Optimal mean-variance selling strategies

Assuming that the stock price X follows a geometric Brownian motion with drift µ ∈ IR and volatility σ> 0, and letting Px denote a probability measure under which X starts at x> 0, we study the dynamic version of the nonlinear mean-variance optimal stopping problem sup τ EXt(Xτ) − cVarXt(Xτ) where t runs from 0 onwards, the supremum is taken over stopping times τ of X, and c> 0 is a given and fixed constant. Using direct martingale arguments we first show that when µ ≤ 0 it is optimal to stop at once and when µ ≥ σ2/2 it is optimal not to stop at all. By employing the method of Lagrange multipliers we then show that the nonlinear problem for 0 < µ < σ2/2 can be reduced to a family of linear problems. Solving the latter using a free-boundary approach we find that the optimal stopping time is given by τ ∗ = inf t ≥ 0 | Xt ≥ γc(1−γ) where γ = µ/(σ2/2). The dynamic formulation of the problem and the method of solution are applied to the constrained problems of maximising/minimising the mean/variance subject to the upper/lower bound on the variance/mean from which the nonlinear problem above is obtained by optimising the Lagrangian itself. 1