The Fundamental Theorem of Asset Pricing with either Frictionless or Frictional Security Markets ()
Keywords:The First Fundamental Valuation Theorems; Transaction Costs; Weak Arbitrage-Freeness; Strict Arbitrage-Freeness; Arbitrage-Free Pricing Theory; Locally Convex Topological Space; Separable Banach Space
1. Introduction
Arbitrage-free asset pricing theory is of fundamental importance in neo-classical finance. In this paper, we consider it in a setting of general state spaces, in particular, locally convex topological space for weakly arbitrage-free security markets, and separable Banach space for strictly arbitrage-free security markets. We deal with both frictionless and frictional security markets.
In the study of mathematical economics, arbitrage-free conditions have always been an important first step toward the general equilibrium theorem with incomplete asset markets (Duffie and Shafer [1,2]; Geanakoplos [3]; Geanakoplos and Shafer [4]; Hirsch, Magill and Mas-Colell [5]; Husseini, Lasry and Magill [6]; and Magill and Shafer [7]). Since the 1980s, for finite period economies, no-arbitrage pricing theory has been applied by various authors to prove the existence of general equilibrium for stochastic economies with incomplete financial markets (Duffie [8-10]; Florenzano and Gourdel [11]; Magill and Shafer [7]; Werner [12,13]; and Zhang [14]). In those works, the finite number of possible states of nature and the finite-dimensional commodity space are usually assumed in order for the proofs to be carried out for the general equilibrium model with incomplete financial markets. Our model considers a general state space which may have an infinite number of states.
Many mathematical economists usually apply Stiemke’s Lemma, a strict version of Farkas-Minkowski’s Lemma, to study the asset pricing theory with no-arbitrage conditions, for example, discrete-time models of dynamic asset pricing theory (Duffie [9,10]) and the theory of economic equilibrium with incomplete asset markets (Geanakoplos [3]; Geanakoplos and Shafer [4]; Hirsch, Magill and Mas-Colell [5]; Husseini, Lasry and Magill [6]; and Magill and Shafer [7]) where the commodity space is of finite dimension. Farkas-Minkowski’s Lemma and Stiemke’s Lemma are in essence the mathematical counter part of the asset pricing theory with no-arbitrage conditions. In this paper, we obtain extensions (to the general state space in our discussion) of Farkas-Minkowski’s Lemma and Stiemke’s Lemma by applying Clark’s separating hyperplane theorems (Clark [15,16]), and thus establish our main results.
Harrison and Kreps [17] initiated the study of martingales and arbitrage in multiperiod security markets. They first introduced general theory of arbitrage in a two-period economy with uncertainty, and then extended it to the models of multiperiod security markets and the models of continuous-time securities markets. Kreps [18] studied arbitrage and equilibrium in economies with infinitely many commodities and presented an abstract analysis of “arbitrage” in economies that have infinite dimensional commodity space. Harrison and Pliska [19] studied martingales and stochastic integrals in the theory of continuous trading. Dalang, Morton and Willinger [20] studied equivalent martingale measures and no-arbitrage in stochastic securities market models. Back and Pliska [21] studied the fundamental theorem of asset pricing with an infinite state space and showed some equivalent relations on arbitrage. Jacod and Sgiryaev [22] studied local martingales and the fundamental asset pricing theorems in the discrete-time case. These papers studied fundamental theorems of asset pricing in multiperiod financial models with the help of techniques from stochastic analysis. Our work is based on separating hyperplane theorems and does not rely on assumptions made for stochastic analysis to be able to carry out in the above models.
Friction in markets has attracted attention of several works in this field recently. Chen [23] examined the incentives and economic roles of financial innovation and at the same time studied the effectiveness of the replication-based arbitrage valuation approach in frictional economies (the friction means holding constraints). Jouini and Kallal [24] derived the implications of the absence of arbitrage in securities markets models where traded securities are subject to short-sales constraints and where the borrowing and lending rates differ, and showed that a securities price system is arbitrage free if and only if there exists a numeraire and an equivalent probability measure for which the normalized (by the numeraire) price processes of traded securities are supermartingales. Jouini and Kallal [25] derived the implications from the absence of arbitrage in dynamic securities markets with bid-ask spreads. The absence of arbitrage is equivalent to the existence of at least an equivalent probability measure that transforms some process between the bid and the ask price processes of traded securities into a martingale. Pham and Touzi [26] addressed the problem of characterization of no arbitrage (strictly arbitrage-free) in the presence of friction in a discrete-time financial model, and extended the fundamental theorem of asset pricing under a non-degeneracy assumption. The friction is described by the transaction cost rates for purchasing and selling the securities. We follow the model for transaction costs from Pham and Touzi [26], and then extend the first fundamental valuation theorems of asset pricing from frictionless security markets to frictional security markets, for general state space. Again, their stochastic analysis method requires stronger assumptions than ours. In addition, their arbitrage-free conditions are slightly different from ours based on that of Duffie [9].
