Arbitrage-Free Conditions and Hedging Strategies for Markets with Penalty Costs on Short Positions

Abstract

We consider a discrete-time financial model in a general sample space with penalty costs on short positions. We consider a friction market closely related to the standard one except that withdrawals from the portfolio value proportional to short positions are made. We provide necessary and sufficient conditions for the nonexistence of arbitrages in this situation and for a self-financing strategy to replicate a contingent claim. For the finite-sample space case, this result leads to an explicit and constructive procedure for obtaining perfect hedging strategies.

1. Introduction

In recent years, applications of stochastic analysis and control have entered in the field of financial engineering in an effective and rapid way, due mainly to the powerful tools that can be brought from these disciplines into almost all aspects of fields like, for instance, in the study of arbitrage, hedging, pricing, and portfolio optimization. One of the classical problems in portfolio optimization is the mean-variance portfolio selection problem, which was transformed with the seminal work of Markowitz in [1] (see also [2]). Since then, the amount of research on this subject has increased in order to provide the development of sophisticated analytical and numerical methods for financial engineering models with more realistic assumptions see, for instance, [3–8], among others. More recently, the multiperiod mean-variance problem was tackled by [9] and later extended in several directions see, for instance, [10–25]. The pioneering work of Black and Scholes [26] and Merton [27] provided a major change in the area of pricing of derivative securities, showing that the analysis should be based on nonarbitrage considerations rather than on preference-related concepts such as expected values. From the works of Harrison and Kreps [28] and Harrison and Pliska [29], it became apparent that semimartingale theory provided a natural framework for the analysis of financial markets and pricing.

Another fundamental result on the study of arbitrage, hedging, and pricing of financial markets is the Dalang-Morton-Willinger theorem, also known as the fundamental theorem of asset pricing. It states that in a frictionless security market, the existence of an equivalent martingale measure for the discounted price process is equivalent to the absence of arbitrage (see, e.g., [30]). Recently, there has been a number of papers dealing with contingent claim valuation and extending versions of the aforementioned theorem in several directions (see, e.g., [31–38]). The subject of pricing derivatives with transaction costs and portfolio selection with transaction costs is of practical importance and has been in evidence over the last years. Two types of transaction costs are considered; fixed costs, which are paid whenever there is a change of position, and proportional costs, which are charged according to the volume traded. Different approaches to the problem of pricing derivatives with transaction costs and the portfolio choice problem under transaction costs can be found in the literature see, for instance, [32, 37, 39–44]. The results in [45] provided a version of the fundamental theorem of asset pricing within a short sales constraints framework and possible infinite number of transactions within a finite period of time, using the free-lunch notion, a stronger notion of the no arbitrage condition. The case of closed cone constraints on the amount invested in the risky assets, which includes restrictions on short sales, has been studied in [46] for the case in which the price process is positive and under a nondegeneracy hypothesis on the price process. In [47], these results were generalized, and the fundamental theorem of asset pricing was stated under polyhedral convex cone constraints and using the classical notion of no arbitrage instead of free lunch. The general short sales constraints in [45, 47] were considered by separating the price process into two sort of securities; those which cannot be held in negative amounts and those that can only be held in negative amounts. The no arbitrage condition in this case is shown in [47] to be equivalent to the existence of a positive interest rate process and an equivalent probability measure 𝐐 under which the discounted price processes of securities that cannot be sold short are supermartingales, and the discounted price processes of securities that can only be sold short are submartingales.

In this paper, we consider a model closely related to the standard one (see [30, 48, 49]), except that a withdrawal directly proportional to the amount on short positions is made from the portfolio. As far as the authors are aware of, this model has not been studied before (see also Remark 3.3). Theorem 3.1 provides necessary and sufficient conditions for the nonexistence of arbitrages directly in terms of the price process and penalty costs at time 𝑡 and can be seen as a natural extension of the standard fundamental theorem of asset pricing (see, e.g., [30, 48, 49]). When the penalty costs go to zero, our result reduces to that presented in [30]. From this result we derive a sufficient condition for nonarbitrage and for a self-financing strategy to consistently replicate a contingent claim (e.g., any other superreplicating self-financing strategy will have an initial value greater than that of the replicanting strategy). For the finite sample space case, this result yields an explicit and constructive procedure for obtaining perfect hedging strategies.

This paper is organized in the following way. Section 2 presents some notation, definitions, and the financial model. Section 3 contains the main results of the paper. In Section 4, we present an explicit and constructive procedure for obtaining perfect hedging strategies for the case in which the sample space is finite, as well as some numerical examples. Section 5 concludes the paper. The proof of some auxiliary results and the main results are presented in the appendix.

2. Notation, Definitions and Problem Formulation

Let the real 𝑑 -dimensional vector space be denoted by ℝ^𝑑 and for 𝑥∈ℝ^𝑑 we will write 𝑥_𝑖 for the 𝑖th component of the vector. The superscript will be omitted for the case𝑑=1. For𝑥, 𝑦 in ℝ^𝑑, we set 𝑥⋅𝑦=^𝑑_𝑖=1𝑥_𝑖𝑦_𝑖. (2.1) We write 𝑥≥0 to denote that all components of 𝑥 are positive, that is, 𝑥_𝑖≥0 for 𝑖=1,…,𝑑. For 𝑥∈ℝ^𝑑, we set the following vectors in ℝ^𝑑: 𝑥⁺≥0 such that its 𝑖th component 𝑥⁺_𝑖 is equal to 𝑥_𝑖 if 𝑥_𝑖≥0, zero otherwise, 𝑥⁻=(−𝑥)⁺≥0 (therefore, 𝑥=𝑥⁺−𝑥⁻ and 𝑥⁺_𝑖𝑥⁻_𝑖=0 for each 𝑖=1,…,𝑑). The vector formed by 1 in all components will be represented by 𝑒, and the vector with 1 at the 𝑖th component and 0 elsewhere by 𝑏^𝑖. For a real number 𝑎, we define 𝑎^⊕=1/𝑎 if 𝑎≠0, zero otherwise.

Let (Ω,ℱ,𝐏) be a complete probability space equipped with a filtration {ℱ_𝑡},𝑡=0,1,…,𝑇. For 𝒢 a sub-𝜎 -algebra of ℱ, we denote by ℒ^𝑑₀(Ω,𝒢,𝐏) (or simply ℒ^𝑑₀(𝒢) ) the space of 𝒢 -measurable random variables with values in ℝ^𝑑, which is a complete topological vector space if equipped with the topology of convergence measure. As any sequence converging in probability contains a subsequence converging almost surely (a.s.), we can assume without loss of generality that any convergent sequence in ℒ^𝑑₀(𝒢) will converge a.s. For any probability measure𝐐, 𝐸_Q(⋅) denotes the expectation with respect to 𝐐, and we write 𝐐∼𝐏(𝐐<<𝐏) whenever the probability measure 𝐐 is equivalent to (absolutely continuous with respect to) 𝐏. For any 𝐴∈ℱ, 1_𝐴 denotes the indicator function of the set 𝐴. Let ℒ₀^𝑑+(𝒢) be the set of random vectors 𝑋∈ℒ^𝑑₀(𝒢) such that 𝐏(𝑋≥0)=1. The space of integrable random vectors in ℒ^𝑑₀(𝒢) will be denoted by ℒ^𝑑₁(𝒢) and the space of essentially bounded random vectors in ℒ^𝑑₀(𝒢) by ℒ^𝑑_∞(𝒢).

Consider given stochastic processes 𝑆={𝑆(𝑡);𝑡=0,…,𝑇} and 𝐷={𝐷(𝑡);𝑡=1,…,𝑇} taking values in ℝ^𝑑 with 𝑆(𝑡)∈ℒ^𝑑₀(ℱ_𝑡) and 𝑆₁(𝑡)=1 for each 𝑡=0,…,𝑇, and 𝐷(𝑡)∈ℒ₀^𝑑+(ℱ_𝑡) for each 𝑡=1,…,𝑇. We define for 𝑡=1,…,𝑇,Δ𝑆(𝑡)∶=𝑆(𝑡)−𝑆(𝑡−1). A trading strategy 𝐻=(𝐻(1),…,𝐻(𝑇)) is defined such that each 𝐻(𝑡) is a 𝑑×2 -dimensional random matrix with columns 𝐻⁺(𝑡)∈ℒ₀^𝑑+(ℱ_𝑡−1) and 𝐻⁻(𝑡)∈ℒ₀^𝑑+(ℱ_𝑡−1). It describes an investor's portfolio as carried forward from time 𝑡=0 to time 𝑡=𝑇. In the model of a security market, 𝑆 describes the evolution of the prices of 𝑑 securities, 𝐻⁺(𝑡) represents the number of units of each security hold in a long position from time 𝑡−1 to time 𝑡, 𝐻⁻(𝑡) represents the number of units of each security hold in a short selling position from time 𝑡−1 to time 𝑡, and 𝐷 the evolution of the penalty costs and possible spread costs between borrowing and lending rates due to short selling positions on each security 𝑖.

