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Optimal portfolio selection under vanishing fixed transaction costs

Part of: Stochastic systems and control Mathematical finance

Published online by Cambridge University Press:  17 November 2017

Sören Christensen ,
Albrecht Irle  and
Andreas Ludwig
Sören Christensen*
Affiliation:
Hamburg University
Albrecht Irle*
Affiliation:
Christian-Albrechts-University Kiel
Andreas Ludwig*
Affiliation:
Christian-Albrechts-University Kiel
*
* Postal address: Department of Mathematics, Hamburg University, SPST, Bundesstraße 55, 20146 Hamburg, Germany. Email address: [email protected]
** Postal address: Mathematisches Seminar, Christian-Albrechts-University Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany.
** Postal address: Mathematisches Seminar, Christian-Albrechts-University Kiel, Ludewig-Meyn-Str. 4, D-24098 Kiel, Germany.
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Abstract

In this paper asymptotic results in a long-term growth rate portfolio optimization model under both fixed and proportional transaction costs are obtained. More precisely, the convergence of the model when the fixed costs tend to 0 is investigated. A suitable limit model with purely proportional costs is introduced and the convergence of optimal boundaries, asymptotic growth rates, and optimal risky fraction processes is rigorously proved. The results are based on an in-depth analysis of the convergence of the solutions to the corresponding Hamilton–Jacobi–Bellman equations.

Keywords

Impulse controlHunt processesthreshold ruleexcessive function

MSC classification

Primary: 91G10: Portfolio theory
Secondary: 93E20: Optimal stochastic control
Type
Research Article
Information
Advances in Applied Probability , Volume 49 , Issue 4 , December 2017 , pp. 1116 - 1143
Copyright
Copyright © Applied Probability Trust 2017 