Section 2 presents our model without transaction costs, we define the weakly arbitrage-free security markets and strictly arbitrage-free security markets, following Duffie [9]. In the next two sections, we establish the first fundamental valuation theorem of asset pricing (a necessary and sufficient condition for arbitrage-freeness) with weakly arbitrage-free security markets in Section 3 and strictly arbitrage-free security markets in Section 4. In the rest of the paper we extend our work to markets with transaction costs, following the model of Pham and Touzi [26]. The transaction cost model is presented in Section 5, and the corresponding first fundamental valuation theorems of asset pricing are done in Sections 6 and 7. Section 8 concludes our article with some remarks.
2. Frictionless Security Markets
We consider a two-period model (dates 0 and 1) with uncertainty over the states of nature in the date 1. The unknown nature of the future is represented by a general set 
of possible states of nature, one of which will be revealed as true. Here we make no assumption about the probability of these states. The 
securities are given by a return “matrix”
, where 
denotes the number of units of account paid by security
. Let 
denote the vector of prices of 
securities. A portfolio 
has market value 
and payoff
.
Let 
be a topological space consisting of processes in
, 
is the positive cone of
. Let 
be the dual space composed of the continuous linear functionals on
, 
the positive cone of the space 
(the space of all positive continuous linear functionals on
) and 
the interior of the cone 
(the space of all strictly positive continuous linear functionals on
):
.
In this paper, we assume 
for
. Then
. Our proof must adopt the following notation 
and
.
[Definition ]">1] The frictionless market 
is weakly arbitrage-free if any portfolio 
of securities has a positive market value 
whenever it has a positive payoff
.
[Definition ]">2] The frictionless market 
is strictly arbitrage-free if (1) any portfolio 
of securities has a strictly positive market value 
whenever it has a positive non-zero payoff
; and (2) any portfolio 
of securities has a zero market value 
whenever it has a zero payoff
.
Definition 2 implies Definition 1 obviously. We follow the definition for arbitrage opportunity as provided by Duffie [9,]">10]. An arbitrage is a portfolio 
with either (1) 
and
, or (2) 
and
. That is to say, the frictionless market 
admits an arbitrage opportunity if there exists an portfolio 
of securities such that either (1) 
and
, or (2) 
and
. Consequently, we can define the strictly arbitrage-free (no arbitrage) frictionless market 
as follows.
[Definition 2¢] The frictionless market 
is strictly arbitrage-free if (1) any portfolio 
of securities has a strictly positive market value 
whenever it has a positive non-zero payoff
; and (2) any portfolio 
of securities has a positive market value 
whenever it has a positive payoff
.
[Lemma ]">1] Definitions 2 and 2’ are equivalent.
An arbitrage is therefore, in effect, a portfolio offering “something for nothing”. Not surprisingly, an arbitrage is naturally ruled out in reality. And this fact gives a characterization of security prices as follows: A valuation functional is a functional 
for the weakly arbitrage-free frictionless market 
with consistency
; and a functional 
for the strictly arbitrage-free frictionless market 
with consistency
, where
. The valuation functional is called to be a positive linear consistent valuation operator for the weakly arbitrage-free frictionless market, a strictly positive linear consistent valuation operator for the strictly arbitrage-free frictionless market, respectively.
The idea of arbitrage and the absence of arbitrage opportunities is fundamental in finance. The strict arbitrage-freeness is important in the study of general equilibrium theory with incomplete asset markets (Husseini, Lasry and Magill [6]; Werner [13]; and Magill and Shafer [7]). Theorem 2 to be presented in the following is an important step in the study of equilibrium for economies with general state spaces considered in our work. The principal mathematical tool applied here is the Separating Hyperplane Theorems of Clark [15] and [16].
[Fact ]">1] (Clark [16]) Suppose 
and 
are non-empty disjoint convex cones in a locally convex topological vector space
. Then there exists a non-zero continuous linear functional 
separating
from
: 
for all 
and 
for all 
if and only if
. Moreoverif
, then for any 
we may select 
so that
.
