## 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.

Original language | English (US) |
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Pages (from-to) | 97-122 |

Number of pages | 26 |

Journal | Applied Mathematics and Optimization |

Volume | 67 |

Issue number | 1 |

DOIs | |

State | Published - Feb 2013 |

Externally published | Yes |

## All Science Journal Classification (ASJC) codes

- Control and Optimization
- Applied Mathematics

## Keywords

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