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

Petr Dostál; Jana Klůjová

Kybernetika (2015)

  • Volume: 51, Issue: 4, page 588-628
  • ISSN: 0023-5954

Abstract

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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].

How to cite

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Dostál, Petr, and Klůjová, Jana. "Log-optimal investment in the long run with proportional transaction costs when using shadow prices." Kybernetika 51.4 (2015): 588-628. <http://eudml.org/doc/271825>.

@article{Dostál2015,
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].},
author = {Dostál, Petr, Klůjová, Jana},
journal = {Kybernetika},
keywords = {proportional transaction costs; logarithmic utility; shadow prices},
language = {eng},
number = {4},
pages = {588-628},
publisher = {Institute of Information Theory and Automation AS CR},
title = {Log-optimal investment in the long run with proportional transaction costs when using shadow prices},
url = {http://eudml.org/doc/271825},
volume = {51},
year = {2015},
}

TY - JOUR
AU - Dostál, Petr
AU - Klůjová, Jana
TI - Log-optimal investment in the long run with proportional transaction costs when using shadow prices
JO - Kybernetika
PY - 2015
PB - Institute of Information Theory and Automation AS CR
VL - 51
IS - 4
SP - 588
EP - 628
AB - 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].
LA - eng
KW - proportional transaction costs; logarithmic utility; shadow prices
UR - http://eudml.org/doc/271825
ER -