[Fact ]">2] (Clark [15]) Suppose 
and 
are non-empty convex cones (with vertices at the origin) in a separating Banach space
. Then there exists a non-zero continuous linear functional 
strictly separating N from M: 
for all 
and 
for all 
if and only if
.
Fact 1 and 2 will be used to prove Theorems 1 and 2 in Sections 3 and 4, Theorems 3 and 4 in Sections 6 and 7, respectively. We assume
, which is a topological space, then 
is the positive cone of
, which is a positive closed convex cone of 
with its vertex at the origin. The marketed subspace
is a linear subspace of the space
.
3. Weakly Arbitrage-Free Security Valuation Theorem with Frictionless Security Markets
In this section, we assume that 
is a locally convex topological space.
[Proposition 1] The frictionless market 
is weakly arbitrage-free if and only if
.
Proof. 
implies that there exists 
such that 
and
. On the other hand, 
implies that 
from Definition 1, then
. Thus 
and
, that is,
.
Conversely, if there exists 
such that 
and
, then
. Thus
and
.
This is a contradiction! Therefore the frictionless market 
is weakly arbitrage-free. Q.E.D.
[Theorem 1] The frictionless market 
is weakly arbitrage-free if and only if there exists a positive functional 
satisfying
.
Proof. Both 
and 
are closed and convex cones of 
with their vertices at the origin.
The frictionless market 
is weakly arbitrage-free if and only if
. Note that
is a convex set. Let
, then 
is a (non-empty) convex set. Both 
and 
are non-empty disjoint convex cones
. 
implies
. Clark [16] stated that there exists a non-zero continuous linear functional 
separating 
from
, that is, 
for all 
and 
for all
. Moreover,
.
is a positive continuous linear functional on
. In fact, for any 
and natural number
, 
and
. Therefore,
. Thus
.
That is, 
is a positive continuous linear functional on
.
Thus 
is represented by some 
and 
by 
for any
. In fact, 
is represented by some 
and 
by 
for any
. The fact that
implies
. If
, then there exists 
such that
. Take
, then 
and
. The contradiction means
.
Since 
is a linear space, 
for all
, that is, 
for all
. Then
. And therefore, the vector 
is that we want.
The converse is obvious. Q.E.D.
[Remark 1] If 
is a finite set, then 
is an Euclidean space, Definition 1 is the usual concept of weak arbitrage-freeness. Theorem 1 is the well-known Farkas-Minkowski’s Lemma (Duffie [9]; Farkas [27] and Franklin [28]).
[Remark 2] Theorem 1 (The Extension of Farkas-Minkowski’s Lemma) means that the present value of the securities prices at date 0 is the value of their returns at date 1.
4. Strictly Arbitrage-Free Security Valuation Theorem with Frictionless Security Markets
In this section, we assume that 
is a separable Banach space. We prove Proposition 2 and Theorem 2 by using Definition 2 of the strictly arbitrage-free frictionless market
.
[Proposition 2] The frictionless market 
is strictly arbitrage-free if and only if 
and 
intersect precisely at
, that is,
.
Proof. 
implies that there exists 
such that 
and
. If
, then 
from Definition 2 (1), 
, a contradiction! If
, then 
from Definition 2 (2), 
and
. Thus
. And hence,
.
For sufficiency, we must consider two cases: (1) If there exists 
such that 
and
, then 
and
, a contradiction. Therefore, any portfolio 
of securities has a positive non-zero market value 
whenever it has a positive payoff
. (2)
If there exists 
such that 
and
, we discuss the problem for the following two settings. (2.1) When
, then
, thus
. This is a contradiction! (2.2)
When
, then 
and
, thus
. This is a contradiction! The two contradictions mean that the frictionless market 
is strictly arbitrage-free. Q.E.D.
[Theorem 2] The frictionless market 
is strictly arbitrage-free if and only if there exists a strictly positive functional 
satisfying
.
Proof. Both 
and 
are closed and convex cones of 
with their vertices at the origin.
Since 
is a polyhedral (convex) cone in 
and 
is a closed convex cone of
, then 
is a closed convex cone from Lemma of Clark [15], that is,
. Thus 
is a closed convex cone
.
The frictionless market 
is strictly arbitrage-free if and only if
. We claim that
. On the contrary, we assume that there exists
then 
and 
for 
and
. Thus
contradicting the conclusion 
and
.
