Journal of Applied Probability

Optimal stopping problems in diffusion-type models with running maxima and drawdowns

Pavel V. Gapeev and Neofytos Rodosthenous

Full-text: Access denied (no subscription detected)

We're sorry, but we are unable to provide you with the full text of this article because we are not able to identify you as a subscriber. If you have a personal subscription to this journal, then please login. If you are already logged in, then you may need to update your profile to register your subscription. Read more about accessing full-text


We study optimal stopping problems related to the pricing of perpetual American options in an extension of the Black-Merton-Scholes model in which the dividend and volatility rates of the underlying risky asset depend on the running values of its maximum and maximum drawdown. The optimal stopping times of the exercise are shown to be the first times at which the price of the underlying asset exits some regions restricted by certain boundaries depending on the running values of the associated maximum and maximum drawdown processes. We obtain closed-form solutions to the equivalent free-boundary problems for the value functions with smooth fit at the optimal stopping boundaries and normal reflection at the edges of the state space of the resulting three-dimensional Markov process. We derive first-order nonlinear ordinary differential equations for the optimal exercise boundaries of the perpetual American standard options.

Article information

J. Appl. Probab., Volume 51, Number 3 (2014), 799-817.

First available in Project Euclid: 5 September 2014

Permanent link to this document

Digital Object Identifier

Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 60G40: Stopping times; optimal stopping problems; gambling theory [See also 62L15, 91A60] 34K10: Boundary value problems 91B70: Stochastic models
Secondary: 60J60: Diffusion processes [See also 58J65] 34L30: Nonlinear ordinary differential operators 91B25: Asset pricing models

Multidimensional optimal stopping problem Brownian motion running maximum and running maximum drawdown process free-boundary problem instantaneous stopping and smooth fit normal reflection change-of-variable formula with local time on surfaces perpetual American option


Gapeev, Pavel V.; Rodosthenous, Neofytos. Optimal stopping problems in diffusion-type models with running maxima and drawdowns. J. Appl. Probab. 51 (2014), no. 3, 799--817. doi:10.1239/jap/1409932675.

Export citation