References

top
  1. Algoet, P. H., Cover, T. M., 10.1214/aop/1176991793, Ann. Probab. 16 (1988), 2, 876-898. MR0929084DOI10.1214/aop/1176991793
  2. Akian, M., Menaldi, J. L., Sulem, A., 10.1137/s0363012993247159, SIAM J. Control Optim. 34 (1996), 1, 329-364. Zbl1035.91505MR1372917DOI10.1137/s0363012993247159
  3. Akian, M., Sulem, A., Taksar, M. I., 10.1111/1467-9965.00111, Math. Finance 11 (2001), 2, 153-188. Zbl1055.91016MR1822775DOI10.1111/1467-9965.00111
  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. MR3224442
  5. Benedetti, G., Campi, L., Kallsen, J., Muhle-Karbe, J., 10.1007/s00780-012-0201-4, Finance Stoch. 17 (2013), 801-818. Zbl1280.91070MR3105934DOI10.1007/s00780-012-0201-4
  6. Choi, J. H., Sirbu, M., Zitkovix, G., 10.1137/120881373, SIAM J. Control Optim. 51 (2013), 6, 4414-4449. MR3141745DOI10.1137/120881373
  7. Czichowsky, Ch., Muhle-Karbe, J., Schachermayer, W., 10.1137/130925864, SIAM J. Financial Math. 5 (2014), 1, 258-277. Zbl1318.91179MR3194656DOI10.1137/130925864
  8. Bell, R. M., Cover, T. M., 10.1287/moor.5.2.161, Math. Oper. Res. 5 (1980), 2, 161-166. Zbl0442.90120MR0571810DOI10.1287/moor.5.2.161
  9. Bell, R., Cover, T. M., 10.1287/mnsc.34.6.724, Management Sci. 34 (1998), 6, 724-733. Zbl0649.90014MR0943277DOI10.1287/mnsc.34.6.724
  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. MR0135630
  11. Browne, S., Whitt, W., 10.2307/1428168, Adv. in Appl. Probab. 28 (1996), 4, 1145-1176. Zbl0867.90010MR1418250DOI10.2307/1428168
  12. Davis, M., Norman, A., 10.1287/moor.15.4.676, Math. Oper. Res. 15 (1990), 4, 676-713. Zbl0717.90007MR1080472DOI10.1287/moor.15.4.676
  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. Zbl1184.91199
  15. Dostál, P., 10.1080/14697680802039873, Quant. Finance 9 (2009), 2, 231-242. MR2512992DOI10.1080/14697680802039873
  16. Gerhold, S., Muhle-Karbe, J., Schachermayer, W., 10.1080/17442508.2011.619699, Stochastics 84 (2012), 5-6, 625-641. Zbl1276.91093MR2995515DOI10.1080/17442508.2011.619699
  17. Gerhold, S., Muhle-Karbe, J., Schachermayer, W., 10.1007/s00780-011-0165-9, Finance Stoch. 17 (2013), 2, 325-354. Zbl1319.91142MR3038594DOI10.1007/s00780-011-0165-9
  18. Goll, T., Kallsen, J., 10.1016/s0304-4149(00)00011-9, Stochastic Process. Appl. 89 (2000), 1, 31-48. Zbl1048.91064MR1775225DOI10.1016/s0304-4149(00)00011-9
  19. Herczegh, A., Prokaj, V., Shadow Price in the Power Utility Case., arXiv: 1112.4385. MR3375886
  20. Janeček, K., Optimal Growth in Gambling and Investing., MSc Thesis, Charles University Prague 1999. 
  21. Janeček, K., Shreve, S. E., 10.1007/s00780-003-0113-4, Finance Stoch. 8 (2004), 2, 181-206. MR2048827DOI10.1007/s00780-003-0113-4
  22. Janeček, K., Shreve, S. E., Futures trading with transaction costs., Illinois J. Math. 54 (2010), 4, 1239-1284. Zbl1276.91094MR2981847
  23. Kallenberg, O., Foundations of Modern Probability., Springer Verlag, Heidelberg 1997. Zbl0996.60001MR1464694
  24. Kallsen, J., Muhle-Karbe, J., 10.1214/09-aap648, Ann. Appl. Probab. 20 (2010), 4, 1341-1358. MR2676941DOI10.1214/09-aap648
  25. Kallsen, J., Muhle-Karbe, J., 10.1007/s00186-011-0345-6, Math. Methods Oper. Res. 73 (2011), 2, 251-262. Zbl1217.91170MR2776563DOI10.1007/s00186-011-0345-6
  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., 10.1002/j.1538-7305.1956.tb03809.x, Bell Sys. Tech. J. 35 (1956), 4, 917-926. MR0090494DOI10.1002/j.1538-7305.1956.tb03809.x
  28. Magill, M. J. P., Constantinides, G. M., 10.1016/0022-0531(76)90018-1, J. Econom. Theory 13 (1976), 2, 245-263. MR0469196DOI10.1016/0022-0531(76)90018-1
  29. Merton, R. C., 10.1016/0022-0531(71)90038-X, J. Econom. Theory 3 (1971), 4, 373-413. Erratum 6 (1973), 2, 213-214, Zbl1011.91502MR0456373DOI10.1016/0022-0531(71)90038-X
  30. Morton, A. J., Pliska, S., 10.1111/j.1467-9965.1995.tb00071.x, Math. Finance 5 (1995), 4, 337-356. Zbl0866.90020DOI10.1111/j.1467-9965.1995.tb00071.x
  31. Revuz, D., Yor, M., 10.1007/978-3-662-06400-9, Springer Verlag, Heidelberg, Berlin, New York 1999. Zbl1087.60040MR1725357DOI10.1007/978-3-662-06400-9
  32. Rokhlin, D. B., 10.1007/s00780-013-0206-7, Finance Stoch. 17 (2013), 4, 819-838. Zbl1279.91150MR3105935DOI10.1007/s00780-013-0206-7
  33. Rotando, L. M., Thorp, E. O., 10.2307/2324484, Amer. Math. Monthly 99 (1992), 10, 922-931. Zbl0768.90105MR1190557DOI10.2307/2324484
  34. Samuelson, P. A., 10.1073/pnas.68.10.2493, Proc. Natl. Acad. Sci. 68 (1971), 10, 2493-2496. MR0295739DOI10.1073/pnas.68.10.2493
  35. Sass, J., Schäl, M., 10.1007/s10203-012-0132-8, Decis. Econ. Finance 37 (2014), 2, 195-234. MR3260886DOI10.1007/s10203-012-0132-8
  36. Shreve, S. E., Soner, H. M., 10.1214/aoap/1177004966, Ann. Appl. Probab. 4 (1994), 3, 609-692. Zbl0813.60051MR1284980DOI10.1214/aoap/1177004966
  37. Skorokhod, A., 10.1137/1106035, Theory Probab. Appl. 6 (1961), 3, 264-274. Zbl0201.49302DOI10.1137/1106035
  38. Skorokhod, A., 10.1137/1107002, Theory Probab. Appl. 7 (1962), 1, 3-23. Zbl0201.49302DOI10.1137/1107002
  39. Thorp, E., 10.1016/b978-0-12-780850-5.50051-4, In: Stochastic Optimization Models in Finance (W. T. Ziemba and R. G. Vickson, eds.), Acad. Press, Bew York 1975, pp. 599-619. DOI10.1016/b978-0-12-780850-5.50051-4
  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. 

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