Advertisement

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

  • Christoph Kühn
Article
  • 20 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. 1.
    Auerbach, A., Bradford, D.: Generalized cash-flow taxation. J. Public Econ. 88, 957–980 (2004)CrossRefGoogle Scholar
  2. 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. 3.
    Black, F.: The dividend puzzle. J. Portf. Manag. 2, 5–8 (1976)CrossRefGoogle Scholar
  4. 4.
    Bradford, D.: Taxation, Wealth, and Saving. MIT Press, Cambridge (2000)Google Scholar
  5. 5.
    Constantinides, G.M.: Capital market equilibrium with personal taxes. Econometrica 51, 611–636 (1983)MathSciNetCrossRefGoogle Scholar
  6. 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. 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. 8.
    Dybvig, P., Koo, H.: Investment with taxes. Working paper, Washington University, St. Louis, MO (1996)Google Scholar
  9. 9.
    Dybvig, P., Ross, S.: Tax clienteless and asset pricing. J. Finance 41, 751–762 (1986)Google Scholar
  10. 10.
    Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. Walter de Gruyter, Berlin (2011)CrossRefGoogle Scholar
  11. 11.
    Gallmeyer, M., Srivastava, S.: Arbitrage and the tax code. Math. Financ. Econ. 4, 183–221 (2011)MathSciNetCrossRefGoogle Scholar
  12. 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. 13.
    He, S., Wang, J., Yan, J.: Semimartingale Theory and Stochastic Calculus. CRC Press, Boca Raton (1992)zbMATHGoogle Scholar
  14. 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. 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. 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. 17.
    Kabanov, Y., Safarian, M.: Markets with Transaction Costs. Springer, Berlin (2009)zbMATHGoogle Scholar
  18. 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. 19.
    Napp, C.: The Dalang–Morton–Willinger theorem under cone constraints. J. Math. Econ. 39, 111–126 (2003)MathSciNetCrossRefGoogle Scholar
  20. 20.
    Pham, H., Touzi, N.: The fundamental theorem of asset pricing with cone constraints. J. Math. Econ. 31, 265–279 (1999)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Ross, S.: Arbitrage and martingales with taxation. J. Polit. Econ. 95, 371–393 (1987)CrossRefGoogle Scholar
  22. 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. 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

Copyright information

© Springer-Verlag GmbH Germany, part of Springer Nature 2018

Authors and Affiliations

  1. 1.Institut für MathematikGoethe-Universität FrankfurtFrankfurt amGermany

Personalised recommendations