Kybernetika 51 no. 4, 588-628, 2015

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

Petr Dostál and Jana KlůjováDOI: 10.14736/kyb-2015-4-0588

Abstract:

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].

Keywords:

proportional transaction costs, logarithmic utility, shadow prices

Classification:

60H30, 60G44, 91B28

paper.pdf

References:

  1. P. H. Algoet and T. M. Cover: Asymptotic optimality and asymptotic equipartition properties of log-optimum investment. Ann. Probab. 16 (1988), 2, 876-898.   DOI:10.1214/aop/1176991793
  2. M. Akian, J. L. Menaldi and A. Sulem: On an investment-consumption model with transaction costs. SIAM J. Control Optim. 34 (1996), 1, 329-364.   DOI:10.1137/s0363012993247159
  3. M. Akian, A. Sulem and M. I. Taksar: Dynamic optimization of long-term Growth rate for a portfolio with transaction costs and logarithmic utility. Math. Finance 11 (2001), 2, 153-188.   DOI:10.1111/1467-9965.00111
  4. Ch. Bayer and B. Veliyev: Utility Maximization in a Binomial Model with Transaction Costs: a Duality Approach Based on the Shadow Price Process. arXiv: 1209.5175.   CrossRef
  5. G. Benedetti, L. Campi, J. Kallsen and J. Muhle-Karbe: On the existence of shadow prices. Finance Stoch. 17 (2013), 801-818.   DOI:10.1007/s00780-012-0201-4
  6. J. H. Choi, M. Sirbu and G. Zitkovix: 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:10.1137/120881373
  7. Ch. Czichowsky, J. Muhle-Karbe and W. Schachermayer: Transaction costs, shadow prices, and duality in discrete time. SIAM J. Financial Math. 5 (2014), 1, 258-277.   DOI:10.1137/130925864
  8. R. M. Bell and T. M. Cover: Competitive optimality of logarithmic investment. Math. Oper. Res. 5 (1980), 2, 161-166.   DOI:10.1287/moor.5.2.161
  9. R. Bell and T. M. Cover: Game-theoretic optimal portfolios. Management Sci. 34 (1998), 6, 724-733.   DOI:10.1287/mnsc.34.6.724
  10. L. Breiman: 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.   CrossRef
  11. S. Browne and W. Whitt: Portfolio choice and the Bayesian Kelly criterion. Adv. in Appl. Probab. 28 (1996), 4, 1145-1176.   DOI:10.2307/1428168
  12. M. Davis and A. Norman: Portfolio selection with transaction costs. Math. Oper. Res. 15 (1990), 4, 676-713.   DOI:10.1287/moor.15.4.676
  13. P. Dostál: Almost optimal trading strategies for small transaction costs and a HARA utility function. J. Comb. Inf. Syst. Sci. 38 (2010), 257-291.   CrossRef
  14. P. Dostál: 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.   CrossRef
  15. P. Dostál: Investment strategies in the long run with proportional transaction costs and HARA utility function. Quant. Finance 9 (2009), 2, 231-242.   DOI:10.1080/14697680802039873
  16. S. Gerhold, J. Muhle-Karbe and W. Schachermayer: Asymptotics and duality for the Davis and Norman problem. Stochastics 84 (2012), 5-6, 625-641.   DOI:10.1080/17442508.2011.619699
  17. S. Gerhold, J. Muhle-Karbe and W. Schachermayer: The dual optimizer for the growth-optimal portfolio under transaction costs. Finance Stoch. 17 (2013), 2, 325-354.   DOI:10.1007/s00780-011-0165-9
  18. T. Goll and J. Kallsen: Optimal portfolios for logarithmic utility. Stochastic Process. Appl. 89 (2000), 1, 31-48.   DOI:10.1016/s0304-4149(00)00011-9
