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International Journal of Stochastic Analysis
Volume 2012 (2012), Article ID 687376, 14 pages
Consistent Price Systems in Multiasset Markets
Department of Mathematical Sciences, Worcester Polytechnic Institute, Worcester, MA 01609, USA
Received 27 May 2012; Accepted 9 July 2012
Academic Editor: Lukasz Stettner
Copyright © 2012 Florian Maris and Hasanjan Sayit. 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.
Let be any d-dimensional continuous process that takes values in an open connected domain in . In this paper, we give equivalent formulations of the conditional full support (CFS) property of in . We use them to show that the CFS property of X in implies the existence of a martingale M under an equivalent probability measure such that M lies in the neighborhood of for any given under the supremum norm. The existence of such martingales, which are called consistent price systems (CPSs), has relevance with absence of arbitrage and hedging problems in markets with proportional transaction costs as discussed in the recent paper by Guasoni et al. (2008), where the CFS property is introduced and shown sufficient for CPSs for processes with certain state space. The current paper extends the results in the work of Guasoni et al. (2008), to processes with more general state space.
We consider a financial market with risky assets and a risk-free asset which is used as a numéraire and therefore assumed to be equal to one. We assume that the price processes of the risky assets are given by an -valued process , where , , and the -dimensional process is defined on a filtered probability space and adapted to the filtration that satisfies the usual assumptions. We assume that there are transaction costs in the market and they are fully proportional in the sense that each cost is equal to the actual dollar amount being traded beyond the riskless asset, multiplied by a fixed constant. In the presence of such transaction costs, it is reasonable to assume that purchases and sales do not overlap to avoid dissipation of wealth. In general, in markets with proportional transaction costs trading strategies are given by the difference of two processes and representing respectively the cumulative number of shares purchased and sold up to time , namely, . We are also required to start and end without any position in the risky assets to and this requirement corresponds to .
For each such trading strategy , the corresponding wealth process, after taking into account the incurred transaction costs, is given by where is the total variation of in for each and is the proportion of the transaction costs. In our model (1.1), transaction costs between risky assets and cash are permitted and all transaction costs are charged to the cash account. Next, we introduce the class of trading strategies that we consider in this paper.
Definition 1.1. An admissible trading strategy is a predictable -valued process of finite variation with such that the corresponding wealth process satisfies for some deterministic and for all .
In the next definition, we state the absence of arbitrage condition for the market.
Definition 1.2. We say that the market does not admit arbitrage with -sized transaction costs if there is no admissible trading strategy such that the corresponding value process satisfies
The absence of arbitrage condition excludes trading strategies that enables the investors to have nonnegative payoff with the possibility of positive payoff with zero initial investment. The purpose of this note is to study the sufficient conditions on that ensure absence of arbitrage in the market . It is clear that if the stock price process is a martingale under a measure that is equivalent to the original measure , then the model (1.1) does not admit arbitrage. This can easily be seen from the fundamental theorem of asset pricing (see ) that states that martingale price processes do not admit arbitrage in frictionless markets (i.e., ). In the absence of such martingale measure for , the existence of a process which is a martingale under an equivalent measure and which has the following property: also implies absence of arbitrage for the model (1.1). To see this simple fact, observe the following: Note that because of (1.3), we have a.s. This implies that The financial interpretation of (1.5) is that trading at price process without transaction costs is always at least as profitable as trading at price process with transaction costs. The martingale property of implies that trading on is arbitrage free, and therefore trading on with transaction costs is also arbitrage-free.
The process is called consistent price systems (CPSs) for the price process . The origin of CPSs is due to  and the name consistent price system first appeared in . In the following, we write down the formal definition of CPSs.
Definition 1.3. Let . We say that is an -consistent price system for , if there exists a measure such that is a martingale under , and
The existence of such pricing functions is a central question in markets with proportional transaction costs and their existence was extensively studied in the past literature. For example, the papers [4, 5] studied CPSs for semimartingale models and the papers [6–11] studied CPSs for non-semi-martingale models. Other papers that studied similar problems include [4, 8, 12–17]. Particularly, the recent paper  introduced a general condition, conditional full support (CFS), for price processes and showed that if a continuous process with state space has the CFS property, then the exponential process admits -CPS for any . The proof of this result is based on a clever approximation of by a discrete process which is called random walk with retirement (see ). In this paper, we consider continuous processes with general state space , where is any connected open set in . Unlike the original paper , where the random walk with retirement is constructed by using geometric grids, in this paper we choose to work on arithmetic grid. As a consequence, we show that if the process with the state space has the corresponding CFS property, then for any given there exists a martingale , under an equivalent change of measure, such that By an abuse of language we call such a -consistent price system for the process . To achieve this goal, we first provide a few of equivalent formulations of the CFS property. We use these equivalent formulations in the proof of our result. The advantage is that with our approach the proofs become more transparent and also it enables us to state some stronger results than the original paper. For example, our Lemma 2.10 is a stronger result than the corresponding result in  that states that the CFS property is equivalent to the so-called strong CFS property which is stated in terms of stopping times.
