Journal of Applied Probability

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

Pavel V. Gapeev and Neofytos Rodosthenous

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Abstract

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

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

Dates
First available in Project Euclid: 5 September 2014

Permanent link to this document
https://projecteuclid.org/euclid.jap/1409932675

Digital Object Identifier
doi:10.1239/jap/1409932675

Mathematical Reviews number (MathSciNet)
MR3256228

Zentralblatt MATH identifier
1312.60044

Subjects
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

Keywords
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

Citation

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. https://projecteuclid.org/euclid.jap/1409932675


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