Associated to a trading strategy 𝐻, we have the value process 𝑉^𝐻∶=(𝑉^𝐻(0),…,𝑉^𝐻(𝑇)) describing the total value of the portfolio at each time 𝑡. For notational simplicity, we will omit the superscript 𝐻 whenever no confusion arises. The portfolio value can be written, at time 𝑡=0, as 𝑉(0)=𝐻⁺(1)−𝐻⁻(1)⋅𝑆(0), (2.2) and at times 𝑡=1,…,𝑇, as 𝑉(𝑡)=𝐻⁺(𝑡)−𝐻⁻(𝑡)⋅𝑆(𝑡)−𝐻⁻(𝑡)⋅𝐷(𝑡). (2.3) The quantity 𝑉(𝑡) represents the value of the portfolio at time 𝑡 just before any change of ownership positions take place at that time. The penalty costs due to short selling positions are represented by the costs: 𝐻⁻(𝑡)⋅𝐷(𝑡). (2.4) The value of the portfolio at time 𝑡+1 just after the change of ownership positions is (𝐻⁺(𝑡+1)−𝐻⁻(𝑡+1))⋅𝑆(𝑡). We consider in this paper self-financing trading strategies, so that no money is added or withdrawn from the portfolio between times 𝑡=0 to time 𝑡=𝑇. Any change in the portfolio's value is due to a gain or loss in the investments, and penalty costs due to the short selling positions. Thus, we must have 𝑉(𝑡)=𝐻⁺(𝑡+1)−𝐻⁻(𝑡+1)⋅𝑆(𝑡). (2.5) From (2.2)–(2.5), we have for 𝑡=1,…,𝑇 that 𝑉(𝑡)=𝑉(𝑡−1)+𝐻⁺(𝑡)−𝐻⁻(𝑡)⋅Δ𝑆(𝑡)−𝐻⁻(𝑡)⋅𝐷(𝑡). (2.6) We notice that the penalties can be seen as withdrawals from the portfolio value proportional to the short selling positions.

We conclude this section with the definition of an arbitrage opportunity. We say that there is an arbitrage opportunity if for some self-financing trading strategy 𝐻, we have a.s. that (i)𝑉(0)≤0, (ii)𝑉(𝑇)≥0, and (iii)𝐸(𝑉(𝑇))>0.

3. Main Results

In this section, we present the main results of the paper. We start with Theorem 3.1, which provides necessary and sufficient conditions for the nonexistence of arbitrages and can be seen as a natural extension of the standard fundamental theorem of asset pricing (see, e.g., [30, 48, 49]). As pointed out in Remark 3.2, when the penalty costs go to zero, our result reduces to that presented in [30]. In Remark 3.3, we point out the differences between our result and previous results presented in the literature. As usual in this kind of problems, the hardest part of the proof is to show that a certain set is closed (see Proposition A.1 in the appendix). In the sequence, we derive a sufficient condition for nonarbitrage and for a self-financing strategy to consistently replicate a contingent claim. In Section 4, we consider the finite sample space case so that the results in this section yield an explicit and constructive procedure for obtaining perfect hedging strategies.

The following theorem provides necessary and sufficient conditions for the nonexistence of arbitrages. The proof can be found in the appendix. In what follows, we recall that ℒ⁺_∞(ℱ_𝑡) represents the space of essentially bounded ℱ_t -measurable random variables 𝑍 such that 𝐏(𝑍≥0)=1.

Theorem 3.1. The following statements are equivalent:(i)there are no arbitrage opportunities,(ii)for any self-financing strategy 𝐻, one has a.s. that 𝐻⁺(𝑡)−𝐻⁻(𝑡)()𝐻⋅𝑆𝑡−1≤0,₊(𝑡)−𝐻⁻(𝑡)⋅Δ𝑆(𝑡)−𝐻⁻(𝑡)(𝑡)⟹𝐻⋅𝐷≥0⁺(𝑡)−𝐻⁻(𝑡)⋅Δ𝑆(𝑡)−𝐻⁻(𝑡)⋅𝐷(𝑡)=0, (3.1)(iii)there exists a stochastic process {𝑟(𝑡)} with 𝑟(𝑡)∈ℒ⁺_∞(ℱ_𝑡) for each 𝑡=0,1,…,𝑇−1 and a probability measure 𝐐∼𝐏 such that 0<𝑑𝐐/𝑑𝐏∈ℒ⁺_∞(ℱ), 𝑆(𝑡),𝐷(𝑡) are integrable with respect to 𝐐 and for each 𝑡=0,…,𝑇−1, 𝐸_𝐐Δ𝑆(𝑡+1)∣ℱ_𝑡≤𝑟(𝑡)𝑆(𝑡)≤𝐸_𝐐Δ𝑆(𝑡+1)+𝐷(𝑡+1)∣ℱ_𝑡𝑎.𝑠. (3.2)

Remark 3.2. For the case in which 𝐷(𝑡)=0, our results reduce to the well-known fundamental theorem of asset pricing with finite-discrete time and infinite state space, see [30] (recall that 𝑆₁(𝑡)=1,Δ𝑆₁(𝑡)=0, and if 𝐷₁(𝑡)=0, then (3.2) implies that 𝑟(𝑡)=0).

Remark 3.3. In [45, 47], the authors consider a financial market with two sort of securities, those that cannot be held in negative amounts and represented by 𝑆(𝑡), and those that can only be held in negative amounts and represented by 𝑆(𝑡). To write our problem in the above set-up, we would need to define the fictitious price processes 𝑆(𝑡), 𝑆(𝑡) as: 𝑆(0)=𝑆(0)=𝑆(0) and for 𝑡=1,…,𝑇, 𝑆(𝑡)=𝑆(𝑡), 𝑆(𝑡)=𝑆(𝑡)+𝐷(1)+⋯+𝐷(𝑡). Notice that there is no discounting to be applied since we are considering that 𝑆₁(𝑡)=1. By doing this, we would have that for 𝑡=1,…,𝑇, Δ𝑆(𝑡)=Δ𝑆(𝑡), Δ𝑆(𝑡)=Δ𝑆(𝑡)+𝐷(𝑡) and this would yield that 𝑉(𝑡)=𝑉(𝑡−1)+(𝐻⁺(𝑡)−𝐻⁻(𝑡))⋅Δ𝑆(𝑡)−𝐻⁻(𝑡)⋅𝐷(𝑡), which is similar to (2.6). Note, however, that the models are different since we cannot guarantee that a self-financing strategy 𝐻 for the above model will be self-financing for our model, and vice versa. Indeed, for the above model, the self-financing condition would read as (𝐻⁺(𝑡+1)−𝐻⁻(𝑡+1))⋅𝑆(𝑡)−𝐻⁻(𝑡+1)⋅∑^𝑡_𝑘=1𝐷(𝑘)=(𝐻⁺(𝑡)−𝐻⁻(𝑡))⋅𝑆(𝑡)−𝐻⁻(𝑡)⋅∑^𝑡_𝑘=1𝐷(𝑘), while for our model it would be (𝐻⁺(𝑡+1)−𝐻⁻(𝑡+1))⋅𝑆(𝑡)=(𝐻⁺(𝑡)−𝐻⁻(𝑡))⋅𝑆(𝑡)−𝐻⁻(𝑡)⋅𝐷(𝑡). This also occurs with the nonarbitrage conditions of the two models. Indeed, the nonarbitrage condition presented in [45, 47] states that 𝑆(𝑡) is a supermartingale and 𝑆(𝑡) is a submartingale, which would involve the sum of the terms (𝑘). On the other hand, for our model, the nonarbitrage condition (3.2) involves only the state price 𝑆(𝑡), 𝑆(𝑡+1) and penalty costs 𝐷(𝑡+1) emphasizing the difference between the two models.

In what follows, we define 𝒦=({𝑟(𝑡)},𝐐);𝑟(𝑡)and𝐐satisfyingcondition(iii)ofTheorem3.1. (3.3) We recall next that a contingent claim (random variable) 𝑋∈ℒ₀(ℱ_𝑇) is marketable if for some self-financing strategy 𝐻 we have that a.s. 𝑋=𝑉^𝐻(𝑇) and, in this case, 𝐻 is said to replicate 𝑋. We say that 𝐻 superreplicates 𝑋 if a.s. we have that 𝑉^𝐻(𝑇)≥𝑋. We have the following corollary (see the proof in the appendix).

Corollary 3.4. Suppose that 𝐻 superreplicates 𝑋 and there is no arbitrage. Then, for any ({𝑟(𝑡)},𝐐)∈𝒦 one has a.s. that 𝐸_𝐐𝑋(1+𝑟(𝑇))⋯(1+𝑟(𝑡))∣ℱ_𝑡≤𝑉(𝑡). (3.4) Writing Γ(𝑡)={𝜔;𝑉(𝑡)(𝜔)−𝐸_𝐐(𝑋/(1+𝑟(𝑇))⋯(1+𝑟(𝑡))∣ℱ_𝑡)(𝜔)>0} one has that if 𝐏(Γ(𝑇))>0 then 𝐏(Γ(𝑡))>0 for every 𝑡=𝑇−1,…,0.

We will be interested now in deriving a condition such that the pricing of a marketable contigent claim 𝑋∈ℒ₀(ℱ_𝑇) is obtained from a self-financing strategy 𝐻 that replicates 𝑋 with 𝐻⁺(𝑡)⋅𝐻⁻(𝑡)=0for𝑡=1,…,𝑇, so that logical pricing can be obtained in this way. Let us define 𝕁∶={𝑎={𝑎(𝑡)}^𝑇−1_𝑡=0;for𝑡=0,…,𝑇−1,𝑎(𝑡)∈ℒ₀^𝑑+(ℱ_𝑡),𝑎_𝑖(𝑡)=0or1}. For a self-financing strategy 𝐻 satisfying 𝐻⁺(𝑡)⋅𝐻⁻(𝑡)=0, we set 𝑎^𝐻={𝑎^𝐻(𝑡)}∈𝕁 as 𝑎^𝐻_𝑖(𝑡)=1_{𝐻⁻_{𝑖(𝑡+1)>0}}.