Note that
. It follows that
. Thus
. Since both 
and 
are non-empty convex cones with vertices at the origin in the separable Banach space
, Theorem 5 in Clark [15] states that there exists a non-zero continuous linear functional 
on 
strictly separating 
from
, that is, 
for all 
and 
for all
.
The condition 
for all 
implies that 
is a strictly positive continuous linear functional on
. Thus 
is represented by some 
and 
by 
for any
.
Since 
is a linear space, 
for all
, that is, 
for all
. It follows that
. The vector 
is that we want.
The converse is again obvious. Q.E.D.
[Remark 1] If 
is a finite set, then 
is an Euclidean space, Definition 1 is the usual concept of strict arbitrage-freeness. Theorem 2 is the well-known Stiemke’s Lemma (Duffie [9,10]).
[Remark 2] Theorem 2 (The Extension of Stiemke’s Lemma) means that the present value of the securities prices at date 0 is the value of their returns at date 1.
[Remark 3] If 
where 
is a linear functional on
, we can prove Theorem 2 by Kreps [18]. The sufficient condition is obvious, we only need prove the necessary condition. Take
, then any element 
is of the form 
where
. Take 
as a linear functional on 
solving the equation 
for all
, i.e. 
and take
. If the frictionless market 
is strictly arbitrage-free, then the pair 
admits no free lunches. Kreps’ conditions (4.7) and (4.10) are applied, then the pair 
is 
-viable from Theorem 3 (Kreps [18]), that is, the pair 
has the extension property for
. Thus the necessary condition is obtained.
5. Frictional Security Markets
Suppose that there are transaction costs in the trading, the coefficients 
and 
are respectively the transaction cost rates for purchasing and selling the security
. Then the algebraic cost induced by (buying) a position 
units of security 
is 
and the algebraic gain induced by (selling) a position 
units of security 
is
. We introduce the functions 
defined by
and the functions 
defined by
Then
.
For any integer
, a function 
is subliner if, for any
, 
, 
and
,
The function 
is sublinear, and hence convex. Therefore the function 
is also sublinear, and hence convex.
The total cost or gain induced by (trading) a portfolio 
is
. We define the function 
by
Then the total cost or gain induced by (trading) a portfolio 
is
. As we know, the function 
is sublinear, and hence convex.
[Definition ]">3] The frictional market 
is weakly arbitrage-free if any portfolio 
of securities has a positive total cost or gain 
whenever it has a positive payoff
.
We note that Definitions 2 and 2’ are equivalent. However, in the presence of transaction costs, the corresponding Definitions 4 and 4’ (as follows) are not. We establish Theorem 4 for Definition 4’ of the strictly arbitrage-free frictional market
.
[Definition ]">4] The frictional market 
is strictly arbitrage-free if (1) any portfolio 
of securities has a strictly positive total cost or gain 
whenever it has a positive non-zero payoff
; and (2) any portfolio 
of securities has a zero total cost or gain 
whenever it has a zero payoff
.
[Definition 4’] The frictional market 
is strictly arbitrage-free if (1) any portfolio 
of securities has a strictly positive total cost or gain 
whenever it has a positive non-zero payoff
; and (2) any portfolio 
of securities has a positive total cost or gain 
whenever it has a positive payoff
.
Definition 4 obviously implies Definition 4’. Definition 4’ does not imply Definition 4 because of the presence of friction. In the frictionless model, we define the marketed subspace
of the space 
to prove the first fundamental theorems of asset pricing. In the frictional model, we can’t consider the corresponding marketed “subspace”
. In fact, this marketed “subspace”
isn’t a subspace of the space
. Instead, we define the subset 
in the space 
as follows
[Lemma 2] 
is a closed and convex cone in the space
.
Proof. 
is a closed cone obviously since the function 
is continuous and sublinear.
For any 
and
, there exist 
and 
such that
,
and
,
. Thus 
since the function 
is sublinear, and
. Therefore
. Since 
is a cone, 
is convex. Q.E.D.
For simplicity, we use the following notations in the subsequent sections.
We define the box product of two vectors 
and 
by
6. Weakly Arbitrage-Free Security Valuation Theorem with Frictional Security Markets
In this section, we assume that 
is a locally convex topological space.
[Proposition ]">3] The frictional market 
is weakly arbitrage-free if and only if
.