Our main result in this paper is Theorem 2.6 which states that the CFS property of in any open connected domain implies the existence of CPSs. To prove this result, we first prove Lemmas 2.7, 2.8, 2.9, 2.10, and 2.11. In Lemma 2.7, we show that the CFS property implies the necessary properties of a random walk with retirement (see  for the formal definition of random walk with retirement). In Lemma 2.8, we prove that our approximating discrete time process is a martingale under an equivalent martingale measure. The proof of this Lemma gives an alternative and elementary proof for the corresponding result in the paper . In Lemma 2.9, we prove that the approximating discrete time process is in fact a uniformly integrable martingale. The proof of this lemma is standard and similar to the corresponding proofs of the papers [10, 11]. In Lemma 2.10, we show the equivalence of the -stickiness with the weak -stickiness for each given . In Lemma 2.11, we show that the CFS property is equivalent to the seemingly weaker linear stickiness property.
2. Main Results
Let , be a -dimensional continuous process that takes values in an open connected domain . For simplicity of our discussion, we assume that . We also assume that the process is defined on a probability space and adapted to a filtration that satisfies the usual assumptions in this space. Let denote the set of continuous functions defined on the interval and with values in and, for any , let denote the set of functions in with .
Definition 2.1. An adapted continuous process satisfies the CFS property in , if for any and for almost all ,
The CFS condition requires that, at any given time, the conditional law of the future of the process, given the past, must have the largest possible support. An equivalent formulation of this property is given in the following definition.
Definition 2.2. Let be an adapted continuous process that takes values in an open and connected domain . We say that is linear sticky if for any , , and any deterministic ,
on the set .
The equivalence of the CFS and the linear stickiness properties will be established in Lemmas 2.10 and 2.11. We also need the following definition.
Definition 2.3. Let be an adapted continuous process that takes values in an open and connected domain . (a)We say that is -sticky for if on the set for any and any stopping time .(b)We say that is weak -sticky for if on the set for any and any deterministic time .
Remark 2.4. It is clear that the CFS property of in is equivalent to the weak -stickiness of for all . The linear stickiness of is seemingly weaker condition than the weak -stickiness of for all . However, this is not the case and in Lemma 2.11 we will show that linear stickiness is equivalent to weak -stickiness of for all . This, in turn, implies that the linear stickiness property is equivalent to the CFS property.
Remark 2.5. When a process is -sticky as in in Definition 2.3, we say that is jointly sticky and this property was studied in the recent paper . The -stickiness roughly means that starting from any stopping time on, the process has paths that are as close as one wants to the path . As it was shown in , the -stickiness holds for any for the process , where are independent fractional Brownian motions with respective Hurst parameters . From , the -stickiness also holds for any continuous Markov process with the full support property in for any .
The following is the main result of this paper. This result is an extension of the main result in  to processes with more general state space. We use  as a road map in the proof of this result.
Theorem 2.6. Let be a continuous process that takes values in a connected domain in . If is linear sticky, then admits CPSs for all .
To show this result, one fix any and define the following increasing sequence of stopping times associated with the process : with . One should mention that the paper  defined the corresponding stopping times in a slightly different way, see the proof of Theorem 1.2 in .
In addition, for each we define Let for every . Note that is bounded and measurable.
In the following, we use the notation to denote the smallest closed set of that contains the values of the random variable with probability one. We use to denote the open ball in with center and radius . When the center is , we simply write . We first prove the following lemma.
Lemma 2.7. If is -sticky in for all , then the process in (2.6) satisfies the following three properties:
Proof. Property is obvious since and is increasing. Property follows from the fact that almost surely each path of is contained in a compact set of and therefore almost surely . To prove property , let us assume that and let be any measurable set such that . Then, it is clear that there exist , and such that and , where is the distance of with the boundary of . It is also clear that on the set we have .
First we show that a.s. on . To see this, define the following stopping time: The -stickiness of implies that But and since was an arbitrary measurable set with , we have Next we show that almost surely on . Too see this, take any , and define and note that By -stickiness of , we obtain or equivalently , where We claim that . Indeed, if , we get Hence on . Also, for we have So for all , . Since this is true for any small , and any arbitrary measurable set with , we conclude that almost surely on . Note that the other direction is clear from the definition of .
