We consider a simple multi-asset discrete-time model of a currency market with transaction costs assuming the finite number of states of the nature. Defining two kinds of arbitrage opportunities we study necessary and sufficient conditions for the absence of arbitrage. Our main result is a natural extension of the Harrison–Pliska theorem on asset pricing. We prove also a hedging theorem without supplementary hypotheses.
The famous result of Harrison and Pliska (1981), known also as the ‘fundamental theorem’ on asset (or arbitrage) pricing (FTAP) asserts that a frictionless financial market is free of arbitrage if and only if the price process is a martingale under a probability measure equivalent to the objective one. The original formulation involved the assumption that the underlying probability space (Ω, View the MathML source, P) (in other words, the number of states of the nature) is finite; it has been removed in the subsequent study of Dalang et al. (1990). It is worth to note that the passage from finite to infinite Ω is by no means trivial: instead of purely geometric considerations (which make the Harrison–Pliska theorem so attractive for elementary courses in financial economics) much more delicate topological or measure-theoretical arguments must be used. These mathematical aspects were investigated in details by a number of authors (e.g. Stricker, 1990, Schachermayer, 1992, Kabanov and Kramkov, 1994, Rogers, 1994 and Jacod and Shiryayev, 1998). The aim of this note is to present an extension of the arbitrage pricing theorem for a multi-asset multi-period model with finite Ω and proportional transaction costs. We use the geometric formalism developed in Kabanov (1999), Delbaen et al. (2001) and Kabanov and Last (2001). Our result makes clear that the concept of the equivalent martingale measure, though useful in the context of frictionless market models, has no importance (and even misleading) in a more realistic situation of transaction costs. We track how the dual variables in our more general model “degenerate” into densities of martingale measures. Our paper contains also a hedging theorem which is free from any auxiliary assumptions (cf. with results in Kabanov (1999), Delbaen et al. (2001) and Kabanov and Last (2001)). In spite the results being mathematically simple, they are not deductible, to the best of our knowledge, from the existing literature. Their extensions for the arbitrary probability space and, further, to the continuous-time setting, require more sophisticated tools and will be published elsewhere.
The reader is invited to compare the suggested approach to that of the important article by Jouini and Kallal (1995), conceptually different not only at the level of modeling (continuous-time setting with bid and ask prices) but also in the formulation of the no-arbitrage criteria, see the end of our note. An attempt to find the arbitrage pricing theorem (for the binomial model) can be found in the preprint (Shirakawa and Konno, 1995).Do not assuming that the reader has any background in random processes above the definitions and elementary properties of martingales and supermartingales, we explain standard (and very convenient) notations of stochastic calculus used throughout the paper. Namely, for X=(Xt) and Y=(Yt) we define X−≔(Xt−1), ΔXt≔Xt−Xt−1, and, at last,
View the MathML source
for the discrete-time integral (here X and Y can be scalar or vector-valued). For finite Ω, if X is a predictable process (i.e. X− is adapted) and Y is a martingale, then XY is a martingale. The product formula View the MathML source is obvious. The books, Rockafellar (1970) and Pshenychnyi (1980), may serve as references in convex analysis.
One can observe that the different matrices of transaction costs coefficients may generate
the same geometric structures of the problem (i.e. the solvency cone). Moreover, in
multi-asset models sometimes is tacitly assumed that all exchanges are done through the
money and only transaction costs coefficients for buying and selling assets (1i and i1) are
specified. In the need, one can complete the matrix by assigning sufficiently large values
of the remaining transaction costs coefficients to make the direct transfers prohibitively
expensive. As an alternative, a purely geometric approach seems to be useful. The model
considered in this paper allows easily for the following generalization. Assume that the portfolio
dynamics is given by relation (1) (or relation (3)) where the adapted process B is such
that its increments 1Bt take values in the polyhedral cones −Mt (eventually, depending on
! in a causal way). Defining the solvency cones Kt :D Mt C Rd
C, we can prove Theorems1 and 2 in this purely geometric framework assuming only that intKt Rd
Cnf0g. The mapping
in this context will be the projection on the quotient space Rd=F . Such ramifications
may have a certain importance also for comparing various models with transaction costs
including those where baskets of currencies are exchanged.
At last, let us consider the model where S1 1, i.e. the first asset (“money”) is the
numéraire, and for all i and j
.1 C i1/.1 C 1j / 1 C ij:
This means that the direct exchanges are not less expensive than those via money; they can
be excluded at all (as it is usually done in stock market models). According to Eq. (6) the
cone K consists of all w 2 Rd
C satisfying the inequalities
1
1 C i1w1 wi .1 C 1i/w1; i>1:
Indeed, it follows that for any pair i, j we have
wj .1 C 1j/w1 .1 C i1/.1 C 1j/wi .1 C ij/wi :
By Theorem 1 the condition NAw
T holds if and only if there is a process Z 2 D with Zi
T > 0.
In particular, OZ
1 D Z1 is a martingale; we can always assume that EZ1
T
D 1 and define the
probability Q P D Z1
T P. The condition that OZ
evolves in K reads as
1
1 C i1Z1 Zi
Si
.1 C 1i/Z1; i>1:
Putting QS
i :D Zi=Z1 and introducing the selling and buying prices
S¯
i :D 1
1 C i1 Si ; NS
i :D .1 C 1i/Si ;
we conclude that NAw
T holds if and only if there is a process QS
and an equivalent probability
measure Q P such that QS
is a martingale with respect to Q P and
S¯
i QS
i NS
i; i>1:
Thus, in this case our criteria for NAw
T coincides with that suggested in Jouini and Kallal
(1995).
