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Mathematical Problems in Engineering
Volume 2012 (2012), Article ID 937324, 20 pages
Arbitrage-Free Conditions and Hedging Strategies for Markets with Penalty Costs on Short Positions
Departamento de Engenharia de Telecomunicações e Controle, Escola Politécnica da Universidade de São Paulo, 05508-900 São Paulo, SP, Brazil
Received 25 October 2011; Accepted 13 December 2011
Academic Editor: Weihai Zhang
Copyright © 2012 O. L. V. Costa and E. V. Queiroz Filho. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.
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.
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  (see also ). 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  and later extended in several directions see, for instance, [10–25]. The pioneering work of Black and Scholes  and Merton  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  and Harrison and Pliska , 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., ). 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  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  for the case in which the price process is positive and under a nondegeneracy hypothesis on the price process. In , 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  to be equivalent to the existence of a positive interest rate process and an equivalent probability measure 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 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 . 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 -dimensional vector space be denoted by and for we will write for the component of the vector. The superscript will be omitted for the case. For, in , we set We write to denote that all components of are positive, that is, for . For , we set the following vectors in : such that its component is equal to if , zero otherwise, (therefore, and for each ). The vector formed by 1 in all components will be represented by , and the vector with 1 at the component and 0 elsewhere by . For a real number , we define if , zero otherwise.
Let be a complete probability space equipped with a filtration . For a sub- -algebra of , we denote by (or simply ) the space of -measurable random variables with values in , 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 will converge a.s. For any probability measure, denotes the expectation with respect to , and we write whenever the probability measure is equivalent to (absolutely continuous with respect to) . For any , denotes the indicator function of the set . Let be the set of random vectors such that . The space of integrable random vectors in will be denoted by and the space of essentially bounded random vectors in by .
Consider given stochastic processes and taking values in with and for each , and for each . We define for ,. A trading strategy is defined such that each is a -dimensional random matrix with columns and . It describes an investor's portfolio as carried forward from time to time . In the model of a security market, describes the evolution of the prices of securities, represents the number of units of each security hold in a long position from time to time , represents the number of units of each security hold in a short selling position from time to time , and the evolution of the penalty costs and possible spread costs between borrowing and lending rates due to short selling positions on each security .
Associated to a trading strategy , we have the value process describing the total value of the portfolio at each time . For notational simplicity, we will omit the superscript whenever no confusion arises. The portfolio value can be written, at time , as and at times , as The quantity represents the value of the portfolio at time 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: The value of the portfolio at time just after the change of ownership positions is . We consider in this paper self-financing trading strategies, so that no money is added or withdrawn from the portfolio between times to time . 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 From (2.2)–(2.5), we have for that 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 , we have a.s. that (i), (ii), and (iii).
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 . 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 represents the space of essentially bounded -measurable random variables such that .
Theorem 3.1. The following statements are equivalent:(i)there are no arbitrage opportunities,(ii)for any self-financing strategy , one has a.s. that (iii)there exists a stochastic process with for each and a probability measure such that , are integrable with respect to and for each ,
Remark 3.2. For the case in which , our results reduce to the well-known fundamental theorem of asset pricing with finite-discrete time and infinite state space, see  (recall that , and if , then (3.2) implies that ).
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 , and those that can only be held in negative amounts and represented by . To write our problem in the above set-up, we would need to define the fictitious price processes , as: and for , , . Notice that there is no discounting to be applied since we are considering that . By doing this, we would have that for , , and this would yield that , which is similar to (2.6). Note, however, that the models are different since we cannot guarantee that a self-financing strategy 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 , while for our model it would be . This also occurs with the nonarbitrage conditions of the two models. Indeed, the nonarbitrage condition presented in [45, 47] states that is a supermartingale and is a submartingale, which would involve the sum of the terms . On the other hand, for our model, the nonarbitrage condition (3.2) involves only the state price , and penalty costs emphasizing the difference between the two models.
In what follows, we define We recall next that a contingent claim (random variable) is marketable if for some self-financing strategy we have that a.s. and, in this case, is said to replicate . We say that superreplicates if a.s. we have that . We have the following corollary (see the proof in the appendix).
Corollary 3.4. Suppose that superreplicates and there is no arbitrage. Then, for any one has a.s. that Writing one has that if then for every .
We will be interested now in deriving a condition such that the pricing of a marketable contigent claim is obtained from a self-financing strategy that replicates with , so that logical pricing can be obtained in this way. Let us define . For a self-financing strategy satisfying , we set as .
