Vadesiz amerikan tipi opsiyonların doğrusal programlama ile fiyatlandırılması ve en iyi kullanım değerlerinin belirlenmesi

An American option is the right but not the obligation to purchase or sell anunderlying equity at any time up to a predetermined expiration date for a predeterminedamount. A perpetual American option differs from a plain Americanoption in that it does not expire. In this study, we solve the optimal stoppingproblem of a perpetual American option with methods from the linear programmingliterature. Under the assumption that the underlying's price follows a discretetime and discrete state Markov process, we formulate the problem with aninfinite dimensional linear program using the excessive and majorant properties ofthe value function. This formulation allows us to solve complementary slacknessconditions efficiently, revealing an optimal stopping strategy which highlights theset of stock-prices for which the option should be exercised. Under two differentstock-price movement scenarios (simple and geometric random walks), we showthat the optimal strategy is to exercise the option when the stock-price hits a specialcritical value. The analysis also reveals that such a critical value exists onlyfor some special cases under the geometric random walk, dependent on a combinationof state-transition probabilities and the economic discount factor. Wefurther demonstrate that the method is useful for determining the optimal stoppingtime for combinations of plain vanilla options, by solving the same problemfor spread and strangle positions under simple random walks.