorem <cit.>:

  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 <ref> 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:

    E[U(ϕΔ S_1)] →max!    ϕ∈ℤ^d, ϕΔ S_1 ≥ -c.

Then, Theorem <ref> easily yields a sufficient criterion to ensure that the admissibility
set contains a non-zero portfolio. Refinements of Theorem <ref> give
several linearly independent portfolios.



































siam



