Previous |  Up |  Next


dynamic stochastic portfolio optimization; Hamilton-Jacobi-Bellman equation; Conditional value-at-risk; $CVaRD$-based Sharpe ratio
In this paper we investigate the expected terminal utility maximization approach for a dynamic stochastic portfolio optimization problem. We solve it numerically by solving an evolutionary Hamilton-Jacobi-Bellman equation which is transformed by means of the Riccati transformation. We examine the dependence of the results on the shape of a chosen utility function in regard to the associated risk aversion level. We define the Conditional value-at-risk deviation ($CVaRD$) based Sharpe ratio for measuring risk-adjusted performance of a dynamic portfolio. We compute optimal strategies for a portfolio investment problem motivated by the German DAX 30 Index and we evaluate and analyze the dependence of the $CVaRD$-based Sharpe ratio on the utility function and the associated risk aversion level.
[1] Abe, R., Ishimura, N.: Existence of solutions for the nonlinear partial differential equation arising in the optimal investment problem. Proc. Japan Acad. Ser. A 84 (2008), 1, 11-14. DOI 10.3792/pjaa.84.11 | MR 2381178
[2] Agarwal, V., Naik, N. Y.: Risk and portfolio decisions involving hedge funds. Rev. Financ. Stud. 17 (2004), 1, 63-98. DOI 10.1093/rfs/hhg044
[3] Andrieu, L., Lara, M. De, Seck, B.: Conditional Value-at-Risk Constraint and Loss Aversion Utility Functions.
[4] Arrow, K. J.: Aspects of the theory of risk bearing. In: The Theory of Risk Aversion. Helsinki: Yrjo Jahnssonin Saatio. (Reprinted in: Essays in the Theory of Risk Bearing, Markham Publ. Co., Chicago, 1971), (1965), pp. 90-109. MR 0363427
[5] Aubin, J. P.: Lipschitz behavior of solutions to convex minimization problems. Math. Oper. Res. 9 (1984), 87-111. DOI 10.1287/moor.9.1.87 | MR 0736641
[6] Bank, B., Guddat, J., Klatte, D., Kummer, B., Tammer, K.: Non-linear Parametric Optimization. Licensed ed. Birkhauser Verlag, Basel-Boston, Mass., 1983. DOI 10.1007/978-3-0348-6328-5 | MR 0701243
[7] Bertsekas, D. P.: Dynamic Programming and Stochastic Control. Academic Press, New York 1976. DOI 10.1016/s0076-5392(08)x6050-3 | MR 0688509
[8] Biglova, A., Ortobelli, S., Rachev, S., Stoyanov, S.: Different Approaches to Risk Estimation in Portfolio Theory. J. Portfolio Management 31 (2004), 1, 103-112. DOI 10.3905/jpm.2004.443328
[9] Browne, S.: Risk-constrained dynamic active portfolio management. Management Sci. 46 (2000), 9, 1188-1199. DOI 10.1287/mnsc.46.9.1188.12233
[10] Denuit, M., Dhaene, J., Goovaerts, M., Kaas, R., Laeven, R.: Risk measurement with equivalent utility principles. Statist. Decisions 24 (2006), 1-25. DOI 10.1524/stnd.2006.24.1.1 | MR 2323186
[11] Farinelli, S., Ferreira, M., Rosselloc, D., Thoeny, M., Tibiletti, L.: Beyond Sharpe ratio: Optimal asset allocation using different performance ratios. J. Banking Finance 32 (2008), 10, 2057-2063. DOI 10.1016/j.jbankfin.2007.12.026
[12] Huang, Y., Forsyth, P. A., Labahn, G.: Combined fixed point and policy iteration for Hamilton-Jacobi-Bellman equations in finance. SIAM J. Numer. Anal. 50 (2012), 4, 1861-1882. DOI 10.1137/100812641 | MR 3022201
[13] Ishimura, N., Koleva, M. N., Vulkov, L. G.: Numerical solution via transformation methods of nonlinear models in option pricing. AIP Conference Proceedings 1301 (2010), 1, 387-394. DOI 10.1063/1.3526637
[14] Ishimura, N., Ševčovič, D.: On traveling wave solutions to a Hamilton-Jacobi-Bellman equation with inequality constraints. Japan J. Ind. Appl. Math. 30 (2013), 1, 51-67. DOI 10.1007/s13160-012-0087-8 | MR 3022806
[15] Karatzas, I., Lehoczky, J. P., Sethi, S. P., Shreve, S.: Explicit solution of a general consumption/investment problem. Math. Oper. Res. 11 (1986), 2, 261-294. DOI 10.1287/moor.11.2.261 | MR 0844005
[16] Kilianová, S., Ševčovič, D.: A transformation method for solving the Hamilton-Jacobi-Bellman equation for a constrained dynamic stochastic optimal allocation problem. ANZIAM J. 55 (2013), 14-38. DOI 10.21914/anziamj.v55i0.6816 | MR 3144202
[17] Kilianová, S., Trnovská, M.: Robust portfolio optimization via solution to the Hamilton-Jacobi-Bellman equation. Int. J. Comput. Math. 93 (2016), 725-734. DOI 10.1080/00207160.2013.871542 | MR 3483306
[18] Klatte, D.: On the {L}ipschitz behavior of optimal solutions in parametric problems of quadratic optimization and linear complementarity. Optim. J. Math. Program. Oper. Res. 16 (1985), 6, 819-831. DOI 10.1080/02331938508843080 | MR 0814211