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References

[1] Akian, M., Menaldi, J. L. and Sulem, A. (1996). On an investment-consumption model with transaction costs. SIAM J. Control Optimization 34, 329–364. Google Scholar
[2] Akian, M., Sulem, A. and Taksar, M. I. (2001). Dynamic optimization of long-term growth rate for a portfolio with transaction costs and logarithmic utility. Math. Finance 11, 153–188. Google Scholar
[3] Altarovici, A., Muhle-Karbe, J. and Soner, H. M. (2015). Asymptotics for fixed transaction costs. Finance Stoch. 19, 363–414. Google Scholar
[4] Belak, C. and Christensen, S. (2017). Utility maximization in a factor model with constant and proportional costs. Preprint. Available at https://ssrn.com/abstract=2774697. Google Scholar
[5] Bensoussan, A. and Lions, J.-L. (1982). Applications of Variational Inequalities in Stochastic Control (Studies Math. Appl. 12). North-Holland, Amsterdam. Google Scholar
[6] Bensoussan, A. and Lions, J.-L. (1984). Impulse Control and Quasivariational Inequalities. Gauthier-Villars, Montrouge. Google Scholar
[7] Bielecki, T. R. and Pliska, S. R. (2000). Risk sensitive asset management with transaction costs. Finance Stoch. 4, 1–33. Google Scholar
[8] Christensen, S. and Wittlinger, M. (2012). Optimal relaxed portfolio strategies for growth rate maximization problems with transaction costs. Preprint. Available at https://arxiv.org/abs/1209.0305. Google Scholar
[9] Davis, M. H. A. and Norman, A. R. (1990). Portfolio selection with transaction costs. Math. Operat. Res. 15, 676–713. Google Scholar
[10] Eastham, J. F. and Hastings, K. J. (1988). Optimal impulse control of portfolios. Math. Operat. Res. 13, 588–605. Google Scholar
[11] Ethier, S. N. and Kurtz, T. G. (1986). Markov Processes: Characterization and Convergence. John Wiley, New York. CrossRefGoogle Scholar
[12] Feodoria, M.-R. (2016). Optimal investment and utility indifference pricing in the presence of small fixed transaction costs. Doctoral thesis. University of Kiel. Google Scholar
[13] Gīihman, Ĭ. Ī. and Skorokhod, A. V. (1972). Stochastic Differential Equations. Springer, New York. CrossRefGoogle Scholar
[14] Guasoni, P. and Weber, M. (2015). Nonlinear price impact and portfolio choice. Preprint. Available at https://ssrn.com/abstract=2613284. Google Scholar
[15] Irle, A. and Prelle, C. (2009). A renewal theoretic result in portfolio theory under transaction costs with multiple risky assets. Statist. Decisions 27, 211–233. Google Scholar
[16] Irle, A. and Sass, J. (2006). Good portfolio strategies under transaction costs: a renewal theoretic approach. In Stochastic Finance, Springer, New York, pp. 321–341. Google Scholar
[17] Irle, A. and Sass, J. (2006). Optimal portfolio policies under fixed and proportional transaction costs. Adv. Appl. Probab. 38, 916–942. Google Scholar
[18] Jacod, J. and Shiryaev, A. N. (2003). Limit Theorems for Stochastic Processes, 2nd edn. Springer, Berlin. Google Scholar
[19] Kelly, J. L., Jr. (1956). A new interpretation of information rate. Bell. System Tech. J. 35, 917–926. Google Scholar
[20] Korn, R. (1998). Portfolio optimisation with strictly positive transaction costs and impulse control. Finance Stoch. 2, 85–114. CrossRefGoogle Scholar
[21] Kruk, L., Lehoczky, J., Ramanan, K. and Shreve, S. (2007). An explicit formula for the Skorokhod map on [0, a]. Ann. Prob. 35, 1740–1768. Google Scholar
[22] Ludwig, A. (2012). Comparisons and asymptotics in the theory of portfolio optimization under fixed and proportional transaction costs. Doctoral thesis. University of Kiel. Google Scholar
[23] Magill, M. J. P. and Constantinides, G. M. (1976). Portfolio selection with transactions costs. J. Econom. Theory 13, 245–263. CrossRefGoogle Scholar
[24] Melnyk, Y. and Seifried, F. T. (2017). Small-cost asymptotics for long-term growth rates in incomplete markets. To appear in Math. Finance.. Google Scholar
[25] Merton, R. C. (1969). Lifetime portfolio selection under uncertainty: The continuous-time case. Rev. Econom. Statist. 51, 247–257. CrossRefGoogle Scholar
[26] Morton, A. J. and Pliska, S. R. (1995). Optimal portfolio management with fixed transaction costs. Math. Finance 5, 337–356. CrossRefGoogle Scholar
[27] Nagai, H. (2004). Risky fraction processes and problems with transaction costs. In Stochastic Processes and Applications to Mathematical Finance, World Scientific, River Edge, NJ, pp. 271–288. Google Scholar
[28] Shreve, S. E. and Soner, H. M. (1994). Optimal investment and consumption with transaction costs. Ann. Appl. Prob. 4, 609–692. CrossRefGoogle Scholar
[29] Sulem, A. (1997). Dynamic optimization for a mixed portfolio with transaction costs. In Numerical Methods in Finance (Publ. Newton Inst. 13), Cambridge University Press, pp. 165–180. CrossRefGoogle Scholar
[30] Taksar, M., Klass, M. J. and Assaf, D. (1988). A diffusion model for optimal portfolio selection in the presence of brokerage fees. Math. Operat. Res. 13, 277–294. Google Scholar
[31] Tamura, T. (2006). Maximizing the growth rate of a portfolio with fixed and proportional transaction costs. Appl. Math. Optimization 54, 95–116. CrossRefGoogle Scholar
[32] Tamura, T. (2008). Maximization of the long-term growth rate for a portfolio with fixed and proportional transaction costs. Adv. Appl. Prob. 40, 673–695. Google Scholar
[33] Von Landau, E. (1914). Einige Ungleichungen für zweimal differentiierbare Funktionen. Proc. London Math. Soc. s2-13, 43–49. CrossRefGoogle Scholar