Definition 3.5. For 𝑎={𝑎(𝑡)}∈𝕁, set Θ^𝑎=𝐐∼𝐏,𝑑𝐐𝑑𝐏∈ℒ⁺_∞(ℱ);𝑆(𝑡),𝐷(𝑡)areintegrablewithrespectto𝐐,for𝑡=0,…,𝑇,𝐸_𝐐Δ𝑆_𝑖(𝑡+1)∣ℱ_𝑡=0a.s.on𝑎_𝑖(𝑡)𝐸=0,and_𝐐Δ𝑆_𝑖(𝑡+1)+𝐷_𝑖(𝑡+1)∣ℱ_𝑡=0a.s.on𝑎_𝑖(𝑡)=1. (3.5) We have the following proposition showing that any 𝑎={𝑎(𝑡)}∈𝕁 will lead to an element in 𝒦 (see the proof in the appendix).

Proposition 3.6. If for some 𝑎={𝑎(𝑡)}∈𝕁 one has Θ^𝑎≠∅, then there are no arbitrages.

Finally, we have the following result, presenting a sufficient condition for a self-financing strategy to consistently replicate a contingent claim (i.e., any other superreplicating self-financing strategy will have an initial value greater than that of the replicating strategy). The proof can be found in the appendix.

Proposition 3.7. If 𝐻 is a self-financing strategy that replicates 𝑋 with 𝐻⁺(𝑡)⋅𝐻⁻(𝑡)=0 for 𝑡=1,…,𝑇 and Θ^𝑎𝐻≠∅, then for any superreplicating strategy 𝐻 for 𝑋 one has a.s. for 𝑡=0,1,…,𝑇 that 𝑉𝐻(𝑡)≥𝑉^𝐻(𝑡) and if 𝐏(𝑉𝐻(𝑇)>𝑋)>0, then (V𝐻(𝑡)>𝑉^𝐻(𝑡))>0.

4. A Numerical Procedure

In this section, we consider the finite-state space case and present an algorithm for obtaining the hedging strategy for a marketable claim 𝑋 satisfying the conditions of Proposition 3.7. We assume here that Ω={𝜔₁,…,𝜔_𝜅} and that 𝐷(𝑡) is ℱ_𝑡−1 -measurable. We consider the single period case only, and suppress the time dependence whenever it is possible. The multiperiod case follows in a similar way, by using the information structure described in [50] or [51], and by applying backwards in time, the procedure described here for the single period case and each node of the information structure. Define the following matrix 𝐴: 𝐴=(𝐴₁𝐴₂), where A₁=1𝑆₂(1)𝜔₁…𝑆_𝑑(1)𝜔₁⋮⋮⋱⋮1𝑆₂(1)𝜔_𝜅…𝑆_𝑑(1)𝜔_𝜅,𝐴₂=−𝐴₁+𝐷, (4.1) with 𝐷=𝑒𝐝 (recall that 𝑒 is the vector formed by 1 in all components), 𝐝=(𝑔₁…𝑔_𝑑). Let the vector 𝑥∈ℝ^𝜅 be such that 𝑥_𝑗=𝑋(𝜔_𝑗), 𝑗=1,…,𝜅. We have that 𝑋 is marketable if and only if there exists 𝐻⁺,𝐻⁻ that satisfy the system: 𝐴𝐻⁺𝐻⁻=𝐴₁𝐻⁺−𝐻⁻−𝐷𝐻⁻𝐻=𝑥,₊≥0,𝐻⁻≥0. (4.2)(4.3) For the case in which 𝜅=𝑑 and 𝐴₁ has an inverse, we have the following explicit and constructive procedure for obtaining a trading strategy 𝐻 that replicates 𝑋 with 𝐻⁺⋅𝐻⁻=0. Since 𝐷=𝑒𝐝 and 𝐴₁⁻¹𝑒=𝑏¹ (recall that 𝑏¹ is the vector formed by 1 at the 1st component, and 0 elsewhere), we have premultiplying (4.2) by 𝐴₁⁻¹ that 𝐻⁺, 𝐻⁻ satisfy (4.2) if and only if satisfy 𝐻⁺−𝐻⁻−𝐴₁⁻¹𝑒𝐝𝐻⁻=𝐻⁺−𝐻⁻−𝑏¹𝐝𝐻⁻=𝐴₁⁻¹𝑥. (4.4) Set 𝑦=𝐴₁⁻¹𝑥. Let us obtain 𝐻 that satisfies (4.2), (4.3) with 𝐻⁺⋅𝐻⁻=0. Define for𝑖=2,…,𝑑: 𝐻⁺_𝑖=𝑦_𝑖, 𝐻⁻_𝑖=0 if 𝑦_𝑖≥0, otherwise, 𝐻⁻_𝑖=−𝑦_𝑖, 𝐻⁺_𝑖=0. For 𝑖=1, calculate 𝑧=𝑦₁+^𝑑_𝑖=2𝑔_𝑖𝐻⁻_𝑖. (4.5) In order to have (4.4) satisfied, we must have (𝐻⁺₁−𝐻⁻₁)−𝑔₁𝐻⁻₁=𝑧. If 𝑧≥0, then set 𝐻⁺₁=𝑧, 𝐻⁻₁=0, otherwise, set 𝐻⁺₁=0 and 𝐻⁻₁=−𝑧/(1+𝑔₁). Thus, we have obtained in this way a trading strategy 𝐻 that replicates 𝑋 with 𝐻⁺⋅𝐻⁻=0.

Finally, notice that for the discrete sample space, the set 𝕁 is finite, and thus if we assume that for every 𝑎∈𝕁, Θ^𝑎≠∅, then the conditions of Proposition 3.7 will be satisfied. With the above procedure, we have a seller price and a buyer price for each contingent claim. The seller price, denoted by 𝑉_𝑠(0), is obtained by applying to 𝑋 backwards in time the algorithm presented above. The buyer price, denoted by 𝑉_𝑏(0), is obtained by applying the backward algorithm to −𝑋, and taking 𝑉_𝑏(0)=−𝑉(0). We illustrate this procedure next for the binomial case.

Example 4.1. Let us consider the binomial model, which consists of a single risky security satisfying 𝑆₂(𝑡)=1𝐵(𝑡)𝑢^𝑁(𝑡)𝑑^{𝑡−𝑁(𝑡)}𝑆₂(0), (4.6)𝑡=1,…,𝑇, where 0<𝑑<1<𝑢 and 𝑁={𝑁(𝑡);𝑡=1,…,𝑇} is a binomial process with parameter𝑝, 0<𝑝<1, and the bank account is given by 𝐵(𝑡)=(1+𝑟_𝑓)^𝑡, 𝑡=0,1,…,𝑇. The penalty costs are assumed to be of the form: 𝐷₁(𝑡)=𝛼₁11+𝑟_𝑓,𝐷₂(𝑡)=𝛼₂𝑆₂(𝑡−1)1+𝑟_𝑓. (4.7) It is easy to see that in this case 𝕁={(0,0),(0,1),(1,0),(1,1)}, and we have the following possibilities for Θ^𝑎={𝜋₁,𝜋₂}, where 𝜋₁ is associated to the probability measure 𝐐 when the stocks goes up, 𝜋₂ when the stocks goes down: (i)𝑎=(0,0); in this case, 𝜋₁=1+𝑟_𝑓−𝑑𝑢−𝑑,𝜋₂=𝑢−1+𝑟_𝑓𝑢−𝑑; (4.8)(ii)𝑎=(1,0); in this case, 𝜋₁=1+𝑟_𝑓−𝛼₁−𝑑𝑢−𝑑,𝜋₂=𝑢−1+𝑟_𝑓−𝛼₁𝑢−𝑑; (4.9)(iii)𝑎=(0,1); in this case, 𝜋₁=1+𝑟_𝑓+𝛼₀−𝑑𝑢−𝑑,𝜋₂=𝑢−1+𝑟_𝑓+𝛼₀𝑢−𝑑; (4.10)(iv)𝑎=(1,1); in this case, 𝜋₁=1+𝑟_𝑓+𝛼₀−𝛼₁−𝑑𝑢−𝑑,𝜋₂=𝑢−1+𝑟_𝑓+𝛼₀−𝛼₁𝑢−𝑑. (4.11)

From above, it is clear that the condition which guarantees that Θ^𝑎≠∅, and thus that the conditions of Proposition 3.7, will be satisfied, is given by 𝑢>1+𝑟_𝑓+𝛼₁ and 𝑑<1+𝑟_𝑓−𝛼₂.

Let us consider the following numerical example. Suppose that 𝑆₂(0)=5, 𝑢=4/3, 𝑑=8/9, 𝛼₁=𝛼₂=1/30, 𝑟_𝑓=1/9. For this case, we have 1+𝑟_𝑓+𝛼₁=103/90<𝑢=4/3, and 1+𝑟_𝑓−𝛼₂=97/90>𝑑=8/9, and the conditions of Proposition 3.7 will be verified. Let us consider the following option: 𝑋=max{𝑆(2)−5,0}. By applying the backward procedure described above, we obtain that the seller price for 𝑋 is 𝑉_𝑠(0)=1.3272, with the following hedging strategy: 𝐻⁺₁(0)=0, 𝐻⁻₁(0)=2.796, 𝐻⁺₂(0)=0.8246, 𝐻⁻₂(0)=0, and for the case in which the risky security goes up, 𝐻⁺₁(1)=0, 𝐻⁻₂(1)=3.932, 𝐻⁺₂(1)=1.0, 𝐻⁻₂(1)=0, 𝑉(1)=2.2977, while for the case in which it goes down, 𝐻⁺₁(0)=0, 𝐻⁻₁(0)=1.4563, 𝐻⁺₂(0)=0.4687, 𝐻⁻₂(0)=0, and 𝑉(1)=0.4652.

