Better than pre-commitment mean-variance portfolio allocation strategies: a semi-self-financing Hamilton-Jacobi-Bellman equation approach.

*(English)*Zbl 1348.91250Summary: We generalize the idea of semi-self-financing strategies, originally discussed in [H. Ehrbar, J. Econ. Theory 50, No. 1, 214–218 (1990; Zbl 0686.90008)], and later formalized in [X. Cui et al., Math. Finance 22, No. 2, 346–378 (2012; Zbl 1278.91131)], for the pre-commitment mean-variance (MV) optimal portfolio allocation problem. The proposed semi-self-financing strategies are built upon a numerical solution framework for Hamilton-Jacobi-Bellman equations, and can be readily employed in a very general setting, namely continuous or discrete re-balancing, jump-diffusions with finite activity, and realistic portfolio constraints. We show that if the portfolio wealth exceeds a threshold, an MV optimal strategy is to withdraw cash. These semi-self-financing strategies are generally non-unique. Numerical results confirming the superiority of the efficient frontiers produced by the strategies with positive cash withdrawals are presented. Tests based on estimation of parameters from historical time series show that the semi-self-financing strategy is robust to estimation ambiguities.

##### MSC:

91G10 | Portfolio theory |

35Q91 | PDEs in connection with game theory, economics, social and behavioral sciences |

49L20 | Dynamic programming in optimal control and differential games |

60H30 | Applications of stochastic analysis (to PDEs, etc.) |

62P05 | Applications of statistics to actuarial sciences and financial mathematics |

93E20 | Optimal stochastic control |

PDF
BibTeX
XML
Cite

\textit{D. M. Dang} and \textit{P. A. Forsyth}, Eur. J. Oper. Res. 250, No. 3, 827--841 (2016; Zbl 1348.91250)

Full Text:
DOI

**OpenURL**

##### References:

[1] | Ait-Sahalia, Y.; Jacod, J., Analysing the spectrum of asset returns: jump and volatilty of components of high frequency data, Journal of Economic Literature, 50, 1007-1050, (2012) |

[2] | Almgren, R., Optimal trading with stochastic liquidity and volatility, SIAM Journal of Financial Mathematics, 3, 163-181, (2012) · Zbl 1256.49031 |

[3] | Basak, S.; Chabakauri, G., Dynamic mean-variance asset allocation, The Review of Financial Studies, 23, 2970-3016, (2010) |

[4] | Bauerle, N., Benchmark and mean-variance problems for insurers, Mathematical Methods of Operations Research, 62, 159-162, (2005) · Zbl 1101.93081 |

[5] | Bauerle, N.; Grether, S., Complete markets do not allow free cash flow streams, Mathematical Methods of Operations Research, 81, 137-145, (2015) · Zbl 1318.91079 |

[6] | Bielecki, T.; Pliska, S.; Zhou, X., Continuous time mean-variance portfolio selection with bankruptcy prohibition, Mathematical Finance, 15, 213-244, (2005) · Zbl 1153.91466 |

[7] | Björk, T. Murgoci, A. (2010). A general theory of Markovian time inconsistent stochastic control problems. Available at SSRN: http://ssrn.com/abstract=1694759. |

[8] | Clewlow, L.; Strickland, C., Energy derivatives - pricing and risk management, (2000), Lacima London |

[9] | Cont, R.; Mancini, C., Nonparametric tests for pathwise properties of semimartingales, Bernoulli, 17, 781-813, (2011) · Zbl 1345.62074 |

[10] | Cui, X. Li, D. (2010). Better than dynamic mean-variance policy in market with all risky assets. In Slides from 6th World Congress of the Bachelier Finance Society, Toronto, 2010. |

[11] | Cui, X.; Li, D.; Wang, S.; Zhu, S., Better than dynamic mean-variance: time-inconsistency and free cash flow stream, Mathematical Finance, 22, 346-378, (2012) · Zbl 1278.91131 |

[12] | Dang, D. M.; Forsyth, P. A., Continuous time mean-variance optimal portfolio allocation under jump diffusion: an numerical impulse control approach, Numerical Methods for Partial Differential Equations, 30, 664-698, (2014) · Zbl 1284.91569 |

[13] | Dang, D. M., & Forsyth, P. A. (2015). Better than pre-commitment mean-variance portfolio allocation strategies: a semi-self-financing Hamilton-Jacobi-Bellman equation approach. Working paper. Cheriton School of Computer Science, University of Waterloo (Expanded version). · Zbl 1348.91250 |

[14] | Dang, D. M.; Forsyth, P. A.; Li, Y., Convergence of the embedded mean-variance optimal points with discrete sampling, Numerische Mathematik, (2015) |

