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

- 100 Downloads

## 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

- 1.Auerbach, A., Bradford, D.: Generalized cash-flow taxation. J. Public Econ.
**88**, 957–980 (2004)CrossRefGoogle Scholar - 2.Ben Tahar, I., Soner, M., Touzi, N.: The dynamic programming equation for the problem of optimal investment under capital gains taxes. SIAM J. Control Optim.
**46**, 1779–1801 (2007)MathSciNetCrossRefGoogle Scholar - 3.Black, F.: The dividend puzzle. J. Portf. Manag.
**2**, 5–8 (1976)CrossRefGoogle Scholar - 4.Bradford, D.: Taxation, Wealth, and Saving. MIT Press, Cambridge (2000)Google Scholar
- 5.Constantinides, G.M.: Capital market equilibrium with personal taxes. Econometrica
**51**, 611–636 (1983)MathSciNetCrossRefGoogle Scholar - 6.Dalang, R., Morton, A., Willinger, W.: Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stoch. Stoch. Rep.
**29**, 185–201 (1990)MathSciNetCrossRefGoogle Scholar - 7.Dammon, R., Green, R.: Tax arbitrage and the existence of equilibrium prices for financial assets. J Finance
**42**, 1143–1166 (1987)CrossRefGoogle Scholar - 8.Dybvig, P., Koo, H.: Investment with taxes. Working paper, Washington University, St. Louis, MO (1996)Google Scholar
- 9.Dybvig, P., Ross, S.: Tax clienteless and asset pricing. J. Finance
**41**, 751–762 (1986)Google Scholar - 10.Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. Walter de Gruyter, Berlin (2011)CrossRefGoogle Scholar
- 11.Gallmeyer, M., Srivastava, S.: Arbitrage and the tax code. Math. Financ. Econ.
**4**, 183–221 (2011)MathSciNetCrossRefGoogle Scholar - 12.Grigoriev, P.: On low dimensional case in the fundamental asset pricing theorem with transaction costs. Stat. Decis.
**23**, 33–48 (2005)MathSciNetzbMATHGoogle Scholar - 13.He, S., Wang, J., Yan, J.: Semimartingale Theory and Stochastic Calculus. CRC Press, Boca Raton (1992)zbMATHGoogle Scholar
- 14.Jensen, B.: Valuation before and after tax in the discrete time, finite state no arbitrage model. Ann. Finance
**5**, 91–123 (2009)CrossRefGoogle Scholar - 15.Jouini, E., Koehl, P.-F., Touzi, N.: Optimal investment with taxes: an optimal control problem with endogenous delay. Nonlinear Anal.
**37**, 31–56 (1999)MathSciNetCrossRefGoogle Scholar - 16.Jouini, E., Koehl, P.-F., Touzi, N.: Optimal investment with taxes: an existence result. J. Math. Econ.
**33**, 373–388 (2000)MathSciNetCrossRefGoogle Scholar - 17.Kabanov, Y., Safarian, M.: Markets with Transaction Costs. Springer, Berlin (2009)zbMATHGoogle Scholar
- 18.Kühn, C., Ulbricht, B.: Modeling capital gains taxes for trading strategies of infinite variation. Stoch. Anal. Appl.
**33**, 792–822 (2015)MathSciNetCrossRefGoogle Scholar - 19.Napp, C.: The Dalang–Morton–Willinger theorem under cone constraints. J. Math. Econ.
**39**, 111–126 (2003)MathSciNetCrossRefGoogle Scholar - 20.Pham, H., Touzi, N.: The fundamental theorem of asset pricing with cone constraints. J. Math. Econ.
**31**, 265–279 (1999)MathSciNetCrossRefGoogle Scholar - 21.Ross, S.: Arbitrage and martingales with taxation. J. Polit. Econ.
**95**, 371–393 (1987)CrossRefGoogle Scholar - 22.Schachermayer, W.: A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insur. Math. Econ.
**11**, 249–257 (1992)MathSciNetCrossRefGoogle Scholar - 23.Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Finance
**14**, 19–48 (2004)MathSciNetCrossRefGoogle Scholar