Proof. 
implies that there exists 
such that 
and
. On the other hand, 
implies that 
from Definition 3. Then 
and
, that is,
.
Conversely, if there exists 
such that 
and
, then
. Thus
and
.
This is a contradiction! Therefore, any portfolio 
of securities has a positive total cost or gain 
whenever it has a positive payoff
. Q.E.D.
[Remark] The proof of Proposition 3 only requires the property that 
is a closed and convex cone in the space 
and doesn’t need the explicit definition of the function
, so does Proposition 3.
[Theorem ]">3] The frictional market 
is weakly arbitrage-free if and only if there exists a positive functional 
satisfying
Proof. Both 
and 
are closed and convex cones of 
with their vertices at the origin.
The frictional market 
is weakly arbitrage-free if and only if
. Note that 
is a convex set. Let
, then 
is a (non-empty) convex set. Both 
and 
are non-empty disjoint convex cones,
. 
implies
. Clark [16] stated that there exists a non-zero continuous linear functional 
separating 
from
, that is, 
for all 
and 
for all
. Moreover,
.
From the proof of Theorem 1, 
is a positive continuous linear functional on
, and 
is represented by some 
and 
by 
for any
.
On the other hand, the fact that 
for all 
implies 
for all 
and 
for
. Then 
for all
. And hence we have
for all
. For each
, assigning
, we have
that is,
. Thus
For each
, we assign 
instead, it follows that
that is,
. Thus
Therefore, for any
,
The vector 
is thus what we want:
Conversely, assume that there exists a positive functional 
satisfying
For any
,
For any
,
And for any
,
Thus
Summarying over
, we then have, for any
,
Therefore the frictional market 
is weakly arbitrage-free. Q.E.D.
[Remark] Theorem 3 means the first fundamental theorem of weakly arbitrage-freeness with frictional security markets. If there exists no transaction costs, that is, 
and
, then Theorem 3 reduces to Theorem 1.
7. Strictly Arbitrage-Free Security Valuation Theorem with Frictional Security Markets
In this section, we assume that 
is a separable Banach space. We prove the following Proposition 4 and Theorem 4 for Definition 4’ of the strictly arbitrage-free frictional market
.
[Proposition ]">4] The frictional market 
is strictly arbitrage-free if and only if 
and 
intersect precisely at
, that is,
.
Proof. 
implies that there exists 
such that 
and
. If
, then 
from Definition 4’(1), 
, which is a contradiction! If
, then 
from Definition 4’(2), 
and
. Thus
and hence
.
For sufficiency, we must consider two cases: (1) If there exists 
such that 
and
then
. Take 
such that 
and
, then
which is a contradiction. It follows that any portfolio 
of securities has a strictly positive total cost or gain 
whenever it has a positive payoff
. (2) If there exists 
such that 
and
, then
. Take 
such that 
and
, then
, which is a contradiction! The two contradictions mean that the frictional market 
is strictly arbitrage-free. Q.E.D.
[Remark] The proof of Proposition 4 doesn’t use the representation of the function
, then Proposition 4 holds for usual total cost or gain function.
[Theorem ]">4] The The frictional market 
is strictly arbitrage-free if and only if there exists a strictly positive functional 
satisfying
Proof. Both 
and 
are closed and convex cones of 
with the vertices at the origin.
Since 
is a polyhedral (convex) cone in 
and 
is a closed convex cone of
, 
is a closed convex cone from Lemma of Clark [15], that is,
. Thus 
is a closed convex cone
.
The frictional market 
is strictly arbitrage-free if and only if
. We claim that
.
On the contrary, we assume that there exists
. Then 
and 
for 
and
. Thus
contradicting the fact that 
(which follows from 
in Proposition 4).
Note that
, then
.
Thus,
. Since both 
and 
are non-empty convex cones with vertices at the origin in the separable Banach space
, Theorem 5 in Clark [15] states that there exists a non-zero continuous linear functional 
on 
strictly separating 
from
. That is, 
for all 
and 
for all
.
The condition 
for all 
implies that 
is a strictly positive continuous linear functional on
. Thus 
is represented by some 
and 
by 
for any
.
The next proof is the same as the corresponding part in Theorem 3. Q.E.D.
[Remark] Theorem 4 means the first fundamental theorem of strict arbitrage-freeness with frictional security markets. If there exists no transaction costs, that is, 
and
, then Theorem 4 reduces to Theorem 2.
