Abstract
Assuming that the stock price X follows a geometric Brownian motion with drift (Formula presented.) and volatility (Formula presented.) , and letting (Formula presented.) denote a probability measure under which X starts at (Formula presented.) , we study the dynamic version of the nonlinear mean–variance optimal stopping problem (Formula presented.) where t runs from 0 onwards, the supremum is taken over stopping times (Formula presented.) of X, and (Formula presented.) is a given and fixed constant. Using direct martingale arguments we first show that when (Formula presented.) it is optimal to stop at once and when (Formula presented.) it is optimal not to stop at all. By employing the method of Lagrange multipliers we then show that the nonlinear problem for (Formula presented.) 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 (Formula presented.) where (Formula presented.). 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.
| Original language | English |
|---|---|
| Journal | Mathematics and Financial Economics |
| Volume | 10 |
| Issue number | 2 |
| Pages (from-to) | 203-220 |
| ISSN | 1862-9679 |
| DOIs | |
| Publication status | Published - 1 Mar 2016 |