Optimal mean–variance selling strategies

Assuming that the stock price X follows a geometric Brownian motion with drift \(\mu \in \mathbb {R}\) and volatility \(\sigma >0\), and letting \(\mathsf {P}_{\!x}\) denote a probability measure under which X starts at \(x>0\), we study the dynamic version of the nonlinear mean–variance optimal stopping problem where t runs from 0 onwards, the supremum is taken over stopping times \(\tau \) of X, and \(c>0\) is a given and fixed constant. Using direct martingale arguments we first show that when \(\mu \le 0\) it is optimal to stop at once and when \(\mu \ge \sigma ^2\!/2\) it is optimal not to stop at all. By employing the method of Lagrange multipliers we then show that the nonlinear problem for \(0 < \mu < \sigma ^2\!/2\) 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 where \(\gamma = \mu /(\sigma ^2\!/2)\). 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.