By repeating the procedure now for −𝑋, we obtain that the buyer price for 𝑋 is 𝑉_𝑏(0)=0.9355, with the following hedging strategy: 𝐻⁺₁(0)=2.6926, 𝐻⁻₁(0)=0, 𝐻⁺₂(0)=0, 𝐻⁻₂(0)=0.7256, and for the case in which the risky security goes up, 𝐻⁺₁(1)=4.23, 𝐻⁻₁(1)=0, 𝐻⁺₂(1)=0, 𝐻⁻₂(1)=1.0, 𝑉(1)=1.9667, while for the case in which it goes down, 𝐻⁺₁(0)=1.5562, 𝐻⁻₁(0)=0, 𝐻⁺₂(0)=0, 𝐻⁻₂(0)=0.4688, and 𝑉(1)=0.3542. As expected, 𝑉_𝑏(0)=0.9355<𝑉_𝑠(0)=1.3272.

5. Conclusions

In this paper, we study a discrete time with infinite sample space financial model with penalty costs on short selling positions. Unlike previous works, we consider only one price structure for both short and long positions, with the penalties being withdrawals from the portfolio proportional to the short selling position. Our main result, Theorem 3.1, provides necessary and sufficient conditions for the nonexistence of arbitrages and can be seen as an extension of the standard fundamental theorem of asset pricing. When the penalty costs go to zero our result reduces to that presented in [30]. We also present a sufficient condition for a self-financing strategy to consistently replicate a contingent claim. For the finite-sample space case, this result leads to an explicit and constructive procedure for obtaining perfect hedging strategies. Some examples are presented to illustrate the possible applications of the model.

Appendices

We present in this appendix the proof of the main results in Section 3. First, we need some auxiliary results, presented next. In what follows, we recall that ℒ^𝑑₀(𝒢) represents the space of 𝒢 -measurable random vectors with values in ℝ^𝑑, ℒ₀^𝑑+(𝒢)⊂ℒ⁺₀(𝒢), the space of 𝒢 -measurable random vectors 𝑍 such that 𝐏(𝑍≥0) and, for simplicity, ℒ₀(𝒢)=ℒ¹₀(𝒢), ℒ⁺₀(𝒢)=ℒ₀¹⁺(𝒢). The definition for ℒ^𝑑₀(ℱ), ℒ₀^𝑑+(ℱ) and ℒ₀(ℱ), ℒ⁺₀(ℱ) is similar.

A. Some Auxiliary Results

Let𝑌∈ℒ^𝑑₀(𝒢), ∈ℒ^𝑑₀(ℱ), and 𝐷∈ℒ₀^𝑑+(ℱ). We set𝐾_𝑌,𝑋=(𝛼⋅𝑌,𝛼⋅𝑋);𝛼∈ℒ₀^𝑑+(𝒢,𝑁_{𝑌,𝑋,𝐷})=𝛽⋅𝑌,𝛽⋅(𝑋+𝐷);𝛽∈ℒ₀^𝑑+(𝒢,𝐽_{𝑌,𝑋,𝐷})=𝐾_𝑌,𝑋−𝑁_{𝑌,𝑋,𝐷},𝐴_{𝑌,𝑋,𝐷}=𝐽_{𝑌,𝑋,𝐷}−{0}×ℒ⁺₀(ℱ). (A.1)(A.2)(A.3)(A.4)

The following propositions will be crucial for the developing of our results and are based on the arguments presented in [47, 48, 52].

Proposition A.1. The following statements are equivalent: (i)𝐴_{𝑌,𝑋,𝐷}∩{{0}×ℒ⁺₀(ℱ)}=(0,0), (ii)𝐴_{𝑌,𝑋,𝐷}∩{{0}×ℒ⁺₀(ℱ)}=(0,0) and 𝐴_{𝑌,𝑋,𝐷} is closed.

