How local in time is the no-arbitrage property under capital gains taxes?

Abstract

In frictionless financial markets, no-arbitrage is a local property in time. This means that a discrete time model is arbitrage-free if and only if there does not exist a one-period-arbitrage. With capital gains taxes, this equivalence fails. For a model with a linear tax and one non-shortable risky stock, we introduce the concept of robust local no-arbitrage (RLNA) as the weakest local condition which guarantees dynamic no-arbitrage. Under a sharp dichotomy condition, we prove (RLNA). Since no-one-period-arbitrage is necessary for no-arbitrage, the latter is sandwiched between two local conditions, which allows us to estimate its non-locality. Furthermore, we construct a stock price process such that two long positions in the same stock hedge each other. This puzzling phenomenon that cannot occur in arbitrage-free frictionless markets (or markets with proportional transaction costs) is used to show that no-arbitrage alone does not imply the existence of an equivalent separating measure if the probability space is infinite. Finally, we show that the model with a linear tax on capital gains can be written as a model with proportional transaction costs by introducing several fictitious securities.

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Correspondence to Christoph Kühn.

Additional information

I would like to thank the editor, Prof. Riedel, and an anonymous associate editor for their valuable comments. I am especially grateful to the anonymous referee for finding a minor error in the previous version of Proposition 2.15 and for many valuable suggestions that lead to a substantial improvement of the presentation of the results.

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Kühn, C. How local in time is the no-arbitrage property under capital gains taxes?. Math Finan Econ 13, 329–358 (2019). https://doi.org/10.1007/s11579-018-0230-7

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Keywords

  • Arbitrage
  • Capital gains taxes
  • Deferment of taxes
  • Proportional transaction costs

Mathematics Subject Classification

  • 91G10
  • 91B60

JEL classification

  • G10
  • H20