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September 2014 Optimal stopping problems in diffusion-type models with running maxima and drawdowns
Pavel V. Gapeev, Neofytos Rodosthenous
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J. Appl. Probab. 51(3): 799-817 (September 2014). DOI: 10.1239/jap/1409932675

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.

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Pavel V. Gapeev. Neofytos Rodosthenous. "Optimal stopping problems in diffusion-type models with running maxima and drawdowns." J. Appl. Probab. 51 (3) 799 - 817, September 2014. https://doi.org/10.1239/jap/1409932675

Information

Published: September 2014
First available in Project Euclid: 5 September 2014

zbMATH: 1312.60044
MathSciNet: MR3256228
Digital Object Identifier: 10.1239/jap/1409932675

Subjects:
Primary: 34K10 , 60G40 , 91B70
Secondary: 34L30 , 60J60 , 91B25

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

Rights: Copyright © 2014 Applied Probability Trust

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Vol.51 • No. 3 • September 2014
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