Discrete-time pricing and optimal exercise of American perpetual warrants in the geometric random walk model
Vanderbei, Robert J. and Pınar, Mustafa C. and Bozkaya, Efe Burak (2013) Discrete-time pricing and optimal exercise of American perpetual warrants in the geometric random walk model. Applied Mathematics & Optimization, 67 (1). pp. 97-122. ISSN 0095-4616 (Print) 1432-0606 (Online)
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Official URL: http://dx.doi.org/10.1007/s00245-012-9182-0
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.
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