On the Market Viability under Proportional Transaction Costs
This paper studies the market viability with proportional transaction costs. Instead of requiring the existence of strictly consistent price systems (SCPS) as in the literature, we show that strictly consistent local martingale systems (SCLMS) can successfully serve as the dual elements such that the market viability can be verified. We introduce two weaker notions of no arbitrage conditions on market models named no unbounded profit with bounded risk (NUPBR) and no local arbitrage with bounded portfolios (NLABP). In particular, we show that the NUPBR and NLABP conditions in the robust sense for the smaller bid-ask spreads is the equivalent characterization of the existence of SCLMS for general market models. We also discuss the implications for the utility maximization problem.
Key words and phrases:Proportional Transaction Costs, (Robust) No Unbounded Profit with Bounded Risk, Strictly Consistent Local Martingale Systems, (Robust) No Local Arbitrage with Bounded Portfolios, Utility Maximization, Market Viability, Numéraire Portfolios
In the Fundamental Theorem of Asset Pricing with proportional transaction costs, consistent price systems (CPS) introduced by  and  take the role of the dual elements instead of the equivalent (local) martingale measures. The CPS is defined as follows:
Given the stock price with transaction cost such that a.s. for all , we call the pair a CPS if
where is a local martingale under and . Moreover, if we have
the pair is said to be a strictly consistent price system (SCPS).
We should note that whether is required to be a local martingale or a true martingale in the above definition depends on the numéraire and numéraire-based admissibility of self-financing portfolios; see section of  and  for details. Sufficient conditions for the existence of CPS for stock price processes with strictly positive and continuous paths have been extensively studied in the literature. One well-known example is the conditional full support condition proposed by . Other related sufficient conditions are discussed in ,  and . Recently, for continuous price processes,  built the equivalence between absence of arbitrage with general strategies for any small constant transaction cost and the existence of CPS for any small transaction cost . Later,  investigated the general càdlàg processes, and they linked two equivalent assertions, i.e., the robust no free lunch with vanishing risk for simple strategies and the existence of a SCPS.
On the other hand, in the market without transaction costs, the existing literature analyzed market models which do not satisfy all the stringent requirements of the fundamental theorem of asset pricing. Compared to No Free Lunch with Vanishing Risk (NFLVR) condition on terminal wealth originally defined by , a weaker condition, which is called in  as the No Unbounded Profit with Bounded Risk (NUPBR) condition, serves as a reasonable substitute using which one can still solve the classical option hedging and utility maximization problems. , , ,  and  showed the equivalence between the NUPBR condition, the existence of a strictly positive local martingale deflator process, the existence of an optimal solution to the utility maximization problem and the existence of a numéraire portfolio.
Motivated by these results obtained in frictionless markets, we aim to determine a similar minimal condition on the market with frictions under which the utility maximization problems still admit optimal solutions. However, due to the special features of the transaction costs, definitions of self-financing and admissibility of working portfolios differ from the usual stochastic integrand for semimartingales. It is revealed in this paper that we need the stock price process to meet two weaker assumptions, i.e., the No Unbounded Profit with Bounded Risk (NUPBR) and No Local Arbitrage with Bounded Portfolios (NLABP) conditions on liquidation value processes at the same time in the robust sense for some strictly smaller bid-ask spreads. It is worth noting that our NUPBR and NLABP conditions are still weaker than the NFLVR requirement in  and even if both NUPBR and NLABP conditions are satisfied, an arbitrage opportunity may still exist in the market. The main contribution of this paper is the equivalence between the NUPBR and NLABP conditions in the robust sense and the existence of SCLMS which is defined as follows.
Given the stock price with transaction cost a.s. for all , we call a pair a consistent local martingale system (CLMS) if is a semimartingale satisfying
and there exists a strictly positive local martingale with such that is a local martingale. We shall denote the set of all CLMS with transaction cost . Moreover, if
we shall call the pair a SCLMS and denote by the set of all SCLMS.
