Constrained Dynamic Optimality and Binomial Terminal Wealth

We assume that the wealth process $X^u$ is self-financing and generated from the initial wealth by holding a fraction $u$ of $X^u$ in a risky stock (whose price follows a geometric Brownian motion) and the remaining fraction $1-u$ of $X^u$ in a riskless bond (whose price compounds exponentially with interest rate $r \in {R}$). Letting $P_{t,x}$ denote a probability measure under which $X^u$ takes value $x$ at time $t,$ we study the dynamic version of the nonlinear optimal control problem $\inf_u\, Var{t,X_t^u}(X_T^u)$ where the infimum is taken over admissible controls $u$ subject to $X_t^u \ge e^{-r(T-t)} g$ and $E{t,X_t^u}(X_T^u) \ge \beta$ for $ t \in [0,T]$. The two constants $g$ and $\beta$ are assumed to be given exogenously and fixed. By conditioning on the expected terminal wealth value, we show that the nonlinear problem can be reduced to a family of linear problems. Solving the latter using a martingale method combined with Lagrange multipliers, we derive the dynamically optimal control $u_*^d$ in closed form and prove that the dynamically optimal terminal wealth $X_T^d$ can only take two values $g$ and $\beta$. This binomial nature of the dynamically optimal strategy stands in sharp contrast with other known portfolio selection strategies encountered in the literature. A direct comparison shows that the dynamically optimal (time-consistent) strategy outperforms the statically optimal (time-inconsistent) strategy in the problem.