Admissible Trading Strategies Under Transaction Costs

  • Walter Schachermayer
Part of the Lecture Notes in Mathematics book series (LNM, volume 2123)


A well known result in stochastic analysis reads as follows: for an \(\mathbb{R}\)-valued super-martingale X = (X t )0 ≤ t ≤ T such that the terminal value X T is non-negative, we have that the entire process X is non-negative. An analogous result holds true in the no arbitrage theory of mathematical finance: under the assumption of no arbitrage, an admissible portfolio process x + (H ⋅ S) verifying x + (H ⋅ S) T  ≥ 0 also satisfies x + (H ⋅ S) t  ≥ 0, for all 0 ≤ t ≤ T. In the present paper we derive an analogous result in the presence of transaction costs. In fact, we give two versions: one with a numéraire-based, and one with a numéraire-free notion of admissibility. It turns out that this distinction on the primal side perfectly corresponds to the difference between local martingales and true martingales on the dual side. A counter-example reveals that the consideration of transaction costs makes things more delicate than in the frictionless setting.


Transaction Cost Trading Strategy Initial Endowment Local Martingale Predictable Process 
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I warmly thank Irene Klein without whose encouragement this note would not have been written and who strongly contributed to its shaping. Thanks go also to Christoph Czichowsky for his advice on some of the subtle technicalities of this note. I thank an anonymous referee for careful reading and for pointing out a number of inaccuracies.


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Copyright information

© Springer International Publishing Switzerland 2014

Authors and Affiliations

  1. 1.Fakultät für MathematikUniversität WienWienAustria

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