It is clear that the definition of CLMS is a generalization of the classical CPS, i.e., any pair of CPS is a CLMS, however the opposite is not necessarily true as can be a strict local martingale. In Section 4, some examples of market models are presented which demonstrate that SCLMS may exist even when we do not have the existence of the CPS.
The second contribution of this paper is the result which shows that NUPBR and NLABP conditions in the robust sense guarantees the existence of a solution to the utility maximization problem defined on the terminal liquidation value. We also discuss the existence of a numéraire portfolio as a corollary. Therefore, NUPBR and NLABP conditions in the robust sense serve as sufficient conditions on the market viability in the sense that optimal portfolio problems admit solutions. Meanwhile, it is also presented that this sense of market viability implies that the corresponding meets the NUPBR condition, although not in the robust sense, which illustrates that our market assumptions are minimal conditions to some extend.
To emphasize the mathematical differences between our setting and the frictionless market models in the literature, we discuss different types of arbitrage opportunities with transaction costs. In particular, we should point out that the NLABP condition in the main theorem is a new feature which appears for the first time. The construction of arbitrage opportunities in our setting with transaction costs is unique because the wealth process in frictionless market models actually has two counterparts, namely, the liquidation value process (see ) and the cost value process (see ). This difference leads to distinct arguments and proofs concerning the absence of arbitrage.
The rest of this paper is organized as follows: In Section 2, we introduce the market model with transaction costs and define the NUPBR condition and NLABP condition on the terminal liquidation value. We state the equivalence between the NUPBR and NLABP conditions in the robust sense and the existence of SCLMS at the end of this section. The proof of the main theorem is given in Section 3. In Section 4, we discuss concrete examples of market models, for both continuous processes and jump processes, in which a CPS fails to exist, but we can find a SCLMS. Section 5 discusses the utility maximization problems under NUPBR and NLABP conditions in the robust sense. The discussion of various types of arbitrage opportunities and the comparison to the frictionless market models are provided in the first part of Section 6. In the second part of this section we discuss our admissibility criterion.
2. Set-up And The Main Result
The financial market consists of one risk-free bond , normalized to be , and one risky asset . We will work with a probability space that satisfies the usual conditions of right continuity and completeness. is assumed to be trivial. The following is a standing assumption which will hold in the rest of the paper:
is adapted to with strictly positive and locally bounded càdlàg paths. The transaction cost process is adapted to with càdlàg paths such that a.s. for all .
We adopt the notion of self-financing admissible strategies defined in:
A self-financing trading strategy starting with zero is a pair of predictable, finite variation processes such that
denoting by and , the canonical decompositions of and into the difference of two increasing processes, starting at , these processes satisfy
where the two integrals in are defined as predictable Stieltjes integrals and
Here we denote and . As discussed in , since is càdlàg , we need to take care of both left and right jumps of the portfolio process . In general, three values , and can be different. If the stopping time is totally inaccessible, the predictability of implies that almost surely. But if the stopping time is predictable, it may happen that both and .
In general, for any càdlàg process and predictable finite variation process , the predictable Stieltjes integral above can be rewritten as
(See Appendix A of  for a detailed discussion on predictable Stieltjes integrals.)
At the initial time, we assume that the investor starts with the position in bond and stock assets for the given constant . The trading strategy is called -admissible if the liquidation value satisfies
-a.s. for . We shall denote (short as ) as the set of all -admissible portfolios with the transaction cost and let . Moreover, we will also denote as the set of the terminal liquidation value under the admissible portfolio .
Parallel to the frictionless market, a weak no arbitrage condition can be defined via the boundedness in probability property of some target subset of . The following definition of NUPBR is parallel to that of .
We say that admits an Unbounded Profit with Bounded Risk (UPBR) with the transaction cost if there exists a sequence of admissible portfolios in and the corresponding terminal liquidation value is unbounded in probability, i.e.,
If no such sequence exists, we say that the stock price process satisfies the NUPBR condition under the transaction cost .
In order to provide the sufficient and necessary conditions on the existence of SCLMS, we also need to introduce another weak no arbitrage condition. To this end, let (short as ) denote the -admissible bounded portfolios such that the position in the stock is uniformly bounded by some constant in the following sense:
Moreover, we denote .