Definition 3.5. For , set We have the following proposition showing that any will lead to an element in (see the proof in the appendix).
Proposition 3.6. If for some one has , 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 is a self-financing strategy that replicates with for and , then for any superreplicating strategy for one has a.s. for that and if , then .
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 satisfying the conditions of Proposition 3.7. We assume here that and that is -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  or , 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 : , where with (recall that is the vector formed by 1 in all components), . Let the vector be such that , . We have that is marketable if and only if there exists that satisfy the system: For the case in which and has an inverse, we have the following explicit and constructive procedure for obtaining a trading strategy that replicates with . Since and (recall that is the vector formed by 1 at the component, and 0 elsewhere), we have premultiplying (4.2) by that , satisfy (4.2) if and only if satisfy Set . Let us obtain that satisfies (4.2), (4.3) with . Define for: , if , otherwise, , . For , calculate In order to have (4.4) satisfied, we must have . If , then set , , otherwise, set and . Thus, we have obtained in this way a trading strategy that replicates with .
Finally, notice that for the discrete sample space, the set is finite, and thus if we assume that for every , , 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 , is obtained by applying to backwards in time the algorithm presented above. The buyer price, denoted by , is obtained by applying the backward algorithm to , and taking . 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 , where and is a binomial process with parameter, , and the bank account is given by , . The penalty costs are assumed to be of the form: It is easy to see that in this case , and we have the following possibilities for , where is associated to the probability measure when the stocks goes up, when the stocks goes down: (i); in this case, (ii); in this case, (iii); in this case, (iv); in this case,
From above, it is clear that the condition which guarantees that , and thus that the conditions of Proposition 3.7, will be satisfied, is given by and .
Let us consider the following numerical example. Suppose that , , , , . For this case, we have , and , and the conditions of Proposition 3.7 will be verified. Let us consider the following option: . By applying the backward procedure described above, we obtain that the seller price for is , with the following hedging strategy: , , , , and for the case in which the risky security goes up, , , , , , while for the case in which it goes down, , , , , and .
By repeating the procedure now for , we obtain that the buyer price for is , with the following hedging strategy: , , , , and for the case in which the risky security goes up, , , , , , while for the case in which it goes down, , , , , and . As expected, .
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 . 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.
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 represents the space of -measurable random vectors with values in , , the space of -measurable random vectors such that and, for simplicity, , . The definition for , and , is similar.
A. Some Auxiliary Results
, , and . We set
Proposition A.1. The following statements are equivalent: (i), (ii) and is closed.
Proof. We have to show that (i) implies that is closed. For this, we consider sequences , in , in , and , such that a.s.,
If we can find and in such that -a.s.
then the result is proved since in this case, setting , we have from (A.7) that and , thus . We set such that the limits (A.5) hold and , , , are always nonnegative. It is easy to see that .
We define next and so that , , , , and on that Set and . From Lemma 2 of , we can find subsequences , of, respectively, , such that on , and for some , in . Set the corresponding subsequence of . It follows that on , from (A.5), and (20), If , then from (A.6) and (A.7) the result is proved. Otherwise, we define . As in [47, 48], we form partitions of , and argue on each separate partition as an autonomous space, considering the appropriate restrictions of the random vectors and traces of the -algebras.
On , we define , , and . From (A.5), (A.8), it follows that on , Since on , and , we have again from Lemma 2 of  that we can find convergent subsequences , of, respectively, , , with limits, respectively, and . We denote by , , the corresponding subsequences of , , , obtained from the association with and . Since for each , , it follows that for each , From (A.10) we have, that on , and from (i), it follows that (A.12) and (A.13) imply that We also have on that We proceed now by applying induction on . Suppose first that . We can find a partition of into 2 disjoint sets, defined by , and . From (A.11) and (A.15) we have that indeed and form a disjoint partition of . From (A.12) and (A.14) we have that on (recalling that in this case ), and , and that (since and ). This implies that on , and as in (A.9), , and (A.6), (A.7) hold with , . Similarly, on , , , and , so that again (A.6), (A.7) hold with , . This completes the proof for .
