Log-optimal investment in the long run with proportional transaction costs when using shadow prices

Dostál, Petr ; Klůjová, Jana

Kybernetika, Tome 51 (2015) no. 4, pp. 588-628

Abstract (VO)

We consider a non-consuming agent interested in the maximization of the long-run growth rate of a wealth process investing either in a money market and in one risky asset following a geometric Brownian motion or in futures following an arithmetic Brownian motion. The agent faces proportional transaction costs, and similarly as in [17] where the case of stock trading is considered, we show how the log-optimal optimal policies in the long run can be derived when using the technical tool of shadow prices. We also provide a brief link between technical tools used in this paper and the ones used in [14,15,17].

Détail

DOI : 10.14736/kyb-2015-4-0588

Classification : 60G44, 60H30, 91B28
Keywords: proportional transaction costs; logarithmic utility; shadow prices

Bibliographie

[1] Algoet, P. H., Cover, T. M.: Asymptotic optimality and asymptotic equipartition properties of log-optimum investment. Ann. Probab. 16 (1988), 2, 876-898. | DOI | MR

[2] Akian, M., Menaldi, J. L., Sulem, A.: On an investment-consumption model with transaction costs. SIAM J. Control Optim. 34 (1996), 1, 329-364. | DOI | MR | Zbl

[3] Akian, M., Sulem, A., Taksar, M. I.: Dynamic optimization of long-term Growth rate for a portfolio with transaction costs and logarithmic utility. Math. Finance 11 (2001), 2, 153-188. | DOI | MR | Zbl

[4] Bayer, Ch., Veliyev, B.: Utility Maximization in a Binomial Model with Transaction Costs: a Duality Approach Based on the Shadow Price Process. arXiv: 1209.5175. | MR

[5] Benedetti, G., Campi, L., Kallsen, J., Muhle-Karbe, J.: On the existence of shadow prices. Finance Stoch. 17 (2013), 801-818. | DOI | MR | Zbl

[6] Choi, J. H., Sirbu, M., Zitkovix, G.: Shadow Prices and well-posedness in the problem of optimal investment and consumption with transaction costs. SIAM J. Control Optim. 51 (2013), 6, 4414-4449. | DOI | MR

[7] Czichowsky, Ch., Muhle-Karbe, J., Schachermayer, W.: Transaction costs, shadow prices, and duality in discrete time. SIAM J. Financial Math. 5 (2014), 1, 258-277. | DOI | MR | Zbl

[8] Bell, R. M., Cover, T. M.: Competitive optimality of logarithmic investment. Math. Oper. Res. 5 (1980), 2, 161-166. | DOI | MR | Zbl

[9] Bell, R., Cover, T. M.: Game-theoretic optimal portfolios. Management Sci. 34 (1998), 6, 724-733. | DOI | MR | Zbl

[10] Breiman, L.: Optimal gambling system for flavorable games. In: Proc. Fourth Berkeley Symp. on Math. Statist. and Prob. 1 (J. Neyman, ed.), Univ. of Calif. Press, Berkeley 1961, pp. 65-78. | MR

[11] Browne, S., Whitt, W.: Portfolio choice and the Bayesian Kelly criterion. Adv. in Appl. Probab. 28 (1996), 4, 1145-1176. | DOI | MR | Zbl

[12] Davis, M., Norman, A.: Portfolio selection with transaction costs. Math. Oper. Res. 15 (1990), 4, 676-713. | DOI | MR | Zbl

[13] Dostál, P.: Almost optimal trading strategies for small transaction costs and a HARA utility function. J. Comb. Inf. Syst. Sci. 38 (2010), 257-291.

[14] Dostál, P.: Futures trading with transaction costs. In: Proc. ALGORITMY 2009. (A. Handlovičová, P. Frolkovič, K. Mikula, and D. Ševčovič, eds.), Slovak Univ. of Tech. in Bratislava, Publishing House of STU, Bratislava 2009, pp. 419-428. | Zbl

[15] Dostál, P.: Investment strategies in the long run with proportional transaction costs and HARA utility function. Quant. Finance 9 (2009), 2, 231-242. | DOI | MR

[16] Gerhold, S., Muhle-Karbe, J., Schachermayer, W.: Asymptotics and duality for the Davis and Norman problem. Stochastics 84 (2012), 5-6, 625-641. | DOI | MR | Zbl

[17] Gerhold, S., Muhle-Karbe, J., Schachermayer, W.: The dual optimizer for the growth-optimal portfolio under transaction costs. Finance Stoch. 17 (2013), 2, 325-354. | DOI | MR | Zbl

[18] Goll, T., Kallsen, J.: Optimal portfolios for logarithmic utility. Stochastic Process. Appl. 89 (2000), 1, 31-48. | DOI | MR | Zbl

[19] Herczegh, A., Prokaj, V.: Shadow\! Price in the Power Utility Case. arXiv: 1112.4385. | MR

[20] Janeček, K.: Optimal Growth in Gambling and Investing. MSc Thesis, Charles University Prague 1999.

