Mean-Variance Portfolio Selection with Random Parameters in a Complete Market

Abstract

This paper concerns the continuous-time, mean-variance portfolio selection problem in a complete market with random interest rate, appreciation rates, and volatility coefficients. The problem is tackled using the results of stochastic linear-quadratic (LQ) optimal control and backward stochastic differential equations (BSDEs), two theories that have been extensively studied and developed in recent years. Specifically, the mean-variance problem is formulated as a linearly constrained stochastic LQ control problem. Solvability of this LQ problem is reduced, in turn, to proving global solvability of a stochastic Riccati equation. The proof of existence and uniqueness of this Riccati equation, which is a fully nonlinear and singular BSDE with random coefficients, is interesting in its own right and relies heavily on the structural properties of the equation. Efficient investment strategies as well as the mean-variance efficient frontier are then analytically derived in terms of the solution of this equation. In particular, it is demonstrated that the efficient frontier in the mean-standard deviation diagram is still a straight line or, equivalently, risk-free investment is still possible, even when the interest rate is random. Finally, a version of the Mutual Fund Theorem is presented.

Implications of Sharpe Ratio as a Performance Measure in Multi-Period Settings

“…This literature includes among others, the following papers: Korn and Trautmann (1995), Bajeux, Portait (1998), Li and Ng (2000), Zhou and Li (2000), Lim and Zhou (2002), Lim (2003), Bielecki, Pliska, Jin and Zhou (2005). A textbook presentation for the case of a single risky asset with constant parameters is available in Cvitanić and Zapatero (2004).…”

Risk-sensitive control with HARA utility

“…Let and be given and fixed. Consider the following special case of (14): (25) where the inputs of player 1 and of player 2 satisfy and . We consider cost functionals of the following form: (26) We assume throughout this section that (A2) holds.…”

“…These results are obtained using results from nonsmooth analysis and viscosity solutions. The inclusion of systems with control dependent diffusions in our analysis has particular relevance to finance applications; see [16], [18], [25], [32].…”

Variance-Optimal Hedging for the Process Based on Non-Extensive Statistical Mechanics and Poisson Jumps

“…The optimal hedging problem is always an important issue in the financial field. Lim [22] and Lim and Zhou [23], using backward stochastic differential equations, studied the problem under the assumption that the asset price process was continuous and driven by the Brownian motion. Using the convex duality and projection Theorem, Gourieroux [24], Laurent and Pham [25], and Schweizer [26] studied that under the assumption that the asset price process was continuous semi-martingale.…”