Proof. We have to show that (i) implies that 𝐴_{𝑌,𝑋,𝐷} is closed. For this, we consider sequences {𝛼^𝑛}, {𝛽^𝑛} in ℒ₀^𝑑+(𝒢), {𝜌^𝑛} in ℒ⁺₀(ℱ), and 𝜒∈ℒ₀(𝒢), 𝜓∈ℒ₀(ℱ) such that a.s., lim_𝑛→∞𝛼^𝑛−𝛽^𝑛⋅𝑌=𝜒,lim_𝑛→∞𝛼^𝑛−𝛽^𝑛⋅𝑋−𝛽^𝑛⋅𝐷−𝜌^𝑛=𝜓. (A.5) If we can find 𝛼 and 𝛽 in ℒ₀^𝑑+(𝒢) such that 𝐏-a.s. 𝛼−𝛽⋅𝑌=𝜒, (A.6)𝛼−𝛽⋅𝑋−𝛽⋅𝐷≥𝜓, (A.7) then the result is proved since in this case, setting 𝜌=(𝛼−𝛽)⋅𝑋−𝛽⋅𝐷−𝜓, we have from (A.7) that 𝜌∈ℒ⁺₀(ℱ) and 𝜓=(𝛼−𝛽)⋅𝑋−𝛽⋅𝐷−𝜌, thus (𝜒,𝜓)∈𝐴_{𝑌,𝑋,𝐷}. We set Ω₀∈ℱ such that the limits (A.5) hold and {𝛼^𝑛}, {𝛽^𝑛}, {𝜌^𝑛}, 𝐷 are always nonnegative. It is easy to see that (Ω₀)=1.
We define next 𝛼^𝑛=(𝛼^𝑛−𝛽^𝑛)⁺ and 𝛽^𝑛=(𝛼^𝑛−𝛽^𝑛)⁻ so that 𝛼^𝑛−𝛽^𝑛=𝛼^𝑛−𝛽^𝑛, 𝛼^𝑛_𝑖𝛽^𝑛_𝑖=0, 𝑖=1,…,𝑑, ‖𝛼^𝑛−𝛽^𝑛‖²=‖𝛼^𝑛‖²+‖𝛽^𝑛‖², and on Ω₀ that 0≤𝛼^𝑛≤𝛼^𝑛,0≤𝛽^𝑛≤𝛽^𝑛,𝛼^𝑛−𝛽^𝑛⋅𝑋−𝛽^𝑛⋅𝐷≥𝛼^𝑛−𝛽^𝑛⋅𝑋−𝛽^𝑛⋅𝐷. (A.8) Set 𝜍=liminf_𝑛→∞‖𝛼^𝑛−𝛽^𝑛‖=liminf_𝑛→∞(‖𝛼^𝑛‖²+‖𝛽^𝑛‖²)^1/2 and Ω₁={𝜔∈Ω₀;𝜍(𝜔)<∞}. From Lemma 2 of [52], we can find subsequences {𝛼^𝑘}, {𝛽^𝑘} of, respectively, {𝛼^𝑛}, {𝛽^𝑛} such that on Ω₁, lim_𝑘→∞𝛼^𝑘=𝛼 and lim_𝑘→∞𝛽^𝑘=𝛽 for some 𝛼, 𝛽 in ℒ₀^𝑑+(𝒢). Set {𝜌^𝑘} the corresponding subsequence of {𝜌^𝑛}. It follows that on Ω₁, lim_𝑘→∞(𝛼^𝑘−𝛽^𝑘)⋅𝑌=(𝛼−𝛽)⋅𝑌=𝜒 from (A.5), and (20), 𝛼−𝛽⋅𝑋−𝛽⋅𝐷=lim_𝑘→∞𝛼^𝑘−𝛽^𝑘⋅𝑋−𝛽^𝑘⋅𝐷≥liminf_𝑘→∞𝛼^𝑘−𝛽^𝑘⋅𝑋−𝛽^𝑘⋅𝐷−𝜌^𝑘≥𝜓. (A.9) If 𝐏(Ω₁)=1, then from (A.6) and (A.7) the result is proved. Otherwise, we define Ω₂={𝜔∈Ω₀;𝜍(𝜔)=∞}. As in [47, 48], we form partitions of Ω, and argue on each separate partition as an autonomous space, considering the appropriate restrictions of the random vectors and traces of the 𝜎 -algebras.
On Ω₂, we define 𝑔^𝑛=‖𝛼^𝑛−𝛽^𝑛‖^⊕𝛼^𝑛, 𝑓^𝑛=‖𝛼^𝑛−𝛽^𝑛‖^⊕𝛽^𝑛, and 𝜈^𝑛=‖𝛼^𝑛−𝛽^𝑛‖^⊕𝜌^𝑛. From (A.5), (A.8), it follows that on Ω₂, lim_𝑛→∞𝑔^𝑛−𝑓^𝑛⋅𝑌=0,liminf_𝑛→∞𝑔^𝑛−𝑓^𝑛⋅𝑋−𝑓^𝑛⋅𝐷−𝜈^𝑛≥lim_𝑛→∞𝛼^𝑛−𝛽^𝑛⊕𝛼^𝑛−𝛽^𝑛⋅𝑋−𝛽^𝑛⋅𝐷−𝜌^𝑛=0. (A.10) Since on Ω₂, ‖𝑔^𝑛‖≤1 and ‖𝑓^𝑛‖≤1, we have again from Lemma 2 of [52] that we can find convergent subsequences {𝑔^𝑘}, {𝑓^𝑘} of, respectively, {𝑔^𝑛}, {𝑓^𝑛}, with limits, respectively, 𝑔≥0 and 𝑓≥0. We denote by {𝛼^𝑘}, {𝛽^𝑘}, {𝜌^𝑘} the corresponding subsequences of {𝛼^𝑛}, {𝛽^𝑛}, {𝜌^𝑛}, obtained from the association with {𝑔^𝑛} and {𝑓^𝑛}. Since for each 𝑖=1,…,𝑑, 𝑔^𝑛_𝑖𝑓^𝑛_𝑖=0, it follows that for each 𝑖=1,…,𝑑, 𝑔_𝑖𝑓_𝑖=0. (A.11) From (A.10) we have, that on Ω₂, 𝑔−𝑓⋅𝑌=lim_𝑘→∞𝑔^𝑘−𝑓^𝑘𝑓⋅𝑌=0,𝑔−⋅𝑋−𝑓⋅𝐷=lim_𝑘→∞𝑔^𝑘−𝑓^𝑘⋅𝑋−𝑓^𝑘⋅𝐷≥0, (A.12)(A.13) and from (i), it follows that (A.12) and (A.13) imply that 𝑔−𝑓⋅𝑋−𝑓⋅𝐷=0. (A.14) We also have on Ω₂ that 1=𝑔−𝑓²=𝑔²+𝑓². (A.15) We proceed now by applying induction on 𝑑. Suppose first that 𝑑=1. We can find a partition of Ω₂ into 2 disjoint sets, defined by Ω₂𝑔={𝜔∈Ω₂;𝑔(𝜔)>0}, and Ω𝑓₂={𝜔∈Ω₂;𝑓(𝜔)>0}. From (A.11) and (A.15) we have that indeed Ω₂𝑔 and Ω𝑓₂ form a disjoint partition of Ω₂. From (A.12) and (A.14) we have that on Ω₂𝑔 (recalling that in this case 𝑓=0), 𝑌=0 and 𝑋=0, and that lim_𝑘→∞𝛽^𝑘=0 (since 𝛼^𝑘𝛽^𝑘=0 and 𝑔>0). This implies that on Ω₂𝑔, lim_𝑘→∞(𝛼^𝑘−𝛽^𝑘)𝑌=0 and as in (A.9), 0=lim_𝑘→∞((𝛼^𝑘−𝛽^𝑘)𝑋−𝛽^𝑘𝐷)≥liminf_𝑘→∞((𝛼^𝑘−𝛽^𝑘)𝑋−𝛽^𝑘𝐷−𝜌^𝑘)≥𝜓, and (A.6), (A.7) hold with 𝛼=0, 𝛽=0. Similarly, on Ω𝑓₂, 𝑌=0, 𝑋+𝐷=0, and lim_𝑘→∞𝛼^𝑘=0, so that again (A.6), (A.7) hold with 𝛼=0, 𝛽=0. This completes the proof for 𝑑=1.
