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

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## 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.

## Keywords

Arbitrage Capital gains taxes Deferment of taxes Proportional transaction costs## Mathematics Subject Classification

91G10 91B60## JEL classification

G10 H20## References

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