Optimal mean–variance selling strategies

Abstract

Assuming that the stock price X follows a geometric Brownian motion with drift µ ∈ IR and volatility σ > 0 , and letting 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 problemwhere 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 ≥ σ 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 < µ < σ 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 bywhere γ = µ/(σ 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.

Optimal mean-variance portfolio selection

“…Moreover, we will see below that in addition to the static formulation of the nonlinear problem (2.4) where the maximisation takes place relative to the initial point (t, x) which is given and fixed, one is also naturally led to consider a dynamic formulation of the nonlinear problem (2.4) in which each new position of the controlled process ((t, X u t )) t∈[0,T ] yields a new optimal control problem to be solved upon overruling all the past problems. We believe that this dynamic optimality is of general interest in the nonlinear problems of optimal control (as well as nonlinear problems of optimal stopping as discussed in [13]). …”

“…2. Apart from the paper [13] where the dynamic optimality was used in a nonlinear problem of optimal stopping, we are not aware of any other paper on optimal control where nonlinear problems were studied using this methodology. The dynamic optimality (Definition 2) appears therefore to be original to the present paper in the context of nonlinear problems of optimal control.…”

“…The purpose of the present paper is to develop a new methodology for solving nonlinear optimal control problems of this kind and demonstrate its use in the optimal mean-variance portfolio selection problem stated above. This is done in parallel to the novel methodology for solving nonlinear optimal stopping problems that was recently developed in [13] when tackling an optimal mean-variance selling problem.…”

On time-inconsistent stochastic control in continuous time

“…Other "quadratic" control problems are considered in [2,6,8], which study mean-variance problems within the present gametheoretic framework. In the papers [19] and [20], the authors study the meanvariance criterion where you are continuously rolling over instantaneously updated pre-committed strategies.…”

Monotone Sharpe Ratios and Related Measures of Investment Performance

“…Nevertheless, the value of this paper is in the new approach to solve them through the monotone Sharpe ratio and buffered probabilities. This approach seems to be simpler than previous ones (the reader can observe how short the solutions presented below compared to [3,4]) and promising for more general Observe that if 0 < µ < σ 2 2 , i.e. γ ∈ (0, 1), then the function f (b) attains its minimum on [x, ∞), since it is continuous with the limit values f (x) = f (∞) = 1.…”