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Optimal Portfolio Selection in an Itô–Markov Additive Market

Author

Listed:
  • Zbigniew Palmowski

    (Department of Applied Mathematics, Faculty of Pure and Applied Mathematics, Wrocław University of Science and Technology, 50-370 Wrocław, Poland)

  • Łukasz Stettner

    (Institute of Mathematics, Polish Academy of Sciences, 00-656 Warsaw, Poland
    Vistula University, 02-787 Warsaw, Poland)

  • Anna Sulima

    (Faculty of Management, Computer Science and Finance, Wrocław University of Economics, 53-345 Wrocław, Poland)

Abstract

We study a portfolio selection problem in a continuous-time Itô–Markov additive market with prices of financial assets described by Markov additive processes that combine Lévy processes and regime switching models. Thus, the model takes into account two sources of risk: the jump diffusion risk and the regime switching risk. For this reason, the market is incomplete. We complete the market by enlarging it with the use of a set of Markovian jump securities, Markovian power-jump securities and impulse regime switching securities. Moreover, we give conditions under which the market is asymptotic-arbitrage-free. We solve the portfolio selection problem in the Itô–Markov additive market for the power utility and the logarithmic utility.

Suggested Citation

  • Zbigniew Palmowski & Łukasz Stettner & Anna Sulima, 2019. "Optimal Portfolio Selection in an Itô–Markov Additive Market," Risks, MDPI, vol. 7(1), pages 1-32, March.
    • Handle: RePEc:gam:jrisks:v:7:y:2019:i:1:p:34-:d:216912
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    References listed on IDEAS

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    1. Robert J. Elliott & Leunglung Chan & Tak Kuen Siu, 2005. "Option pricing and Esscher transform under regime switching," Annals of Finance, Springer, vol. 1(4), pages 423-432, October.
    2. Jun Tu, 2010. "Is Regime Switching in Stock Returns Important in Portfolio Decisions?," Management Science, INFORMS, vol. 56(7), pages 1198-1215, July.
    3. Korn, Ralf & Melnyk, Yaroslav & Seifried, Frank Thomas, 2017. "Stochastic impulse control with regime-switching dynamics," European Journal of Operational Research, Elsevier, vol. 260(3), pages 1024-1042.
    4. Kreps, David M., 1981. "Arbitrage and equilibrium in economies with infinitely many commodities," Journal of Mathematical Economics, Elsevier, vol. 8(1), pages 15-35, March.
    5. Goll, Thomas & Kallsen, Jan, 2000. "Optimal portfolios for logarithmic utility," Stochastic Processes and their Applications, Elsevier, vol. 89(1), pages 31-48, September.
    6. John Buffington & Robert J. Elliott, 2002. "American Options With Regime Switching," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 5(05), pages 497-514.
    7. X. Guo, 2001. "Information and option pricings," Quantitative Finance, Taylor & Francis Journals, vol. 1(1), pages 38-44.
    8. Heston, Steven L, 1993. "A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options," Review of Financial Studies, Society for Financial Studies, vol. 6(2), pages 327-343.
    9. Naik, Vasanttilak, 1993. "Option Valuation and Hedging Strategies with Jumps in the Volatility of Asset Returns," Journal of Finance, American Finance Association, vol. 48(5), pages 1969-1984, December.
    10. Shen, Yang & Siu, Tak Kuen, 2012. "Asset allocation under stochastic interest rate with regime switching," Economic Modelling, Elsevier, vol. 29(4), pages 1126-1136.
    11. Merton, Robert C., 1971. "Optimum consumption and portfolio rules in a continuous-time model," Journal of Economic Theory, Elsevier, vol. 3(4), pages 373-413, December.
    12. Jörn Sass & Ulrich Haussmann, 2004. "Optimizing the terminal wealth under partial information: The drift process as a continuous time Markov chain," Finance and Stochastics, Springer, vol. 8(4), pages 553-577, November.
    13. Bae, Geum Il & Kim, Woo Chang & Mulvey, John M., 2014. "Dynamic asset allocation for varied financial markets under regime switching framework," European Journal of Operational Research, Elsevier, vol. 234(2), pages 450-458.
    14. Ruihua Liu, 2014. "Optimal Investment and Consumption with Proportional Transaction Costs in Regime-Switching Model," Journal of Optimization Theory and Applications, Springer, vol. 163(2), pages 614-641, November.
    15. Niu, Liqun, 2008. "Some stability results of optimal investment in a simple Lévy market," Insurance: Mathematics and Economics, Elsevier, vol. 42(1), pages 445-452, February.
    16. Y.M. Kabanov & D.O. Kramkov, 1998. "Asymptotic arbitrage in large financial markets," Finance and Stochastics, Springer, vol. 2(2), pages 143-172.
    17. Robert J. Elliott & William C. Hunter & Barbara M. Jamieson, 2001. "Financial Signal Processing: A Self Calibrating Model," International Journal of Theoretical and Applied Finance (IJTAF), World Scientific Publishing Co. Pte. Ltd., vol. 4(04), pages 567-584.
    18. Philippe Artzner & David Heath, 1995. "Approximate Completeness With Multiple Martingale Measures," Mathematical Finance, Wiley Blackwell, vol. 5(1), pages 1-11, January.
    19. Goldfeld, Stephen M. & Quandt, Richard E., 1973. "A Markov model for switching regressions," Journal of Econometrics, Elsevier, vol. 1(1), pages 3-15, March.
    20. Merton, Robert C., 1976. "Option pricing when underlying stock returns are discontinuous," Journal of Financial Economics, Elsevier, vol. 3(1-2), pages 125-144.
    21. Fu, Jun & Wei, Jiaqin & Yang, Hailiang, 2014. "Portfolio optimization in a regime-switching market with derivatives," European Journal of Operational Research, Elsevier, vol. 233(1), pages 184-192.
    22. Battig, Robert J & Jarrow, Robert A, 1999. "The Second Fundamental Theorem of Asset Pricing: A New Approach," Review of Financial Studies, Society for Financial Studies, vol. 12(5), pages 1219-1235.
    23. Q. Zhang & G. Yin, 2004. "Nearly-Optimal Asset Allocation in Hybrid Stock Investment Models," Journal of Optimization Theory and Applications, Springer, vol. 121(2), pages 419-444, May.
    24. Elliott, R. J. & Malcolm, W. P. & Tsoi, Allanus H., 2003. "Robust parameter estimation for asset price models with Markov modulated volatilities," Journal of Economic Dynamics and Control, Elsevier, vol. 27(8), pages 1391-1409, June.
    25. Merton, Robert C., 1980. "On estimating the expected return on the market : An exploratory investigation," Journal of Financial Economics, Elsevier, vol. 8(4), pages 323-361, December.
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    2. Sulima Anna, 2021. "The Absence of Arbitrage on the Complete Black-Scholes-Merton Regime-Switching Lévy Market," Econometrics. Advances in Applied Data Analysis, Sciendo, vol. 25(3), pages 72-84, September.

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