  19. A. Herczegh and V. Prokaj: Shadow\! Price in the Power Utility Case. arXiv: 1112.4385.   CrossRef
  20. K. Janeček: Optimal Growth in Gambling and Investing. MSc Thesis, Charles University Prague 1999.   CrossRef
  21. K. Janeček and S. E. Shreve: Asymptotic analysis for optimal investment and consumption with transaction costs. Finance Stoch. 8 (2004), 2, 181-206.   DOI:10.1007/s00780-003-0113-4
  22. K. Janeček and S. E. Shreve: Futures trading with transaction costs. Illinois J. Math. 54 (2010), 4, 1239-1284.   CrossRef
  23. O. Kallenberg: Foundations of Modern Probability. Springer Verlag, Heidelberg 1997.   CrossRef
  24. J. Kallsen and J. Muhle-Karbe: On using shadow in portfolio optimization with transaction costs. Ann. Appl. Probab. 20 (2010), 4, 1341-1358.   DOI:10.1214/09-aap648
  25. J. Kallsen and J. Muhle-Karbe: Existence of shadow prices in finite probability spaces. Math. Methods Oper. Res. 73 (2011), 2, 251-262.   DOI:10.1007/s00186-011-0345-6
  26. J. Kallsen and J. Muhle-Karbe: The General Structure of Optimal Investment and Consumption with Small Transaction Costs. arXiv: 1303.3148.   CrossRef
  27. J. L. Kelly: A new interpretation of information rate. Bell Sys. Tech. J. 35 (1956), 4, 917-926.   DOI:10.1002/j.1538-7305.1956.tb03809.x
  28. M. J. P. Magill and G. M. Constantinides: Portfolio selection with transaction costs. J. Econom. Theory 13 (1976), 2, 245-263.   DOI:10.1016/0022-0531(76)90018-1
  29. R. C. Merton: Optimum consumption and portfolio rules in a continuous-time model. J. Econom. Theory 3 (1971), 4, 373-413. Erratum 6 (1973), 2, 213-214,   CrossRef
  30. A. J. Morton and S. Pliska: Optimal portfolio management with fixed transaction costs. Math. Finance 5 (1995), 4, 337-356.   DOI:10.1111/j.1467-9965.1995.tb00071.x
  31. D. Revuz and M. Yor: Continuous Martingales and Brownian Motion. Springer Verlag, Heidelberg, Berlin, New York 1999.   DOI:10.1007/978-3-662-06400-9
  32. D. B. Rokhlin: On the game interpretation of a shadow price process in utility maximization problems under transaction costs. Finance Stoch. 17 (2013), 4, 819-838.   DOI:10.1007/s00780-013-0206-7
  33. L. M. Rotando and E. O. Thorp: The Kelly criterion and the stock market. Amer. Math. Monthly 99 (1992), 10, 922-931.   DOI:10.2307/2324484
  34. P. A. Samuelson: The ``fallacy" of maximizing the geometric mean in long sequences of investing or gambling. Proc. Natl. Acad. Sci. 68 (1971), 10, 2493-2496.   DOI:10.1073/pnas.68.10.2493
  35. J. Sass and M. Schäl: Numeraire portfolios and utility-based price systems under proportional transaction costs. Decis. Econ. Finance 37 (2014), 2, 195-234.   DOI:10.1007/s10203-012-0132-8
  36. S. E. Shreve and H. M. Soner: Optimal investment and consumption with transaction costs. Ann. Appl. Probab. 4 (1994), 3, 609-692.   DOI:10.1214/aoap/1177004966
  37. A. Skorokhod: Stochastic equations for diffusion processes in a bounded region. Theory Probab. Appl. 6 (1961), 3, 264-274.   DOI:10.1137/1106035
  38. A. Skorokhod: Stochastic equations for diffusion processes in a bounded region II. Theory Probab. Appl. 7 (1962), 1, 3-23.   DOI:10.1137/1107002
  39. E. Thorp: 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:10.1016/b978-0-12-780850-5.50051-4
  40. E. Thorp: 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.   CrossRef