[21] Janeček, K., Shreve, S. E.: Asymptotic analysis for optimal investment and consumption with transaction costs. Finance Stoch. 8 (2004), 2, 181-206. | DOI | MR

[22] Janeček, K., Shreve, S. E.: Futures trading with transaction costs. Illinois J. Math. 54 (2010), 4, 1239-1284. | MR | Zbl

[23] Kallenberg, O.: Foundations of Modern Probability. Springer Verlag, Heidelberg 1997. | MR | Zbl

[24] Kallsen, J., Muhle-Karbe, J.: On using shadow in portfolio optimization with transaction costs. Ann. Appl. Probab. 20 (2010), 4, 1341-1358. | DOI | MR

[25] Kallsen, J., Muhle-Karbe, J.: Existence of shadow prices in finite probability spaces. Math. Methods Oper. Res. 73 (2011), 2, 251-262. | DOI | MR | Zbl

[26] Kallsen, J., Muhle-Karbe, J.: The General Structure of Optimal Investment and Consumption with Small Transaction Costs. arXiv: 1303.3148.

[27] Kelly, J. L.: A new interpretation of information rate. Bell Sys. Tech. J. 35 (1956), 4, 917-926. | DOI | MR

[28] Magill, M. J. P., Constantinides, G. M.: Portfolio selection with transaction costs. J. Econom. Theory 13 (1976), 2, 245-263. | DOI | MR

[29] Merton, R. C.: Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3 (1971), 4, 373-413. Erratum 6 (1973), 2, 213-214, | DOI | MR | Zbl

[30] Morton, A. J., Pliska, S.: Optimal portfolio management with fixed transaction costs. Math. Finance 5 (1995), 4, 337-356. | DOI | Zbl

[31] Revuz, D., Yor, M.: Continuous Martingales and Brownian Motion. Springer Verlag, Heidelberg, Berlin, New York 1999. | DOI | MR | Zbl

[32] Rokhlin, D. B.: On the game interpretation of a shadow price process in utility maximization problems under transaction costs. Finance Stoch. 17 (2013), 4, 819-838. | DOI | MR | Zbl

[33] Rotando, L. M., Thorp, E. O.: The Kelly criterion and the stock market. Amer. Math. Monthly 99 (1992), 10, 922-931. | DOI | MR | Zbl

[34] Samuelson, P. A.: The ``fallacy" of maximizing the geometric mean in long sequences of investing or gambling. Proc. Natl. Acad. Sci. 68 (1971), 10, 2493-2496. | DOI | MR

[35] Sass, J., Schäl, M.: Numeraire portfolios and utility-based price systems under proportional transaction costs. Decis. Econ. Finance 37 (2014), 2, 195-234. | DOI | MR

[36] Shreve, S. E., Soner, H. M.: Optimal investment and consumption with transaction costs. Ann. Appl. Probab. 4 (1994), 3, 609-692. | DOI | MR | Zbl

[37] Skorokhod, A.: Stochastic equations for diffusion processes in a bounded region. Theory Probab. Appl. 6 (1961), 3, 264-274. | DOI | Zbl

[38] Skorokhod, A.: Stochastic equations for diffusion processes in a bounded region II. Theory Probab. Appl. 7 (1962), 1, 3-23. | DOI | Zbl

[39] Thorp, E.: Portfolio choice and the Kelly criterion. In: Stochastic Optimization Models in Finance (W. T. Ziemba and R. G. Vickson, eds.), Acad. Press, Bew York 1975, pp. 599-619. | DOI

[40] Thorp, E.: The Kelly criterion in blackjack, sports betting and the stock market. In: Finding the Edge: Mathematical Analysis of Casino Games (O. Vancura, J. A. Cornelius and W. R. Eadington, eds.), Institute for the Study of Gambling and Commercial Gaming, Reno 2000, pp. 163-213.