Suppose now that the equivalence between (i) and (ii) holds for 𝑑−1, and that (i) holds for. Define the 2𝑑 disjoint sets Ω₂𝑔_𝑖=𝜔∈Ω₂;𝑔_𝑗(𝜔)=0,𝑓_𝑗(𝜔)=0,𝑗≤𝑖−1,𝑔_𝑖(𝜔),Ω𝑓>0_2𝑖=𝜔∈Ω₂;𝑔_𝑗(𝜔)=0,𝑓_𝑗(𝜔)=0,𝑗≤𝑖−1,𝑔_𝑖(𝜔)=0,𝑓_𝑖(𝜔)>0. (A.16) From (A.11) and (A.15), we have that indeed Ω₂𝑔_𝑖 and Ω𝑓_2𝑖, 𝑖=1,…,𝑑, form a disjoint partition of Ω₂. For 𝑖 fixed, we will consider first a disjoint partition of Ω₂𝑔_𝑖. Consider all subsets, indexed by 𝑠, formed as 𝑣^𝑖_𝑠⊆{𝑖+1,…,𝑑}, 𝑢^𝑖_𝑠⊆{𝑖+1,…,𝑑}, with 𝑣^𝑖_𝑠∩𝑢^𝑖_𝑠=∅. Write 𝑤^𝑖_𝑠={𝑖+1,…,𝑑}−(𝑣^𝑖_𝑠∪𝑢^𝑖_𝑠), and consider a disjoint partition of Ω₂𝑔_𝑖 formed by the sets: Ω𝑔_{2𝑖𝑣𝑖𝑠𝑢𝑖𝑠}=𝜔∈Ω₂𝑔_𝑖;𝑔_𝑗(𝜔)>0,𝑗∈𝑣^𝑖_𝑠,𝑓_𝑟(𝜔)>0,𝑟∈𝑢^𝑖_𝑠,𝑔_𝑚(𝜔)=0,𝑓_𝑚(𝜔)=0,𝑚∈𝑤^𝑖_𝑠 (A.17) We fix now 𝑣^𝑖_𝑠, 𝑢^𝑖_𝑠 and for notational simplicity, we write 𝑣=𝑣^𝑖_𝑠, 𝑢=𝑢^𝑖_𝑠, 𝑤=𝑤^𝑖_𝑠, Ω^′=Ω₂𝑔_{𝑖𝑣𝑖𝑠𝑢𝑖𝑠}, and ℱ^′, 𝒢^′, respectively, the corresponding trace of the 𝜎 -algebras ℱ, 𝒢 on Ω^′. Let us consider that 𝐏(Ω^′)>0 (otherwise, it could be discarded) and write 𝐏^′(⋅)=𝐏(⋅)/𝐏(Ω^′). On the set Ω^′, we have from (A.12) and (A.14) that 𝑌_𝑖1=−𝑔_{𝑖𝑗∈𝑣}𝑔_𝑗𝑌_𝑗−_𝑗∈𝑢𝑓_𝑗𝑌_𝑗𝑋_𝑖,1=−𝑔_{𝑖𝑗∈𝑣}𝑔_𝑗𝑋_𝑗−_𝑗∈𝑢𝑓_𝑗𝑋_𝑗+𝐷_𝑗. (A.18)(A.19) Set the 𝑑−1-dimensional random vectors 𝑌^′, 𝑋^′, 𝐷^′ as follows: for 𝑗=1,…,𝑖−1, 𝑌^′_𝑗=𝑌_𝑗, 𝑋^′_𝑗=𝑋_𝑗 and 𝐷^′_𝑗=𝐷_𝑗, for 𝑗=𝑖+1,…,𝑑, 𝑌^′_𝑗−1=𝑌_𝑗, and 𝑋^′_𝑗−1=𝑋_𝑗+𝐷_𝑗𝑋,𝑗∈𝑢,_𝑗𝐷,𝑗∉𝑢,𝑗≥𝑖+1,_′𝑗−1=𝐷0,𝑗∈𝑢∪𝑣,_𝑗,𝑗∉𝑢∪𝑣,𝑗≥𝑖+1. (A.20) For ℓ=1,2,…, define 𝜏^ℓ=inf𝑘;𝑔^𝑘_𝑗>𝑔_𝑗11−ℓ,∀𝑗∈𝑣∪{𝑖},𝑓^𝑘_𝑟>𝑔_𝑟11−ℓ,∀𝑟∈𝑢. (A.21) On Ω^′, we have that 𝜏^ℓ<∞, 𝑓^𝜏ℓ_𝑗=0 for all 𝑗∈𝑣∪{𝑖} and 𝑔^𝜏ℓ_𝑟=0 for all 𝑟∈𝑢, and consequently 𝛽^𝜏ℓ_𝑗=0 for all 𝑗∈𝑣∪{𝑖} and 𝛼^𝜏ℓ_𝑟=0 for all 𝑟∈𝑢. Define 𝛼^ℓ=𝛼^𝜏ℓ, 𝛽^ℓ=𝛽^𝜏ℓ, and 𝜌^ℓ=𝜌^𝜏ℓ. From (A.19), we obtain that 𝛼^ℓ−𝛽^ℓ⋅𝑋−𝛽^ℓ⋅𝐷=𝛿^ℓ−𝜀^ℓ⋅𝑋^′−𝜀^ℓ⋅𝐷^′, (A.22) where for 𝑗=1,…,𝑖−1, 𝛿^ℓ_𝑗=𝛼^ℓ_𝑗, 𝜀^ℓ_𝑗=𝛽^ℓ_𝑗, for 𝑗∈𝑤, 𝛿^ℓ_𝑗−1=𝛼^ℓ_𝑗, 𝜀^ℓ_𝑗−1=𝛽^ℓ_𝑗, for 𝑗∈𝑣, 𝛿^ℓ_𝑗−1=(𝛼^ℓ_𝑗−(𝑔_𝑗/𝑔_𝑖)𝛼^ℓ_𝑖)⁺, 𝜀^ℓ_𝑗−1=(𝛼^ℓ_𝑗−(𝑔_𝑗/𝑔_𝑖)𝛼^ℓ_𝑖)⁻, and for 𝑗∈𝑢, 𝛿^ℓ_𝑗−1=(−𝛽^ℓ_𝑗+(𝑓_𝑗/𝑔_𝑖)𝛼^ℓ_𝑖)⁺, 𝜀^ℓ_𝑗−1=(−𝛽^ℓ_𝑗+(𝑓_𝑗/𝑔_𝑖)𝛼^ℓ_𝑖)⁻. We notice that 𝛿^ℓ and 𝜀^ℓ belong to ℒ₀^𝑑−1+(Ω^′,𝒢^′,𝐏^′). Similarly, from (A.18), we have that 𝛼^ℓ−𝛽^ℓ⋅𝑌=𝛿^ℓ−𝜀^ℓ⋅𝑌^′. (A.23) Next, we show that 𝐽_{𝑌′,𝑋′,𝐷′}⊂𝐽_{𝑌,𝑋,𝐷}. For this, we establish a mapping (𝛼,𝛽)=𝐹(𝛿,𝜀) such that 𝛼 and 𝛽 belong to ℒ₀^𝑑+(Ω^′,𝒢^′,𝐏^′) whenever 𝛿 and 𝜀 belong to ℒ₀^𝑑−1+(Ω^′,𝒢^′,𝐏^′), and that (𝛿−𝜀)⋅𝑌^′=()𝛼−𝛽⋅𝑌,𝛿−𝜀⋅𝑋^′−𝜀⋅𝐷^′=𝛼−𝛽⋅𝑋−𝛽⋅𝐷. (A.24)(A.25) Indeed, setting𝛽_𝑖=0𝛼_𝑖=max0,𝜀_𝑗−1−𝛿_𝑗−1𝑔_𝑖𝑔_𝑗for𝑗∈𝑣,𝛿_𝑟−1−𝜀_𝑟−1𝑔_𝑖𝑓_𝑟for𝑟∈𝑢, (A.26) and for 𝑗=1,…,𝑖−1, 𝛼_𝑗=𝛿_𝑗, 𝛽_𝑗=𝜀_𝑗, for 𝑗∈𝑤, 𝛼_𝑗=𝛿_𝑗−1, 𝛽_𝑗=𝜀_𝑗−1, for 𝑗∈𝑣, 𝛼_𝑗=(𝛿_𝑗−1−𝜀_𝑗−1)+(𝑔_𝑗/𝑔_𝑖)𝛼_𝑖, 𝛽_𝑗=0, and for 𝑗∈𝑢, 𝛼_𝑗=0, 𝛽_𝑗=−(𝛿_𝑗−1−𝜀_𝑗−1)+(𝑓_𝑗/𝑔_𝑖)𝛼_𝑖, we get from (A.26) that 𝛼, 𝛽 belong to ℒ₀^𝑑+(Ω^′,𝒢^′,𝐏^′) whenever 𝛿 and 𝜀 belong to ℒ₀^𝑑−1+(Ω^′,𝒢^′,𝐏^′), and from (A.18) and (A.19), we get that (A.24) and (A.25) are satisfied, yielding the desired inclusion. From this and (i), we can conclude that 𝐴_{𝑌′,𝑋′,𝐷′}∩{{0}×ℒ⁺₀(Ω^′,ℱ^′,𝐏^′)}=(0,0) and by the induction hypothesis, we get that 𝐴_{𝑌′,𝑋′,𝐷′} is closed. Therefore, for some 𝛿 and 𝜀̃ belonging to ℒ₀^𝑑−1+(Ω^′,𝒢^′,𝐏^′), and some 𝑝∈ℒ⁺₀(Ω^′,ℱ^′,𝐏^′), we have from (A.22) and (A.23), and taking (𝛼,𝛽)=𝐹(𝛿,𝜀)̃, that 𝜒=lim_ℓ→∞𝛼^ℓ−𝛽^ℓ⋅𝑌=lim_ℓ→∞𝛿^ℓ−𝜀^ℓ⋅𝑌^′=(𝛿−𝜀)⋅𝑌^′=𝛿^ℓ−𝜀^ℓ̃⋅𝑌^′=𝛼−𝛽⋅𝑌,𝜓=lim_ℓ→∞𝛼^ℓ−𝛽^ℓ⋅𝑋−𝛽^ℓ⋅𝐷−𝜌^ℓ=lim_ℓ→∞𝛿^ℓ−𝜀^ℓ⋅𝑋^′−𝜀^ℓ⋅𝐷^′−𝜌^ℓ=𝛿−𝜀̃⋅𝑋^′−𝜀^ℓ̃⋅𝐷^′−𝑝=𝛼−𝛽⋅𝑋−𝛽⋅𝐷^′−𝑝, (A.27) showing that (A.6) and (A.7) are satisfied on Ω^′.
The proof for the sets Ω𝑓_{2𝑖𝑣𝑖𝑠𝑢𝑖𝑠} goes along the same lines, bearing in mind that (A.18) and (A.19) are replaced, respectively, by 𝑌_𝑖=1/𝑓_𝑖{∑_𝑗∈𝑣𝑔_𝑗𝑌_𝑗−∑_𝑗∈𝑢𝑓_𝑗𝑌_𝑗} and 𝑋_𝑖+𝐷_𝑖=(1/𝑓_𝑖){∑_𝑗∈𝑣𝑔_𝑗𝑋_𝑗−∑_𝑗∈𝑢𝑓_𝑗(𝑋_𝑗+𝐷_𝑗)}, and that the mapping 𝐹(𝛿,𝜀) is defined such that instead of (A.26), we take 𝛼_𝑖=0 and 𝛽_𝑖= max{0,{(𝜀_𝑗−1−𝛿_𝑗−1)(𝑓_𝑖/𝑔_𝑗) for 𝑗∈𝑣},{(𝛿_𝑟−1−𝜀_𝑟−1)(𝑓_𝑖/𝑓_𝑟) for 𝑟∈𝑢}}. This completes the proof of the proposition.

