On the density of properly maximal claims in financial markets with transaction costs

Abstract

We consider trading in a financial market with proportional transaction costs. In the frictionless case, claims are maximal if and only if they are priced by a consistent price process-the equivalent of an equivalent martingale measure. This result fails in the presence of transaction costs. A properly maximal claim is one which does have this property. We show that the properly maximal claims are dense in the set of maximal claims (with the topology of convergence in probability). This is an electronic reprint of the original article published by the Institute of Mathematical Statistics in The Annals of Applied Probability, 2007, Vol. 17, No. 2, 716-740. This reprint differs from the original in pagination and typographic detail. 1 2 S. JACKA AND A. BERKAOUI this is achieved by seeking maximal claims-claims Y which are maximal in A with respect to the partial order(see [6,7,8]). It follows from Kramkov's celebrated result on optional decompositions ([20]) that, at least in a discrete-time context, a claim X in A is maximal if and only if it is priced at 0 by some EMM. It also follows that this is true if and only if [A, X], the cone generated by A and −X, is arbitrage-free, in which case its closure is also arbitrage-free. Consequently (see [13] or [6]), one may obtain a hedging strategy for a maximal claim by martingale representation.Regrettably, when there are transaction costs, just as A may be arbitragefree butĀ contain an arbitrage, so, in this context, a claim X may be maximal and yet the closure of [A, X] contain an arbitrage.In the language of optimization theory, a maximal claim such that the closure of [A, X] is arbitrage-free is said to be proper efficient with respect to L 0,+ . We shall refer to such claims as properly maximal. We shall show in Theorem 2.9 that a properly maximal claim is priced by some consistent price process and that martingale representation can be used to obtain a hedging strategy. It is then of interest (for hedging purposes) as to whether one can approximate maximal claims by properly maximal claims. This is a problem with a long and distinguished history in optimization theory, going back to [1]. We give a positive answer (up to randomization) in Theorem 4.11: the collection of properly maximal claims is dense in the set of maximal claims.In a continuous time framework, the problem is more delicate. Indeed, the task of defining a notion of admissible trading strategy, that has a meaningful financial interpretation, is still in progress. A first solution has been given by Kabanov [15], Kabanov and Last [18] and Kabanov and Stricker [19], where the efficient friction assumption was made. More precisely, an admissible self-financing trading strategy was defined as an adapted, vector-valued, cádlág process of finite variation whose increments lie in the corresponding trading/solvency cones and whose terminal value is bounded from below by a constant with respect to the order induced by the terminal solvency cone. Campi and Schachermayer [4] extend these results to bid-ask pro...

Von Neumann–Gale model, market frictions and capital growth

“…In this chain of relations, the first inequality holds by virtue of (33) and because (x 0 , λ 1 x 1 ) ∈ G 1 (ω) a.s. (see (36)). The second follows from (37).…”

“…The size of the bid-ask spreads for realistic values of transaction costs has been discussed, e.g., by Zhao and Ziemba [75,76] and Leland [49]. An important role in establishing the field was played by the work of Jouini and Kallal [38], Pham and Touzi [61], Kabanov [39], Koehl et al [45,46], Cadenillas and Pliska [10], Cadenillas [9], Chalasani and Jha [12], Akian et al [1], Bouchard and Touzi [7], Delbaen and Kabanov [16], Kabanov et al [40], Janeček and Shreve [32], De Valliere and Kabanov [19], Liu and Loewenstein [51], Jacka and Berkaoui [36], Jacka et al [37], Guasoni et al [29], Albanese and Tompaidis [2], and others. For a comprehensive review of this literature we refer to the monograph "Markets with Transaction Costs" by Kabanov and Safarian [42].…”

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“…However, with transaction costs, the definition of acceptable portfolios is not clear. Indeed, the naive choice of the maximal element from the admissible portfolio processes with transaction costs can not be applied as lower bounds, see [14] for example.…”

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