We say that satisfies No Local Arbitrage with Bounded Portfolios (NLABP) with the transaction cost if there exists a sequence of stopping times as such that for each , we can not find which satisfies
It is noted that the NUPBR condition is defined on the set for a fixed , for instance , which is consistent with definition of the utility maximization problem with a fixed initial position. The NLABP condition is defined for all admissible portfolios on the set . However, these two definitions are consistent since if we have a sequence of portfolios in which leads to UPBR, by rescaling, we also obtain UPBR for any , .
For the completeness of the paper as well as the comparison between different concepts, we shall also introduce the standard no arbitrage condition on liquidation values.
We say that admits arbitrage with the transaction costs if there exits an admissible portfolio such that
If no such portfolio exists, we say that the stock price process satisfies the NA condition under the transaction cost .
Comparing Definition 2.3 and Definition 2.4, it is clear that our NLABP is equivalent to the NA condition with bounded portfolios for a localizing sequence . It is important to note that (NA) (NLABP), however, does not imply in general. In frictionless market, see a discussion and Example in Section of  that NA can hold locally but fail globally. Moreover, the NUPBR condition and the NLABP condition may not imply each other. Given the assumption that NUPBR and NLABP conditions are satisfied, we may still have arbitrage opportunities at time using some unbounded -admissible portfolios . The NFLVR condition in  clearly implies both NUPBR and NLABP conditions in our setting, therefore, we claim that our conditions are weaker assumptions on the market models than the usual conditions in the existing literature.
Some slightly stronger conditions are needed for the main result of this paper.
We say that satisfies the NUPBR and NLABP conditions with the transaction cost in the robust sense if there exist another stock price process and the transaction cost process satisfying Assumption 2.1 such that
and the stock price process satisfies the NUPBR condition and the NLABP condition at the same time with the transaction cost . In particular, if we only consider the case that the stock price process satisfies the NUPBR condition with the transaction cost , we say that satisfies Robust No Unbounded Profit with Bounded Risk (RNUPBR) condition with the transaction cost .
As the main result of this paper, the following theorem provides the equivalence between NUPBR and NLABP conditions in the robust sense and the existence of SCLMS. Its proof is delivered in the next section.
The following two assertions are equivalent.
satisfies NUPBR and NLABP conditions with the transaction cost in the robust sense as in Definition 2.5.
There exists a SCLMS for the market with transaction cost , i.e., .
Compared to the frictionless markets in which we have the equivalence between the NUPBR condition on terminal wealth and the existence of local martingale deflators, see , our equivalence characterization in the markets with transaction costs involves two conditions, i.e., NUPBR and NLABP. The self-financing and admissibility conditions in our framework are more restrictive than those in frictionless markets and different types of convergence are required respectively in two different settings. For example, some convergence results for predictable Stieltjes integrals (Theorem of ) and the integration by parts formula (Proposition of ) play important roles in our proof, however, the literature in frictionless markets relies on the convergence results of stochastic integrals w.r.t. semimartingales.
Actually, in frictionless markets, the NLABP condition on wealth processes may always hold since either there is no local arbitrage for the wealth process or there is a local arbitrage but the portfolio process is not necessarily bounded which is usually only required to be predictable and -integrable. On the other hand, our stock price process is not necessarily a semimartingale and the liquidation value process lacks supermartingale property, which is possessed naturally by each wealth process discounted by local martingale deflators in frictionless markets. For the equivalence between the NUPBR condition and the existence of local martingale deflators in models without transaction costs, the proof in  relies on the fact the numéraire portfolio process is a supermartingale and the trick of change of the numéraire and the proof in  is based on some probability characteristics of the semimartingale price process . However, these results no longer hold in our setting. As discussed in Section 5, we do not expect the numéraire portfolio process to be a supermartingale. Some new ideas to support the proof of the equivalence in Theorem 2.1 are required. In particular, both NUPBR and NLABP conditions are needed to guarantee our main result.
3. Proof of The Theorem 2.1
The proof of Theorem 2.1 is split into several steps. We first show the implication that in the next important proposition.