In the next proposition let us consider that 𝑌∈ℒ^𝑑₀(𝒢) is such that 𝑌₁=1 and 𝑋∈ℒ^d₀(ℱ) is such that 𝑋₁=0. Again we suppose that ∈ℒ₀^𝑑+(ℱ).

Proposition A.2. The following statements are equivalent: (i)for any 𝛼, 𝛽 in ℒ₀^𝑑+(𝒢), one has a.s. that ⟹𝛼−𝛽⋅𝑌≤0𝛼−𝛽⋅𝑋−𝛽⋅𝐷≥0𝛼−𝛽⋅𝑋−𝛽⋅𝐷=0. (A.28)(ii)there exists 𝑟∈ℒ⁺_∞(𝒢) and a probability measure 𝐐∼𝐏 such that 0<𝑅=𝑑𝐐/𝑑𝐏∈ℒ⁺_∞(ℱ), 𝑌,𝑋,𝐷 are integrable with respect to 𝐐, and 𝐸_𝐐(𝑋∣𝒢)≤𝑟𝑌≤𝐸_𝐐(𝑋+𝐷∣𝒢)𝑎.𝑠. (A.29)

Proof. First, we note that (i) is equivalent to 𝐴_{𝑌,𝑋,𝐷}∩{{0}×ℒ⁺₀(ℱ)}=(0,0). Indeed, if (i) holds, then clearly 𝐴_{𝑌,𝑋,𝐷}∩{{0}×ℒ⁺₀(ℱ)}=(0,0). Conversely, suppose that for some 𝛼, 𝛽 in ℒ₀^𝑑+(𝒢), (𝛼−𝛽)⋅𝑌≤0 and (𝛼−𝛽)⋅𝑋−𝛽⋅𝐷≥0 a.s. Then, by taking 𝛼₁=𝛼₁−(𝛼−𝛽)⋅𝑌≥0, 𝛼_𝑖=𝛼_𝑖, and recalling that 𝑌₁=1 and 𝑋₁=0 we get that (𝛼−𝛽)⋅𝑌=0 and (𝛼−𝛽)⋅𝑋−𝛽⋅𝐷=(𝛼−𝛽)⋅𝑋−𝛽⋅𝐷≥0 a.s., which implies that (𝛼−𝛽)⋅𝑋−𝛽⋅𝐷=0 a.s.
Let us show first that (ii) implies (i). Consider 𝛼, 𝛽 in ℒ₀^𝑑+(𝒢) such that (𝛼−𝛽)⋅𝑌=0 and (𝛼−𝛽)⋅𝑋−𝛽𝐷≥0 a.s. From (A.29), we get that a.s., 0≤𝐸_𝐐𝛼−𝛽⋅𝑋−𝛽⋅𝐷∣𝒢=𝛼⋅𝐸_𝐐(𝑋∣𝒢)−𝛽⋅𝐸_𝐐((𝑋+𝐷)∣𝒢)≤𝑟𝛼−𝛽⋅𝑌=0, (A.30) and thus 𝐸_𝐐((𝛼−𝛽)⋅𝑋−𝛽⋅𝐷)=0. From the fact that 𝐐∼𝐏, we get that (𝛼−𝛽)⋅𝑋−𝛽⋅𝐷≥0 a.s., which implies that (𝛼−𝛽)⋅𝑋−𝛽⋅𝐷=0 a.s. and again, from 𝐐∼𝐏, that (𝛼−𝛽)⋅𝑋−𝛽⋅𝐷=0 a.s., showing that (ii) implies (i).
Next, we show that (i) implies (ii). In what follows, we recall that for a complete probability space (Ω,𝒢,𝐏^′), we denote by ℒ^𝑑₁(Ω,𝒢,𝐏^′) the space of integrable 𝒢 -measurable random variables with values in ℝ^𝑑, and 𝑍∈ℒ₁^𝑑+(Ω,𝒢,𝐏^′) if 𝑍∈ℒ^𝑑₁(Ω,𝒢,𝐏^′) and 𝐏^′(𝑍≥0)=1. We remind that if 𝐏<<𝐏 then 𝐏∼𝐏 if and only if 𝑑𝐏/𝑑𝐏>0 a.s., and that for any random variable 𝜂 there exists an equivalent probability measure 𝐏 with bounded density such that 𝜂 is integrable under 𝐏 (see [52]). Consider a change of probability measure 𝑑𝐏^′=𝑝𝑑𝐏 with 0<𝑝∈ℒ_∞(ℱ) such that 𝑌∈ℒ^𝑑₁(Ω,𝒢,𝐏^′), 𝑋∈ℒ^𝑑₁(Ω,ℱ,𝐏^′), and 𝐷∈ℒ₁^𝑑+(Ω,ℱ,𝐏^′). From (i) (which is invariant under equivalent change of probability) and Proposition A.1, we get that 𝑉∶=𝐴_{𝑌,𝑋,𝐷}∩(ℒ₁(Ω,𝒢,𝐏^′)×ℒ₁(Ω,ℱ,𝐏^′)) is a closed convex set of ℒ₁(Ω,𝒢,𝐏^′)×ℒ₁(Ω,ℱ,𝐏^′), and that 𝑉∩{0}×ℒ⁺₁(Ω,ℱ,𝐏^′)=(0,0). Therefore, forany𝐴∈ℱ, with (𝐴)>0, (0,1_𝐴)∈{0}×ℒ⁺₁(Ω,ℱ,𝐏^′), and thus does not belong to the set 𝑉. By the Hahn-Banach Theorem, (0,1_𝐴) can be strongly separated from 𝑉 by a nonzero linear continuous functional, so that there exists 𝑟_𝐴∈ℒ_∞(𝒢), 𝑍_𝐴∈ℒ_∞(ℱ), (𝑟_𝐴,𝑍_𝐴)≠(0,0), such that sup_{(𝜑,𝜃)∈𝑉}𝐸^′𝑟_𝐴𝜑+𝑍_𝐴𝜃<𝐸^′𝑍_𝐴1_𝐴, (A.31) where 𝐸^′ denotes the expectation with respect to 𝐏^′. We note that 𝜑∈ℒ₁(Ω,𝒢,𝐏^′) and 𝜃∈ℒ₁(Ω,ℱ,𝐏^′) are such that 𝜑=(𝛼−𝛽)⋅𝑌and𝜃=(𝛼−𝛽)⋅𝑋−𝛽⋅𝐷−𝜌, with, 𝛽∈ℒ₀^𝑑+(𝒢) and 𝜌∈ℒ⁺₀(𝒢). By taking 𝜑=0 and 𝜃=−𝑛1_𝐵 for any 𝐵∈ℱ and positive integer 𝑛 (just take 𝛼=𝛽=0and𝜌=𝑛1_𝐵), we have from (A.31) that −𝑛𝐸^′(𝑍_𝐴1_𝐵)<𝐸^′(𝑍_𝐴1_𝐴) which implies that 𝑍_𝐴≥0 a.s. Normalizing, we assume that 𝑍_𝐴≤1. Similarly by taking 𝜑=𝑛1_𝐵 for any 𝐵∈𝒢 and 𝜃=0 (this is possible since 𝑌₁=1 and 𝑋₁=0, just take 𝛼₁=𝑛1_𝐵, 𝛼_𝑖=0 for 𝑖=2,…,𝑑,and𝛽=0,𝜌=0), we get from the same reasons as before that −𝑟_𝐴≥0. Considering now 𝜑=𝛼⋅𝑌 and 𝜃=𝛼⋅𝑋 we have from (A.31) and the same arguments as before that for every 𝛼∈ℒ_∞^𝑑+(𝒢), 𝐸^′(𝛼⋅(𝑟_𝐴𝑌+𝑍_𝐴𝑋))≤0, which implies that a.s., 𝐸^′𝑍_𝐴𝑋∣𝐺≤−𝑟_𝐴𝑌. (A.32) Similarly, considering now 𝜑=−𝛽⋅𝑌 and 𝜃=−𝛽⋅(𝑋+𝐷), we have from (A.31) and the same arguments as before that for every 𝛽∈ℒ_∞^𝑑+(𝒢), 𝐸^′(𝛽⋅(𝑟_𝐴𝑌+𝑍_𝐴(𝑋+𝐷)))≥0, which implies that a.s. 𝐸^′𝑍_𝐴(𝑋+𝐷)∣𝐺≥−𝑟_𝐴𝑌. (A.33) Consider the family of measures: 𝑄=𝑄_𝐴;𝑑𝑄_𝐴=𝑍_𝐴𝑑𝐏^′,∀𝐴∈ℱsuchthat𝐏^′(𝐴)>0. (A.34) Clearly, 𝒬 is dominated by 𝐏^′ (i.e., 𝑄_𝐴<<𝐏^′ for every 𝐴∈ℱ such that 𝐏^′(𝐴)>0). From the Halmos-Savage Theorem, 𝒬 contains a countable equivalent family 𝒬={𝑄_𝐴𝑘;𝑘∈𝕀}, where 𝕀 is a countable set. Define Γ={𝜔;𝑍_𝐴𝑘(𝜔)=0forevery𝑘∈𝕀}. Since 𝒬 and 𝒬 are equivalent and 𝑄_𝐴𝑘(Γ)=0 for every 𝑘∈𝕀, it follows that 𝑄_𝐴(Γ)=0 for every 𝐴∈ℱ such that 𝐏^′(𝐴)>0. We show next that 𝐏^′(Γ)=0. Suppose by contradiction that 𝐏^′(Γ)>0, so that 𝑄_Γ(Γ)=𝐸^′(𝑍_Γ1_Γ)=0. From (A.31), sup_{(𝜑,𝜃)∈𝑉}𝐸^′(𝑟_Γ𝜑+𝑍_Γ𝜃)<0 which is clearly an absurd (just take =0, 𝜃=0). This shows that 𝐏^′(Γ)=0. Define 𝑍=𝐶∑_𝑘∈𝕀𝑍_𝐴𝑘2^𝑘∈ℒ⁺_∞Ω,ℱ,𝐏^′,𝑟=𝐶∑_𝑘∈𝕀−𝑟_𝐴𝑘2^𝑘∈ℒ⁺_∞Ω,𝒢,𝐏^′, (A.35) where 𝐶=(∑_𝑘∈𝕀𝐸^′(𝑍_𝐴𝑘)/2^𝑘)⁻¹>0. Note that 𝐏^′(𝑍>0)=1 since 𝐏^′(Γ)=0, and thus 𝐶 is well defined. Define the probability measure 𝐐 as =𝑍𝑑𝐏^′. It is easy to see that 𝐐∼𝐏^′ (since 𝐏^′(𝑍>0)=1). Finally, from (A.32), (A.33), (A.35), and the bounded convergence theorem, we obtain (A.29) with 𝑅=𝑝𝑍, completing the proof of the proposition.

B. Proof of the Main Results

We present next the proof of Theorem 3.1.

