Back
 JMF  Vol.4 No.2 , February 2014
Optimal Investment Strategy for Kinked Utility Maximization: Covered Call Option Strategy
Abstract: This paper describes optimal investment strategies for kinked utility functions. One example is a CRRA utility function with a kink at a maximum wealth, which leads a covered call “like” strategy and the other is a CRRA utility function with a kink at a minimum wealth, which leads a protective put “like” strategy. This paper introduces analytic mathematical solutions providing a mathematical explanation of a dual utility where Black-Sholes assumption is utilized in the solutions. The intuitive solutions are clear for cases of those kinked utilities but minute mathematical explanation is described. Also a numerical simulation is performed for a covered call like strategy case.
Cite this paper: M. Yamashita, "Optimal Investment Strategy for Kinked Utility Maximization: Covered Call Option Strategy," Journal of Mathematical Finance, Vol. 4 No. 2, 2014, pp. 55-74. doi: 10.4236/jmf.2014.42006.
References

[1]   R. C. Merton, “Optimum Consumption and Portfolio Rules in a Continuous-Time Model,” Journal of Economic Theory, Vol. 3, No. 4, 1971, pp. 373-413. http://dx.doi.org/10.1016/0022-0531(71)90038-X

[2]   R. C. Merton, “Continuous-Time Finance,” Blackwell Publishers, Oxford, 1992.

[3]   R. C. Merton, “Lifetime Portfolio Selection under Uncertainty: The Continuous-Time Case,” Review of Economics and Statistics, Vol. 51, No. 3, 1969, pp. 247-257. http://dx.doi.org/10.2307/1926560

[4]   R. C. Merton, “Optimum Consumption and Portfolio Rules in a Continuous-Time Model,” Journal of Economic Theory, Erratum, Vol. 6, No. 2, 1973, pp. 213-214.

[5]   J. C. Cox and C.-F. Huang,”Optimal Consumption and Portfolio Policies When Asset Prices Follow a Diffution Process,” Journal of Economic Theory, Vol. 49, No. 1, 1987, pp. 33-83.
http://dx.doi.org/10.1016/0022-0531(89)90067-7

[6]   B. Bian, S. Miao and H. Zheng, “Smooth Value Functions for a Class of Nonsmooth Utility Maximization Problems,” SIAM Journal of Financial Mathematics, Vol. 2, 2011, pp. 727-747.

[7]   L. Martellini and V. Milhau, “Dynamic Allocation Decisions in the Presence of Funding Ratio Constraints,” Working Paper of EDHEC, Roubaix, 2009.

[8]   M. J. Brennan and R. Solanki, “Optimal Portfolio Insurance,” The Journal of Financial and Quantitative Analysis, Vol. 16, No. 3, 1981, pp. 279-300. http://dx.doi.org/10.2307/2330239

[9]   H. E. Leland, “Who Should Buy Portfolio Insurance?” The Journal of Finance, Vol. 35, No. 2, 1980, pp. 581-594.

[10]   S. Benninga and M. Blume, “On the Optimality of Portfolio Insurance,” The Journal of Finance, Vol. 40, No. 5, 1985, pp. 1314-1352. http://dx.doi.org/10.1111/j.1540-6261.1985.tb02386.x

[11]   J. C. Cox and H. E. Leland, “On Dynamic Investment Strategies,” In: Proceedings of the Seminar on the Analysis of Securities Prices, Center for Research in Security Prices (CRSP), Chicago, 1982, pp. 139-173.

[12]   J. C. Cox and H. E. Leland, “On Dynamic Investment Strategies,” Journal of Economic Dynamics and Control, Vol. 24, No. 11-12, 2000, pp. 1859-1880. http://dx.doi.org/10.1016/S0165-1889(99)00095-0

[13]   N. El Karoui, M. Jeanblanc and V. Lacoste, “Optimal Portfolio Management with American Capital Guarantee,” Journal of Economic Dynamics and Control, Vol. 29, No. 3, 2005, pp. 449-468.
http://dx.doi.org/10.1016/j.jedc.2003.11.005

[14]   W. Schachermayer, “Optimal Investment in Incomplete Markets When Wealth May Become Negative,” The Annals of Applied Probability, Vol. 11, No. 3, 2001, pp. 694-734.
http://dx.doi.org/10.1214/aoap/1015345346

[15]   P. Carr and D. Madan, “Optimal Positioning in Derivative Securities,” Quantitative Finance, Vol. 1, No. 1, 2001, pp. 19-37.
http://dx.doi.org/10.1080/713665549

[16]   J.-L. Prigent and F. Tahar, “Optimal Portfolios with Guarantee at Maturity: Computation and Comparison,” International Journal of Business, Vol. 11, No. 2, 2006, pp. 171-185.

[17]   L. T. Ndounkew, “Stochastic Control: With Applications to Financial Mathematics,” African Institute for Mathematical Sciences Postgraduate Diploma, Muizenberg, 2010.

[18]   H. Pham, “On Some Recent Aspects of Stochastic Control and Their Applications,” Probability Surveys, Vol. 2, No. 2005, 2005, pp. 506-549. http://dx.doi.org/10.1214/154957805100000195

[19]   H. Pham, “Stochastic Control and Applications in Finance,” Lecture Note, Paris Diderot University, LPMA, Paris, 2010.

[20]   J. Sass, “Stochastic Control: With Applications to Financial Mathematics,” Working Paper, Austrian Academy of Sciences, Vienna, 2006.

[21]   J. Sekine, “Long-Term Optimal Portfolios with Floor,” Finance and Stochastics, Vol. 16, No. 3, 2012, pp. 369-401.

[22]   N. El Karoui, S. Peng and M. C. Quenez, “Backward Stochastic Differential Equations in Finance,” Mathematical Finance, Vol. 7, No. 1, 1997, pp. 1-71. http://dx.doi.org/10.1111/1467-9965.00022

 
 
Top