Mean-variance portfolio selection in a complete market with unbounded random coefficients

This paper concerns a mean-variance portfolio selection problem in a complete market with unbounded random coefficients. In particular, we use adapted processes to model market coefficients, and assume that only the interest rate is bounded, while the appreciation rate, volatility and market price of risk are unbounded. Under an exponential integrability assumption of the market price of risk process, we first prove the uniqueness and existence of solutions to two backward stochastic differential equations with unbounded coefficients. Then we apply the stochastic linear-quadratic control theory and the Lagrangian method to solve the problem. We represent the efficient portfolio and efficient frontier in terms of the unique solutions to the two backward stochastic differential equations. To illustrate our results, we derive explicit expressions for the efficient portfolio and efficient frontier in one example with Markovian models of a bounded interest rate and an unbounded market price of risk.

[1]  Hanqing Jin,et al.  Time-Inconsistent Stochastic Linear-Quadratic Control , 2011, SIAM J. Control. Optim..

[2]  Tak Kuen Siu,et al.  Mean-variance portfolio selection under a constant elasticity of variance model , 2014, Oper. Res. Lett..

[3]  X. Zhou,et al.  Continuous-Time Mean-Variance Portfolio Selection: A Stochastic LQ Framework , 2000 .

[4]  Andrew E. B. Lim,et al.  Dynamic Mean-Variance Portfolio Selection with No-Shorting Constraints , 2001, SIAM J. Control. Optim..

[5]  H. Abou-Kandil,et al.  Matrix Riccati Equations in Control and Systems Theory , 2003, IEEE Transactions on Automatic Control.

[6]  Duan Li,et al.  Optimal Dynamic Portfolio Selection: Multiperiod Mean‐Variance Formulation , 2000 .

[7]  Hoi Ying Wong,et al.  Mean-variance portfolio selection of cointegrated assets , 2011 .

[8]  V. Linetsky.  THE SPECTRAL DECOMPOSITION OF THE OPTION VALUE , 2004 .

[9]  Andrew E. B. Lim,et al.  Mean-Variance Portfolio Selection with Random Parameters in a Complete Market , 2002, Math. Oper. Res..

[10]  V. Linetsky,et al.  Black's Model of Interest Rates as Options, Eigenfunction Expansions and Japanese Interest Rates , 2004 .

[11]  S. Peng,et al.  Backward Stochastic Differential Equations in Finance , 1997 .

[12]  Alan G. White,et al.  The Pricing of Options on Assets with Stochastic Volatilities , 1987 .

[13]  Andrew E. B. Lim.  Quadratic Hedging and Mean-Variance Portfolio Selection with Random Parameters in an Incomplete Market , 2004, Math. Oper. Res..

[14]  Yang Shen,et al.  Optimal investment–reinsurance strategy for mean–variance insurers with square-root factor process☆ , 2015 .

[15]  Richard F. Bass,et al.  The Term Structure with Semi-credible Targeting , 2003 .

[16]  Yang Shen,et al.  Optimal investment-consumption-insurance with random parameters , 2016 .

[17]  X. Zhou,et al.  Stochastic Controls: Hamiltonian Systems and HJB Equations , 1999 .

[18]  Xun Yu Zhou,et al.  Constrained Stochastic LQ Control with Random Coefficients, and Application to Portfolio Selection , 2005, SIAM J. Control. Optim..

[19]  J. Harrison,et al.  Brownian motion and stochastic flow systems , 1986 .

[20]  D. Luenberger.  Optimization by Vector Space Methods , 1968 .

[21]  Andrew E. B. Lim.  Mean-Variance Hedging When There Are Jumps , 2005, SIAM J. Control. Optim..

[22]  V. Linetsky.  On the transition densities for reflected diffusions , 2005, Advances in Applied Probability.

[23]  Hai-ping Shi.  Backward stochastic differential equations in finance , 2010 .

[24]  H. Buchholz,et al.  The Confluent Hypergeometric Function: with Special Emphasis on its Applications , 1969 .

[25]  S. Heston.  A Closed-Form Solution for Options with Stochastic Volatility with Applications to Bond and Currency Options , 1993 .