8. Concluding Remarks
In Section 4, we examine the following problem: whether the security prices 
can be represented by strictly positive continuous linear functional 
as
. More precisely, for the security prices 
and the returns
, we study the existence of strictly positive continuous linear functional 
as
. Ross [29] and Kreps [18] studied how to extend a linear functional 
on 
to 
that is strictly positive continuous. However, the existence of the linear functional 
on 
with 
is not at all an assured condition. It holds under certain conditions. For example, if
, our result is an analogue of theirs. In the equation
, each linear functional 
on 
corresponds to a vector 
and each vector 
corresponds to a linear functional 
on 
only if 
is column linear independent. Therefore, our results in Section 4 are the same as theirs under the assumption “
is column linear independent”. But in general, this may not hold.
Dalang, Morton and Willinger [20] and Jacod and Sgiryaev [22] studied arbitrage-free model, weakly arbitrage-free model, and strongly arbitrage-free model, provided simple proofs of the two fundamental theorems of asset pricing theory. Jacod and Sgiryaev [22] proved that the three concepts are equivalent to each another. In fact, the three concepts are strictly arbitrage-free securities’ price-return pair in Section 2 of our paper. Jacod and Sgiryaev [22] assumed that 
is an integrable (finite expectation) random variable (martingale condition). But we only assume 
is in the locally convex topological space 
for weakly arbitragefree frictionless market (Section 3), and 
is in the separable Banach space for strictly arbitrage-free frictionless market (Section 4). Therefore our study in Sections 2, 3 and 4 is more general than Jacod and Sgiryaev [22].
Pham and Touzi [26] studied frictional markets with the transaction cost rates for purchasing and selling the securities, addressed the problem of characterization of no arbitrage opportunity (strict arbitrage-freeness) in the presence of transaction costs in a discrete-time financial model, and extended the fundamental theorem of asset pricing under a non-degeneracy assumption. We also study the first fundamental valuation theorems of asset pricing from frictionless security markets to frictional security markets in Sections 5, 6 and 7. However, our results are different from those in Pham and Touzi [26] in the following two aspects. (1) The definition of no arbitrage opportunity in Pham and Touzi [26] is different from our definition of strict arbitrage-freeness. (2) Our proofs of Theorems 3 and 4 are different from Pham and Touzi [26]. In fact, we provide the proof by using theory of functional analysis, while Pham and Touzi [26] applied stochastic analysis to prove their results. Therefore, our result works for more general space than theirs.
Frictional economies are fundamentally different from their frictionless counterparts. The theory of general economic equilibrium for frictional economies with incomplete financial markets should be studied. We make the first step by establishing the corresponding no-arbitrage (that is, strictly arbitrage-free) pricing theory. From the first fundamental valuation theorems of asset pricing in general state space with transaction costs we have obtained here, one may further study the corresponding existence of general equilibrium for frictional economy with infinite-dimensional commodity space and incomplete financial markets.
In Section 7, we proved the equivalent conditions (Proposition 4 and Theorem 4) of strictly arbitrage-free frictional market by using Definition 4’. There are some difference between Definitions 4 and 4’. In fact, we can obtain the following (weaker) results for strictly arbitrage-free frictional market as defined by Definition 4.
[Proposition] If the frictional market 
is strictly arbitrage-free, then 
and 
intersect precisely at
, that is,
.
[Theorem] If the The frictional market 
is strictly arbitrage-free, then there exists a strictly positive functional 
satisfying
The proofs of Proposition and Theorem are from the proofs of Proposition 4 and Theorem 4. Conversely, we can’t check the sufficiency. The proof of Theorem 4 holds, too. If
where 
is a strictly positive vector, then, for any
,
then
that is,
If any portfolio 
of securities has a zero payoff
, then it has a positive total cost or gain 
for the portfolio
, and a positive total cost or gain 
for the portfolio
, thus
For any portfolio 
of securities, we can’t obtain a zero total cost or gain 
when it has a zero payoff
. That is to say, we can’t make sure that Definition 4 (2) holds. We can’t come to the conclusion that the frictional market 
is strictly arbitrage-free in the sense of Definition 4.
Funding
This work is supported by Social Sciences and Humanities Research Council of Canada (SSHRC Insight Development Grant Number: 430-2012-0698), National Natural Science Foundation of China (NSFC Grant Numbers: 70825003 and 71273271) Major Basic Research Plan of Renmin University of China (Grant Number: 14XNL001)
NOTES
*Corresponding author.