# Arbitrage-Free Conditions and Hedging Strategies for Markets with

Authors: O. L. V. Costa; E. V. Queiroz Filho
Journal: Mathematical Problems in Engineering
 (2012)

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## 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.


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## Body

## 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. Section2 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 reald -dimensional vector space be denoted by ℝd  and for x∈ℝd  we will write xi  for the ith  component of the vector. The superscript will be omitted for the cased=1. Forx, y  in ℝd, we set
(2.1)x⋅y=∑i=1dxiyi.
We write x≥0  to denote that all components of x  are positive, that is, xi≥0  for i=1,…,d. For  x∈ℝd, we set the following vectors in ℝd: x+≥0  such that its ith  component xi+  is equal to xi  if xi≥0, zero otherwise, x-=(-x)+≥0  (therefore, x=x+-x-  and xi+xi-=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 bi. 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  {ℱt},t=0,1,…,T. For 𝒢  a sub-σ -algebra of ℱ,  we denote by ℒ0d(Ω,𝒢,P)  (or simply ℒ0d(𝒢) ) the space of 𝒢 -measurable random variables with values in ℝd, 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 ℒ0d(𝒢)  will converge a.s. For any probability measureQ, EQ(·)  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, 1A  denotes the indicator function of the set  A. Let ℒ0d+(𝒢)  be the set of random vectors X∈ℒ0d(𝒢)  such that P(X≥0)=1. The space of integrable random vectors in ℒ0d(𝒢)  will be denoted by ℒ1d(𝒢)  and the space of essentially bounded random vectors in ℒ0d(𝒢)  by ℒ∞d(𝒢).Consider given stochastic processesS={S(t);t=0,…,T}  and D={D(t);t=1,…,T}  taking values in ℝd  with S(t)∈ℒ0d(ℱt)  and S1(t)=1  for each  t=0,…,T, and D(t)∈ℒ0d+(ℱt)  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)∈ℒ0d+(ℱt-1)  and H-(t)∈ℒ0d+(ℱt-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 strategyH,  we have the value process VH:=(VH(0),…,VH(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
(2.2)V(0)=(H+(1)-H-(1))⋅S(0),
and at times t=1,…,T, as
(2.3)V(t)=(H+(t)-H-(t))⋅S(t)-H-(t)⋅D(t).
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:
(2.4)H-(t)⋅D(t).
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
(2.5)V(t)=(H+(t+1)-H-(t+1))⋅S(t).
From (2.2)–(2.5), we have for t=1,…,T  that
(2.6)V(t)=V(t-1)+(H+(t)-H-(t))⋅ΔS(t)-H-(t)⋅D(t).
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 strategyH,  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 Theorem3.1, 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 thatℒ∞+(ℱt)  represents the space of essentially bounded ℱt -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 strategyH, one has a.s. that
(3.1)(H+(t)-H-(t))⋅S(t-1)≤0,(H+(t)-H-(t))⋅ΔS(t)-H-(t)⋅D(t)≥0}⟹(H+(t)-H-(t))⋅ΔS(t)-H-(t)⋅D(t)=0,(iii)
there exists a stochastic process{r(t)}  with r(t)∈ℒ∞+(ℱt)  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,
(3.2)EQ(ΔS(t+1)∣Ft)≤r(t)S(t)≤EQ(ΔS(t+1)+D(t+1)∣Ft)a.s.Remark 3.2.
For the case in whichD(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 S̃(t). To write our problem in the above set-up, we would need to define the fictitious price processes  S¯(t), S̃(t)  as: S¯(0)=S̃(0)=S(0)  and for  t=1,…,T, S¯(t)=S(t), S̃(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), ΔS̃(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)·∑k=1tD(k)=(H+(t)-H-(t))·S(t)-H-(t)·∑k=1tD(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 S̃(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(3.3)K={({r(t)},Q);r(t)andQsatisfyingcondition(iii)ofTheorem3.1}.
We recall next that a contingent claim (random variable) X∈ℒ0(ℱT)  is marketable if for some self-financing strategy H  we have that a.s. X=VH(T)  and, in this case, H  is said to replicate X. We say that  H  superreplicates X  if a.s. we have that  VH(T)≥X. We have the following corollary (see the proof in the appendix).Corollary 3.4.
Suppose thatH  superreplicates X  and there is no arbitrage. Then, for any ({r(t)},Q)∈𝒦  one has a.s. that
(3.4)EQ(X(1+r(T))⋯(1+r(t))∣Ft)≤V(t).