Suppose now that the equivalence between (i) and (ii) holds for , and that (i) holds for. Define the disjoint sets From (A.11) and (A.15), we have that indeed and , , form a disjoint partition of . For fixed, we will consider first a disjoint partition of . Consider all subsets, indexed by , formed as , , with . Write , and consider a disjoint partition of formed by the sets: We fix now , and for notational simplicity, we write , , , , and , , respectively, the corresponding trace of the -algebras , on . Let us consider that (otherwise, it could be discarded) and write . On the set , we have from (A.12) and (A.14) that Set the -dimensional random vectors , , as follows: for , , and , for , , and For , define On , we have that , for all and for all , and consequently for all and for all . Define , , and . From (A.19), we obtain that where for , , , for , , , for , , , and for , , . We notice that and belong to . Similarly, from (A.18), we have that Next, we show that . For this, we establish a mapping such that and belong to whenever and belong to , and that Indeed, setting and for , , , for , , , for , , , and for , , , we get from (A.26) that , belong to whenever and belong to , 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 and by the induction hypothesis, we get that is closed. Therefore, for some and belonging to , and some , we have from (A.22) and (A.23), and taking , that showing that (A.6) and (A.7) are satisfied on .
The proof for the sets goes along the same lines, bearing in mind that (A.18) and (A.19) are replaced, respectively, by and , and that the mapping is defined such that instead of (A.26), we take and max for , for . This completes the proof of the proposition.
In the next proposition let us consider that is such that and is such that . Again we suppose that .
Proposition A.2. The following statements are equivalent: (i)for any , in , one has a.s. that (ii)there exists and a probability measure such that , are integrable with respect to , and
Proof. First, we note that (i) is equivalent to . Indeed, if (i) holds, then clearly . Conversely, suppose that for some , in , and a.s. Then, by taking , , and recalling that and we get that and a.s., which implies that a.s.
Let us show first that (ii) implies (i). Consider , in such that and a.s. From (A.29), we get that a.s., and thus . From the fact that , we get that a.s., which implies that a.s. and again, from , that 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 , we denote by the space of integrable -measurable random variables with values in , and if and . We remind that if then if and only if a.s., and that for any random variable there exists an equivalent probability measure with bounded density such that is integrable under (see ). Consider a change of probability measure with such that , , and . From (i) (which is invariant under equivalent change of probability) and Proposition A.1, we get that is a closed convex set of , and that . Therefore, , with , , and thus does not belong to the set . By the Hahn-Banach Theorem, can be strongly separated from by a nonzero linear continuous functional, so that there exists , , , such that where denotes the expectation with respect to . We note that and are such that , with, and . By taking and for any and positive integer (just take ), we have from (A.31) that which implies that a.s. Normalizing, we assume that . Similarly by taking for any and (this is possible since and , just take , for ), we get from the same reasons as before that . Considering now and we have from (A.31) and the same arguments as before that for every , , which implies that a.s., Similarly, considering now and , we have from (A.31) and the same arguments as before that for every , , which implies that a.s. Consider the family of measures: Clearly, is dominated by (i.e., for every such that ). From the Halmos-Savage Theorem, contains a countable equivalent family , where is a countable set. Define . Since and are equivalent and for every , it follows that for every such that . We show next that . Suppose by contradiction that , so that . From (A.31), which is clearly an absurd (just take , ). This shows that . Define where . Note that since , and thus is well defined. Define the probability measure as . It is easy to see that (since ). Finally, from (A.32), (A.33), (A.35), and the bounded convergence theorem, we obtain (A.29) with , 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 with , such that a.s., , and , where . Then, we can define an arbitrage opportunity for the multiperiod market as follows: for , for , for , for and , and , in all the other situations. Then, clearly for , , and from (2.5), (2.6), for , , and equal to zero for , showing the result.
Let us show now that (ii) implies (iii). We will show by backward induction on time that we can find random variables , , and random vectors , that satisfy , , , and a.s., , For , the result follows from Proposition A.2 with , , , , , and , . Suppose the result holds for . Define and . Again, the result follows from Proposition A.2 with , , , , , and , , completing the induction argument. Set , and define for , , . It follows that a.s., and similarly, , showing (iii).
Finally, let us show that (iii) implies (i). Indeed, from (2.5), (2.6), and (3.2), we have a.s. that Suppose by contradiction that an arbitrage opportunity exists and (iii) holds. Let us denote by the value process associated to the trading strategy. Then,, and . We have now by backward induction in time that a.s. for all . Indeed, from the definition of an arbitrage, the result is clearly true . Suppose . From (B.3), it follows that