3.1. Proof of
If there exists a SCLMS , then the stock price process satisfies NUPBR and NLABP conditions with the transaction cost in the robust sense.
Since there exists a SCLMS such that a.s., we can define and get that a.s. for all . Moreover, we obtain that
since . We can therefore choose for all and . First of all, it is easy to verify that
as well as a.s.. Also, we can check that for . To see this, we can show that
which is a direct consequence of the definition of .
Thus, it is enough to show that the smaller spread satisfies the NUPBR condition and NLABP condition with the transaction cost . Denote the set of terminal liquidation value under -admissible self-financing portfolios. We first show that is bounded in probability. To this end, we first verify that
By the definition of SCLMS , we have is a local martingale. We claim that for any admissible portfolio , we have
where is interpreted as a stochastic integral. To see this, we rewrite
Using integration by parts (see Proposition of ) we obtain
where the is a predictable Stieltjes integral. By and the fact a.s. for all , we have that and
which implies that a.s. for .
Let . Since is a local martingale, applying integration by parts, we deduce that
Thanks to the Ansel-Stricker Theorem (see ), we get is a local martingale, and since , we deduce that is a supermartingale.
Therefore, it follows that
Since the right hand side is independent of the choice of , we get that holds true.
By and the fact that is strictly positive for all and hence , -a.s., Lemma of  implies that the set is bounded in probability. Therefore, we can conclude that satisfies the NUPBR condition.
On the other hand, to show that satisfies the NLABP condition is straightforward. Thanks to the fact that and are local martingales, there exists a localizing sequence such that and are true martingales. For the same sequence , suppose that for some , there exists some bounded admissible portfolio such that holds for the stopping time . Define the probability measure by . It follows that is a martingale under . Moreover, since a.s. for some , the stochastic integral is a true martingale under . Therefore, we can deduce that
However, this is a contradiction to and by the fact that as well as . Therefore, we obtain a sequence of stopping times which satisfies Definition 2.3 and satisfies the NLABP condition. ∎
3.2. Proof of
The proof of this direction requires more preparation. To begin with, it is noted that the set itself is not convex. We thereby shall consider its solid hull defined by
Clearly and is convex and solid.
If the stock price process satisfies RNUPBR with the transaction cost , the set , where we denoted by the total variation of on , is bounded in probability.
Let be as in Definition 2.5. For any , we have
Let us define and . Observing that for all , it follows from (3.5) that
Since satisfies RNUPBR condition on , we know that is bounded in probability. By assumption and a.s.. Now using Lemma 3.1 of , we obtain that the set is bounded in probability. ∎
The proof of the following result is also crucial in establishing the existence of the optimal solution of the utility maximization problem as well as the existence of a numéraire portfolio in Section 5.
If satisfies RNUPBR with transaction cost , the set is closed under convergence in probability.
Take a sequence such that in probability, and by passing to a subsequence, we can assume without loss of generality that a.s.. We need to verify that . Consider now a sequence satisfying a.s.. By the definition of , there exist a sequence and for all .
Lemma 3.1 states that the set is bounded in probability. Thanks to Lemma of , we can deduce that there exists a sequence of forward convex combinations such that converges pointwise to a predictable and finite variation process such that the sequence also converges to pointwise. The latter convergence implies that the sequence converges to in probability for each which in turn leads to the fact that the set is bounded in probability for each . Similar to the proof of Lemma of , we can write
which gives that
For each , since the set is bounded in probability, we can obtain that
which implies that a.s. and hence a.s. for . As a result we can apply the assertion of Theorem of , to obtain the pointwise convergence of predictable Stieltjes integrals
which holds for any càdlàg process .
Using the same sequence of convex combinations in the definition of , without loss of generality, we can consider as the resulting process after the forward convex combinations. Similarly, we define following the same convex combinations. It follows that
Therefore, we obtain that
We have the following lower-semicontinuity property
for , where we define for all . By definition is a self-financing portfolio. Moreover, since a.s. for all and , we can see that . Since a.s., it follows that