Writing Γ(t)={ω;V(t)(ω)-EQ(X/(1+r(T))⋯(1+r(t))∣ℱt)(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 claimX∈ℒ0(ℱT)  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=0T-1;fort=0,…,T-1,a(t)∈ℒ0d+(ℱt),ai(t)=0or1}. For a self-financing strategy H  satisfying H+(t)·H-(t)=0, we set aH={aH(t)}∈𝕁  as aiH(t)=1{Hi-(t+1)>0}.Definition 3.5.
Fora={a(t)}∈𝕁,  set
(3.5)Θa={Q~P,dQdP∈L∞+(F);S(t),D(t)areintegrablewithrespecttoQ,fort=0,…,T,EQ(ΔSi(t+1)∣Ft)=0a.s.on⁡{ai(t)=0},andEQ(ΔSi(t+1)+Di(t+1)∣Ft)=0a.s.on⁡{ai(t)=1}dQdP}.
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 somea={a(t)}∈𝕁  one has Θa≠∅,  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.
IfH  is a self-financing strategy that replicates X  with H+(t)·H-(t)=0  for t=1,…,T  and ΘaH≠∅,  then for any superreplicating strategy Ĥ  for X  one has a.s. for t=0,1,…,T that VĤ(t)≥VH(t)  and if P(VĤ(T)>X)>0,  then (VĤ(t)>VH(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 claimX  satisfying the conditions of Proposition 3.7. We assume here that Ω={ω1,…,ωκ}  and that D(t)  is ℱt-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
(4.1)A1=(1S2(1)(ω1)…Sd(1)(ω1)⋮⋮⋱⋮1S2(1)(ωκ)…Sd(1)(ωκ)),A2=-(A1+D),
with D=ed  (recall that e  is the vector formed by 1  in all components), d=(g1…gd). Let the vector x∈ℝκ  be such that  xj=X(ωj), j=1,…,κ. We have that X  is marketable if and only if there exists H+,H-  that satisfy the system:
(4.2)A(H+H-)=A1(H+-H-)-DH-=x,(4.3)H+≥0,H-≥0.
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
(4.4)(H+-H-)-A1-1edH-=(H+-H-)-b1dH-=A1-1x.
Set y=A1-1x. Let us obtain H  that satisfies (4.2), (4.3) with H+·H-=0. Define fori=2,…,d: Hi+=yi, Hi-=0  if yi≥0, otherwise, Hi-=-yi, Hi+=0. For i=1,  calculate
(4.5)z=y1+∑i=2dgiHi-.
In order to have (4.4) satisfied, we must have (H1+-H1-)-g1H1-=z. If z≥0,  then set H1+=z, H1-=0, otherwise, set H1+=0  and H1-=-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 set𝕁  is finite, and thus if we assume that for every a∈𝕁, Θa≠∅,  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 Vs(0), is obtained by applying to X  backwards in time the algorithm presented above. The buyer price, denoted by Vb(0), is obtained by applying the backward algorithm to -X, and taking Vb(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(4.6)S2(t)=1B(t)uN(t)dt-N(t)S2(0),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+rf)t, t=0,1,…,T. The penalty costs are assumed to be of the form:
(4.7)D1(t)=α111+rf,D2(t)=α2S2(t-1)1+rf.
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 Θa={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,
(4.8)π1=1+rf-du-d,π2=u-(1+rf)u-d;(ii)
a=(1,0); in this case,
(4.9)π1=1+rf-α1-du-d,π2=u-(1+rf-α1)u-d;(iii)
a=(0,1); in this case,
(4.10)π1=1+rf+α0-du-d,π2=u-(1+rf+α0)u-d;(iv)
a=(1,1); in this case,
(4.11)π1=1+rf+α0-α1-du-d,π2=u-(1+rf+α0-α1)u-d.From above, it is clear that the condition which guarantees thatΘa≠∅, and thus that the conditions of Proposition 3.7, will be satisfied, is given by  u>1+rf+α1  and d<1+rf-α2.Let us consider the following numerical example. Suppose thatS2(0)=5, u=4/3, d=8/9, α1=α2=1/30, rf=1/9. For this case, we have 1+rf+α1=103/90<u=4/3, and 1+rf-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 Vs(0)=1.3272, with the following hedging strategy: H1+(0)=0, H1-(0)=2.796, H2+(0)=0.8246, H2-(0)=0, and for the case in which the risky security goes up, H1+(1)=0, H2-(1)=3.932, H2+(1)=1.0, H2-(1)=0, V(1)=2.2977, while for the case in which it goes down, H1+(0)=0, H1-(0)=1.4563, H2+(0)=0.4687, H2-(0)=0,  and V(1)=0.4652.By repeating the procedure now for-X,  we obtain that the buyer price for X  is Vb(0)=0.9355, with the following hedging strategy: H1+(0)=2.6926, H1-(0)=0, H2+(0)=0, H2-(0)=0.7256, and for the case in which the risky security goes up, H1+(1)=4.23, H1-(1)=0, H2+(1)=0, H2-(1)=1.0, V(1)=1.9667, while for the case in which it goes down, H1+(0)=1.5562, H1-(0)=0, H2+(0)=0, H2-(0)=0.4688,  and V(1)=0.3542. As expected,  Vb(0)=0.9355<Vs(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, Theorem3.1, 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.



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