Abstract
We present closed-form solutions to some discounted optimal stopping problems for the running maximum of a geometric Brownian motion with payoffs switching according to the dynamics of a continuous-time Markov chain with two states. The proof is based on the reduction of the original problems to the equivalent free-boundary problems and the solution of the latter problems by means of the smooth-fit and normal-reflection conditions. We show that the optimal stopping boundaries are determined as the maximal solutions of the associated two-dimensional systems of first-order nonlinear ordinary differential equations. The obtained results are related to the valuation of real switching lookback options with fixed and floating sunk costs in the Black-Merton-Scholes model.
| Original language | English |
|---|---|
| Pages (from-to) | 189-219 |
| Journal | Advances in Applied Probability |
| Volume | 53 |
| Issue number | 1 |
| DOIs | |
| Publication status | Published - Mar 2021 |
Keywords
- Discounted optimal stopping problem
- geometric Brownian motion
- running maximum process
- continuous-time Markov chain
- free-boundary problem
- instantaneous stopping and smooth fit
- normal reflection
- perpetual American and real options
- change-of-variable formula with local time on surfaces
- DIFFUSION-TYPE MODELS
- RENEWABLE ENERGY INVESTMENTS
- PERPETUAL AMERICAN
- RUNNING MAXIMA
- HIDDEN TARGET
- OPTIONS
- POLICY
- TIME
- INEQUALITIES
- UNCERTAINTY