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Discrete-Time Pricing and Optimal Exercise of American Perpetual Warrants in the Geometric Random Walk Model

Author(s): Vanderbei, Robert J.; Pınar, Mustafa Ç; Bozkaya, Efe B

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Abstract: An American option (or, warrant) is the right, but not the obligation, to purchase or sell an underlying equity at any time up to a predetermined expiration date for a predetermined amount. A perpetual American option differs from a plain American option in that it does not expire. In this study, we solve the optimal stopping problem of a perpetual American option (both call and put) in discrete time using linear programming duality. Under the assumption that the underlying stock price follows a discrete time and discrete state Markov process, namely a geometric random walk, we formulate the pricing problem as an infinite dimensional linear programming (LP) problem using the excessive-majorant property of the value function. This formulation allows us to solve complementary slackness conditions in closed-form, revealing an optimal stopping strategy which highlights the set of stock-prices where the option should be exercised. The analysis for the call option reveals that such a critical value exists only in some cases, depending on a combination of state-transition probabilities and the economic discount factor (i.e., the prevailing interest rate) whereas it ceases to be an issue for the put.
Publication Date: Feb-2013
Electronic Publication Date: 26-Sep-2012
Citation: Vanderbei, Robert J, Pınar, Mustafa Ç, Bozkaya, Efe B. "Discrete-Time Pricing and Optimal Exercise of American Perpetual Warrants in the Geometric Random Walk Model" Applied Mathematics & Optimization, 67(1), 97 - 122, doi:10.1007/s00245-012-9182-0
DOI: doi:10.1007/s00245-012-9182-0
ISSN: 0095-4616
EISSN: 1432-0606
Pages: 97 - 122
Type of Material: Journal Article
Journal/Proceeding Title: Applied Mathematics & Optimization
Version: This is the author’s final manuscript. All rights reserved to author(s).

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