Now, define , , as above and let , . The -valued process will be used to construct CPSs for . Next, we prove a lemma that shows that all of , are in fact uniformly integrable martingales under an equivalent change of measure. The proof of this lemma uses Lemma 3.1 of  as a road map (see also Proposition 2.2.14 of ).
Lemma 2.8. There exists a measure equivalent to under which the -valued discrete process is a martingale.
Proof. For any , let be the regular conditional probability of with respect to and let . Let be any strictly increasing convex function defined on with values in such that for every . Define as follows:
Obviously for each , is measurable and convex with respect to . As a consequence, for any fixed , , the image of the function is convex. We first prove that for every and :
By the way of contrary, assume that this is not true, for some and . Then, there exists a sequence with such that is bounded above. We can assume that converges to some (this is a bounded sequence and therefore has a convergent subsequence). We have
for big enough . Therefore, we can conclude that converges to as , which will imply after passing to the limit that and this is a contradiction. From this it follows easily that . If , then using the geometric form of Hahn Banach theorem, there exists a unit vector such that for every . Therefore, . But
and so it contradicts (2.18).
Next, we want to show that is closed. Let , so there exists a sequence such that . But then is unbounded, and therefore this contradicts (2.18). So based on the continuity of , .
Therefore, we conclude that for any and , there exists an , unique, as a consequence of the strict monotonicity of , such that . being continuous with respect to and measurable with respect to , it follows that is measurable. We extend with outside and define: It is easy to check that satisfies Let and . Note that this limit exists almost surely since a.s. on and . From (2.22), we get which shows that and are martingales under . We thus get , and Fatou's lemma gives . We will show that . We have Combining Fatou's lemma with the equation , we obtain . Also, Hence, is the density of a measure under which our discrete process is a martingale. And since ( for all n), is equivalent to .
Lemma 2.9. Under the measure of Lemma 2.8 the process is uniformly integrable for each . In particular, for each .
Proof. For any , set and observe that on we have . Observing that and that we obtain the following:
The two lemmas above uses the -stickiness. The -stickiness is seemingly stronger condition than the weak -stickiness since it involves stopping times. However, the next Lemma 2.10 shows that, in fact, these two conditions are equivalent.
Lemma 2.10. Let be an adapted continuous process with state space and . Then, is weak -sticky if and only if it is -sticky.
Proof. Let us show first that for any weak -stickiness implies -stickiness. Suppose for a contradiction that is weak -sticky but not -sticky. Then there exists a stopping time with , and an such that
Since , there exists a such that for all , , implies . In addition, we can find , , , and such that
For each , let , where Since and , there exists a such that . Note that and . Hence, since is weak -sticky, we obtain Let . Then we claim that which contradicts (2.27). Indeed, if , then for we have by the definition of and the choice of . We will show also that on whenever : Thus, weak -stickiness implies -stickiness. Since the opposite direction is obvious, the proposition is proved.
Lemma 2.11. Let be a continuous adapted process with state space . Then is linear sticky if and only if is -sticky for all .
Proof. We only need to show that linear stickiness implies the weak -stickiness for each . Fix any , . We need to show that
on the set . To do this, for any with , we need to show that
Define for any . From the definition of , it is clear that a.s. on . Let be a constant such that the set has positive probability. Note that and . In the following, we show that
Let and set , and define
for . Let be the smallest positive integer such that . For each , define on to be equal to the linear function that connects the two points and . We can assume that
for some constant vector for each . It is clear that
Because of (2.39), to show (2.36) we only need to show
For each , let Note that is contained in the event in (2.40). Therefore, it is sufficient to prove that has positive probability. When , we have . Therefore, we have the following relation: By the definition of and the above relation, it is easy see that On we have for each and for all . From this, we conclude that . Now, from the linear stickiness and the fact that we conclude . This completes the proof.
Proof of Theorem 2.6. By Lemmas 2.8 and 2.9, there exists an equivalent probability measure such that is a uniformly integrable martingale for each . Let . For each , set . Observe that , and on the set for all . Thus the following equation holds: We write . Note that each of , , and takes values in on the set . Therefore, we have on the set . Since , we conclude that Since is arbitrary, the claim follows.
Example 2.12. Let be a sequence of independent fractional Brownian motions with respective Hurst parameters . Let be a homeomorphism for each , where is an open interval in the real line. Then the new process admits CPSs for each . This can be easily seen from the CFS property of the process which was shown in  and the fact that the map is a homomorphism from to .
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