[19] Koleva, M. N.: Iterative methods for solving nonlinear parabolic problem in pension saving management. AIP Confer. Proc. 1404 (2011), 1, 457-463. DOI 10.1063/1.3659948
[20] Koleva, M. N., Vulkov, L.: Quasilinearization numerical scheme for fully nonlinear parabolic problems with applications in models of mathematical finance. Math. Comput. Modell. 57 (2013), 2564-2575. DOI 10.1016/j.mcm.2013.01.008 | MR 3068748
[21] Kútik, P., Mikula, K.: Finite volume schemes for solving nonlinear partial differential equations in financial mathematics. In: Finite Volumes for Complex Applications VI, Problems and Perspectives (J. Fořt, J. Fürst, J. Halama, R. Herbin, and F. Hubert, eds.), Springer Proc. Math. 4 (2011), pp. 643-651. DOI 10.1007/978-3-642-20671-9_68 | MR 2882342
[22] LeVeque, R. J.: Finite Volume Methods for Hyperbolic Problems. Cambridge Texts in Applied Mathematics, Cambridge University Press, Cambridge 2002. DOI 10.1017/cbo9780511791253 | MR 1925043
[23] Lin, S., Ohnishi, M.: Optimal portfolio selection by CVaR based Sharpe ratio genetic algorithm approach. Sci. Math. Japon. Online e-2006 (2006), 1229-1251. MR 2300341
[24] Macová, Z., Ševčovič, D.: Weakly nonlinear analysis of the Hamilton-Jacobi-Bellman equation arising from pension savings management. Int. J. Numer. Anal. Model. 7 (2010), 4, 619-638. MR 2644295
[25] McNeil, A. J., Frey, R., Embrechts, P.: Quantitattive Risk Management. Princeton Series in Finance, Princeton University Press, 2005. DOI 10.1007/s11408-006-0016-4 | MR 2175089
[26] Merton, R. C.: Optimal consumption and portfolio rules in a continuous time model. J. Economic Theory 71 (1971), 373-413. DOI 10.1016/0022-0531(71)90038-x | MR 0456373
[27] Milgrom, P., Segal, I.: Envelope theorems for arbitrary choice sets. Econometrica 70 (2002), 2, 583-601. DOI 10.1111/1468-0262.00296 | MR 1913824
[28] Musiela, M., Zariphopoulou, T.: An example of indifference prices under exponential preferences. Finance Stochast. 8 (2004), 2, 229-239. DOI 10.1007/s00780-003-0112-5 | MR 2048829
[29] Muthuraman, K., Kumar, S.: Multidimensional portfolio optimization with proportional transaction costs. Math. Finance 16 (2006), 2, 301-335. DOI 10.1111/j.1467-9965.2006.00273.x | MR 2212268
[30] Pflug, G. Ch., Römisch, W.: Modeling, Measuring and Managing Risk. World Scientific Publushing, 2007. DOI 10.1142/6478 | MR 2424523
[31] Post, T., Fang, Y., Kopa, M.: Linear tests for DARA stochastic dominance. Management Sci. 61 (2015), 1615-1629. DOI 10.1287/mnsc.2014.1960 | MR 0668272
[32] Pratt, J. W.: Risk aversion in the small and in the large. Econometrica. 32 (1964), 1-2, 122-136. DOI 10.2307/1913738 | Zbl 0267.90010
[33] Protter, M. H., Weinberger, H. F.: Maximum Principles in Differential Equations. Springer-Verlag, New York 1984. DOI 10.1007/978-1-4612-5282-5 | MR 0762825
[34] Reisinger, C., Witte, J. H.: On the use of policy iteration as an easy way of pricing American options. SIAM J. Financ. Math. 3 (2012), 459-478. DOI 10.1137/110823328 | MR 2968042
[35] Seck, B., Andrieu, L., Lara, M. De: Parametric multi-attribute utility functions for optimal profit under risk constraints. Theory Decision. 72 (2012), 2, 257-271. DOI 10.1007/s11238-011-9255-6 | MR 2878102
[36] Sharpe, W. F.: The Sharpe ratio. J. Portfolio Management 21 (1994), 1, 49-58. DOI 10.3905/jpm.1994.409501
[37] Ševčovič, D., Stehlíková, B., Mikula, K.: Analytical and Numerical Methods for Pricing Financial Derivatives. Nova Science Publishers, Inc., Hauppauge 2011.
[38] Tourin, A., Zariphopoulou, T.: Numerical schemes for investment models with singular transactions. Comput. Econ. 7 (1994), 4, 287-307. DOI 10.1007/bf01299457 | MR 1318095
[39] Vickson, R. G.: Stochastic dominance for decreasing absolute risk aversion. J. Financial Quantitative Analysis 10 (1975), 799-811. DOI 10.2307/2330272
[40] Wiesinger, A.: Risk-Adjusted Performance Measurement State of the Art. Bachelor Thesis of the University of St. Gallen School of Business Administration, Economics, Law and Social Sciences (HSG), 2010.
[41] Xia, J.: Risk aversion and portfolio selection in a continuous-time model. J. Control Optim. 49 (2011), 5, 1916-1937. DOI 10.1137/10080871x | MR 2837505
[42] Zariphopoulou, T.: Consumption-investment models with constraints. SIAM J. Control Optim. 32 (1994), 1, 59-85. DOI 10.1137/s0363012991218827 | MR 1255960
[43] Zheng, H.: Efficient frontier of utility and CVaR. Math. Meth. Oper. Res. 70 (2009), 1, 129-148. DOI 10.1007/s00186-008-0234-9 | MR 2529428
Partner of
EuDML logo