Applied Mathematics & Optimization

, Volume 67, Issue 1, pp 97–122 | Cite as

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

  • Robert J. Vanderbei
  • Mustafa Ç. PınarEmail author
  • Efe B. Bozkaya


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.


American perpetual warrants Pricing Optimal stopping Optimal exercise Random walk Linear programming Duality 


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Copyright information

© Springer Science+Business Media New York 2012

Authors and Affiliations

  • Robert J. Vanderbei
    • 1
  • Mustafa Ç. Pınar
    • 2
    Email author
  • Efe B. Bozkaya
    • 3
  1. 1.Department of Operations Research and Financial EngineeringPrinceton UniversityPrincetonUSA
  2. 2.Department of Industrial EngineeringBilkent UniversityAnkaraTurkey
  3. 3.Faculty of Administrative SciencesSabancı UniversityIstambulTurkey

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