Proof. Let us show that (i) implies (ii). The proof of this fact follows from the contrapositive. Suppose we can find 𝐻=(𝐻⁺𝐻⁻) with 𝐻⁺, 𝐻⁻∈ℒ₀^𝑑+(ℱ_𝑡) such that a.s., (𝐻⁺−𝐻⁻)⋅𝑆(𝑡)=0, ℛ=(𝐻⁺−𝐻⁻)⋅Δ𝑆(𝑡+1)−𝐻⁻⋅𝐷(𝑡+1)≥0 and 𝐏(Γ)>0, where Γ={𝜔;ℛ(𝜔)>0}. Then, we can define an arbitrage opportunity 𝐻 for the multiperiod market as follows: 𝐻(𝑠)=0 for 𝑠≤𝑡, 𝐻(𝑡+1)(𝜔)=𝐻 for 𝜔∈Γ, 𝐻(𝑡+1)(𝜔)=0 for 𝜔∉Γ, 𝐻⁺₁(𝑠)(𝜔)=ℛ(𝜔) for 𝜔∈Γ and 𝑠≥𝑡+2, and 𝐻⁺_𝑖(𝑠)(𝜔)=0, 𝐻⁻_𝑖(𝑠)(𝜔)=0 in all the other situations. Then, clearly 𝑉(𝑠)=0 for 𝑠=0,…,𝑡−1, (𝑡)=0, and from (2.5), (2.6), 𝑉(𝑠)(𝜔)=ℛ(𝜔) for 𝑠=𝑡+1,…,𝑇, 𝜔∈Γ, and equal to zero for 𝜔∉Γ, showing the result.
Let us show now that (ii) implies (iii). We will show by backward induction on time 𝑡=𝑇,…,1 that we can find random variables 0<𝑅(𝑡)∈ℒ⁺_∞(ℱ_𝑡), 0≤𝑟(̃𝑡−1)∈ℒ⁺_∞(ℱ_𝑡−1), and random vectors 𝑋(𝑡)∈ℒ^𝑑₀(ℱ_𝑡), 𝐷(𝑡)∈ℒ₀^𝑑+(ℱ_𝑡) that satisfy 𝐸(‖𝑋(𝑡)‖𝑅(𝑡))<∞, 𝐸(‖𝐷(𝑡)‖𝑅(𝑡))<∞, 𝐸(‖𝑆(𝑡−1)‖𝑅(𝑡))<∞, and a.s., 𝐸(𝑅(𝑡)∣ℱ_𝑡−1)=1, 𝐸𝑋(𝑡)𝑅(𝑡)∣ℱ_𝑡−1≤𝑟̃(𝑡−1)𝑆(𝑡−1)≤𝐸𝑅(𝑡)𝑋(𝑡)+𝐷(𝑡)∣ℱ_𝑡−1. (B.1) For 𝑡=𝑇, the result follows from Proposition A.2 with ℱ=ℱ_𝑇, 𝒢=ℱ_𝑇−1, 𝑌=𝑆(𝑇−1), 𝑋=𝑋(𝑇)=Δ𝑆(𝑇), 𝐷=𝐷(𝑇)=𝐷(𝑇), and 𝑅(𝑇)=𝑅, 𝑟(̃𝑇)=𝑟. Suppose the result holds for 𝑡+1. Define 𝑋(𝑡)=Δ𝑆(𝑡)𝐸(𝑅(𝑇)⋯𝑅(𝑡+1)∣ℱ_𝑡) and 𝐷(𝑡)=𝐷(𝑡)𝐸(𝑅(𝑇)⋯𝑅(𝑡+1)∣ℱ_𝑡). Again, the result follows from Proposition A.2 with ℱ=ℱ_𝑡, 𝒢=ℱ_𝑡−1, 𝑌=𝑆(𝑡−1), 𝑋=𝑋(𝑡), 𝐷=𝐷(𝑡), and 𝑅(𝑡)=𝑅, 𝑟(̃𝑡−1)=𝑟, completing the induction argument. Set 𝑅(0)=1, and define for 𝑡=1,…,𝑇, (𝑡−1)=𝑅(0)⋯𝑅(𝑡−1)𝑟(̃𝑡−1), 𝑑𝐐=𝑅(0)⋯𝑅(𝑇)𝑑𝐏. It follows that a.s., 𝐸_𝐐Δ𝑆(𝑡)∣ℱ_𝑡−1=𝑅(0)⋯𝑅(𝑡−1)𝐸Δ𝑆(𝑡)𝑅(𝑡)⋯𝑅(𝑇)∣ℱ_𝑡−1=𝑅(0)⋯𝑅(𝑡−1)𝐸Δ𝑆(𝑡)𝑅(𝑡)𝐸𝑅(𝑡+1)⋯𝑅(𝑇)∣ℱ_𝑡∣ℱ_𝑡−1=𝑅(0)⋯𝑅(𝑡−1)𝐸𝑅(𝑡)𝑋(𝑡)∣ℱ_𝑡−1≤𝑅(0)⋯𝑅(𝑡−1)𝑟̃(𝑡−1)𝑆(𝑡−1)=𝑟(𝑡−1)𝑆(𝑡−1), (B.2) and similarly, 𝐸_𝐐((Δ𝑆(𝑡)+𝐷(𝑡))∣ℱ_𝑡−1)≥𝑟(𝑡−1)𝑆(𝑡−1), showing (iii).
Finally, let us show that (iii) implies (i). Indeed, from (2.5), (2.6), and (3.2), we have a.s. that 𝐸_𝐐𝑉(𝑡+1)∣ℱ_𝑡=𝑉(𝑡)+𝐻⁺(𝑡+1)⋅𝐸_𝐐Δ𝑆(𝑡+1)∣ℱ_𝑡−𝐻⁻(𝑡+1)⋅𝐸_𝐐Δ𝑆(𝑡+1)+𝐷(𝑡+1)∣ℱ_𝑡≤(1+𝑟(𝑡))𝐻⁺(𝑡+1)−𝐻⁻()(𝑡)=((𝑡𝑉(𝑡).𝑡+1⋅𝑆1+𝑟)) (B.3) Suppose by contradiction that an arbitrage opportunity 𝐻 exists and (iii) holds. Let us denote by 𝑉=(𝑉(0),…,𝑉(𝑇)) the value process associated to the trading strategy𝐻. Then,𝑉(0)=0,𝑉(𝑇)≥0, and (𝑉(𝑇))>0. We have now by backward induction in time that 𝑉(𝑡)≥0 a.s. for all 𝑡=𝑇,…,0. Indeed, from the definition of an arbitrage, the result is clearly true for𝑡=𝑇. Suppose 𝑉(𝑡)≥0. From (B.3), it follows that 𝑉(𝑡)≥0(since𝑟(𝑡)≥0), showing the desired result. We show now by forward induction in time that, in fact, 𝑉(𝑡)=0 a.s. forall𝑡=0,…,𝑇. Indeed, for 𝑡=0, the result follows from the definition of an arbitrage. Suppose𝑉(𝑡)=0. From (B.3) and recalling that 𝑉(𝑡+1)≥0 we get that 𝐸_𝐐[𝑉(𝑡+1)∣ℱ_𝑡]=0. Taking the expected value, we obtain that 𝐸_𝐐[𝑉(𝑡+1)]=0, which shows that 𝑉(𝑡+1)=0 a.s., completing the induction argument. In particular, we have that 𝑉(𝑇)=0, in contradiction with the fact that 𝐸(𝑉(𝑇))>0, showing the desired result.

Next, we present the proof of Corollary 3.4.

Proof. By backward induction on𝑡, the result is true for 𝑡=𝑇 by assumption. Suppose it holds for𝑡+1, that is, a.s. 𝐸_𝐐𝑋(1+𝑟(𝑇))⋯(1+𝑟(𝑡+1))∣ℱ_𝑡+1≤𝑉(𝑡+1). (B.4) Then, from (B.3) and (B.4), we have a.s. that 𝐸_𝐐𝑋(1+𝑟(𝑇))⋯(1+𝑟(𝑡+1))∣ℱ_𝑡11+𝑟(𝑡)=𝐸_𝐐𝐸_𝐐𝑋(1+𝑟(𝑇))⋯(1+𝑟(𝑡+1))∣ℱ_𝑡+1∣ℱ_𝑡11+𝑟(𝑡)≤𝐸_𝐐𝑉(𝑡+1)∣ℱ_𝑡1(𝑡)(𝑡)1+𝑟≤𝑉 (B.5) showing (3.4). We apply backward induction on 𝑡 to show that 𝐏(Γ(𝑡))>0. For 𝑡=𝑇, the result is true by assumption. Suppose it holds for 𝑡+1. Thus we get that 𝐐(Γ(𝑡+1)∣ℱ_𝑡)>0 a.s.. Therefore, we have from (B.3) that a.s., 𝐸_𝐐((𝑡𝑉(𝑡)−𝑋1+𝑟))(1+𝑟(𝑇))⋯(1+𝑟(𝑡+1))∣ℱ_𝑡≥𝐸_𝐐𝑉(𝑡+1)−𝐸_𝐐𝑋(1+𝑟(𝑇))⋯(1+𝑟(𝑡+1))∣ℱ_𝑡+11_Γ(𝑡+1)∣ℱ_𝑡>0, (B.6) which implies, after dividing by (1+𝑟(𝑡) that 𝐸_𝐐(𝑉(𝑡)−𝑋/(1+𝑟(𝑇))⋯(1+𝑟(𝑡))∣ℱ_𝑡)>0 a.s., yielding the desired result.

In what follows, we present the proof of Propositions 3.6 and 3.7.

Proof. We take 𝑟(𝑡)=0and𝐐∈Θ^𝑎 and show that ({𝑟(𝑡)},𝐐)∈𝒦. Indeed, recalling that 𝐷(𝑡+1)≥0 we have a.s. that on {𝑎_𝑖(𝑡)=0}∶𝐸_𝐐Δ𝑆_𝑖(𝑡+1)∣ℱ_𝑡=0≤𝐸_𝐐𝐷_𝑖(𝑡+1)∣ℱ_𝑡=𝐸_𝐐Δ𝑆_𝑖(t+1)+𝐷_𝑖(𝑡+1)∣ℱ_𝑡 (B.7) and similarly on {𝑎_𝑖(𝑡)=1}, 𝐸_𝐐Δ𝑆_𝑖(𝑡+1)+𝐷_𝑖(𝑡+1)∣ℱ_𝑡=0≥𝐸_𝐐Δ𝑆_𝑖(𝑡+1)∣ℱ_𝑡. (B.8)

Proof. Consider 𝑟(𝑡)=0and𝐐∈Θ^𝑎𝐻. As shown in Proposition 3.6, ({𝑟(𝑡)},𝐐)∈𝒦. From the hypothesis that 𝐻⁺_𝑖(𝑡)𝐻⁻_𝑖(𝑡)=0 and (2.6), we have a.s. that 𝐸_𝐐𝑉^𝐻(𝑡)∣ℱ_𝑡−1=𝑉^𝐻(𝑡−1)+_{{𝑖∈{𝑎𝐻(𝑡−1)=0}}}𝐻⁺_𝑖(𝑡)𝐸_𝐐Δ𝑆_𝑖(𝑡)∣ℱ_𝑡−1−_{{𝑖∈{𝑎𝐻(𝑡−1)=1}}}𝐻⁻_𝑖(𝑡)𝐸_𝐐Δ𝑆_𝑖(𝑡)+𝐷_𝑖(𝑡)∣ℱ_𝑡−1=𝑉^𝐻(𝑡−1), (B.9) and thus a.s., 𝑉^𝐻(𝑡)=𝐸_𝐐(𝑋∣ℱ_𝑡) for all 𝑡=0,…,𝑇−1. From Corollary 3.4, we have the result.

Acknowledgment

The authors would like to express their gratitude to the referee for some helpful comments. O. L. V. Costa received financial support from CNPq (Brazilian National Research Council), Grant 301067/09-0, and USP Project MaCLinC.