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.
Similar content being viewed by others
References
Auerbach, A., Bradford, D.: Generalized cash-flow taxation. J. Public Econ. 88, 957–980 (2004)
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)
Black, F.: The dividend puzzle. J. Portf. Manag. 2, 5–8 (1976)
Bradford, D.: Taxation, Wealth, and Saving. MIT Press, Cambridge (2000)
Constantinides, G.M.: Capital market equilibrium with personal taxes. Econometrica 51, 611–636 (1983)
Dalang, R., Morton, A., Willinger, W.: Equivalent martingale measures and no-arbitrage in stochastic securities market models. Stoch. Stoch. Rep. 29, 185–201 (1990)
Dammon, R., Green, R.: Tax arbitrage and the existence of equilibrium prices for financial assets. J Finance 42, 1143–1166 (1987)
Dybvig, P., Koo, H.: Investment with taxes. Working paper, Washington University, St. Louis, MO (1996)
Dybvig, P., Ross, S.: Tax clienteless and asset pricing. J. Finance 41, 751–762 (1986)
Föllmer, H., Schied, A.: Stochastic Finance: An Introduction in Discrete Time, 3rd edn. Walter de Gruyter, Berlin (2011)
Gallmeyer, M., Srivastava, S.: Arbitrage and the tax code. Math. Financ. Econ. 4, 183–221 (2011)
Grigoriev, P.: On low dimensional case in the fundamental asset pricing theorem with transaction costs. Stat. Decis. 23, 33–48 (2005)
He, S., Wang, J., Yan, J.: Semimartingale Theory and Stochastic Calculus. CRC Press, Boca Raton (1992)
Jensen, B.: Valuation before and after tax in the discrete time, finite state no arbitrage model. Ann. Finance 5, 91–123 (2009)
Jouini, E., Koehl, P.-F., Touzi, N.: Optimal investment with taxes: an optimal control problem with endogenous delay. Nonlinear Anal. 37, 31–56 (1999)
Jouini, E., Koehl, P.-F., Touzi, N.: Optimal investment with taxes: an existence result. J. Math. Econ. 33, 373–388 (2000)
Kabanov, Y., Safarian, M.: Markets with Transaction Costs. Springer, Berlin (2009)
Kühn, C., Ulbricht, B.: Modeling capital gains taxes for trading strategies of infinite variation. Stoch. Anal. Appl. 33, 792–822 (2015)
Napp, C.: The Dalang–Morton–Willinger theorem under cone constraints. J. Math. Econ. 39, 111–126 (2003)
Pham, H., Touzi, N.: The fundamental theorem of asset pricing with cone constraints. J. Math. Econ. 31, 265–279 (1999)
Ross, S.: Arbitrage and martingales with taxation. J. Polit. Econ. 95, 371–393 (1987)
Schachermayer, W.: A Hilbert space proof of the fundamental theorem of asset pricing in finite discrete time. Insur. Math. Econ. 11, 249–257 (1992)
Schachermayer, W.: The fundamental theorem of asset pricing under proportional transaction costs in finite discrete time. Math. Finance 14, 19–48 (2004)
Author information
Authors and Affiliations
Corresponding author
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.
Rights and permissions
About this article
Cite this article
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
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s11579-018-0230-7