[15] | Delong, L.; Gerrard, R., Mean-variance portfolio selection for a non-life insurance company, Mathematical Methods of Operations Research, 66, 339-367, (2007) · Zbl 1148.60040 |

[16] | Delong, L.; Gerrard, R.; Haberman, S., Mean-variance optimization problems for an accumulation phase in a defined benefit plan, Insurance: Mathematics and Economics, 42, 107-118, (2008) · Zbl 1141.91501 |

[17] | Ehrbar, H., Mean-variance efficiency when investors are not required to invest all their money, Journal of Economic Theory, 50, 214-218, (1990) · Zbl 0686.90008 |

[18] | Honore, P. (1998). Pitfalls in estimating jump diffusion models. Working paper. Center for Analytical Finance, University of Aarhus. |

[19] | Jose-Fombellida, R.; Rincon-Zapatero, J., Mean variance portfolio and contribution selection in stochastic pension funding, European Journal of Operational Research, 187, 120-137, (2008) · Zbl 1135.91019 |

[20] | Leippold, M.; Trojani, F.; Vanini, P., A geometric approach to mulitperiod mean variance optimization of assets and liabilities, Journal of Economic Dynamics and Control, 28, 1079-1113, (2004) · Zbl 1179.91234 |

[21] | Li, D.; Ng, W.-L., Optimal dynamic portfolio selection: multiperiod mean variance formulation, Mathematical Finance, 10, 387-406, (2000) · Zbl 0997.91027 |

[22] | Li, X.; Xhou, X.; Lim, A. E.B., Dynamic mean-variance portfolio selection with no-shorting constraints, SIAM Journal on Control and Optimization, 40, 1540-1555, (2002) · Zbl 1027.91040 |

[23] | Ma, K., & Forsyth, P. (2015). Numerical solution of the Hamilton-Jacobi-Bellman formulation for continuous time mean variance asset allocation under stochastic volatility. To appear, Journal of Computational Finance. |

[24] | Mancini, C., Non-parametric threshold estimation models with stochastic diffusion coefficient and jumps, Scandinavian Journal of Statistics, 36, 270-296, (2009) · Zbl 1198.62079 |

[25] | Merton, R., Option pricing when underlying stock returns are discontinuous, Journal of Financial Economics, 3, 125-144, (1976) · Zbl 1131.91344 |

[26] | Øksendal, B.; Sulem, A., Applied control of jump diffusions, (2009), Springer |

[27] | Pham, H., Continuous-time stochastic control and optimization with financial applications, (2009), Springer · Zbl 1165.93039 |

[28] | Schweizer, M., Mean-variance hedging, (Cont, R., Encyclopedia of quantitative finance, (2010), Wiley New York), 1177-1181 |

[29] | Shimizu, Y., Threshold estimation for stochastic differential equations with jumps, Proceedings of the 59th ISI world statistics conference, Hong Kong, 747-752, (2013) |

[30] | Tauchen, G.; Zhou, H., Realized jumps on financial markets and predicting credit spreads, Journal of Econometrics, 160, 235-245, (2011) · Zbl 1441.62884 |

[31] | Tse, S.; Forsyth, P.; Li, Y., Preservation of scalarization optimal points in the embedding technique for continuous time mean variance optimization, SIAM Journal on Control and Optimization, 52, 1527-1546, (2014) · Zbl 1297.90148 |

[32] | Vigna, E., On the efficiency of mean-variance based portfolio selection in defined contibution pension schemes, Quantitative finance, 14, 237-258, (2014) · Zbl 1294.91168 |

[33] | Wang, J.; Forsyth, P., Numerical solution of the Hamilton-Jacobi-Bellman formulation for continuous time mean variance asset allocation, Journal of Economic Dynamics and Control, 34, 207-230, (2010) · Zbl 1182.91161 |

[34] | Wang, J.; Forsyth, P., Continuous time mean variance asset allocation: a time consistent strategy, European Journal of Operational Research, 209, 184-201, (2011) · Zbl 1208.91139 |

[35] | Wang, J.; Forsyth, P., Comparison of mean variance like strategies for optimal asset allocation problems, International Journal of Theoretical and Applied Finance, 15, 2, (2012) · Zbl 1282.91312 |

[36] | Yu, P., Cone convexity, cone extreme points, and nondominated solutions in decision problem with multiobjectives, Journal of Optimization Theory and Applications, 14, 319-377, (1974) · Zbl 0268.90057 |

[37] | Zhou, X.; Li, D., Continuous time mean variance portfolio selection: a stochastic LQ framework, Applied Mathematics and Optimization, 42, 19-33, (2000) · Zbl 0998.91023 |

This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. It attempts to reflect the references listed in the original paper as accurately as possible without claiming the completeness or perfect precision of the matching.