Optimal exercise collar type and multiple type perpetual American stock options in discrete time with linear programming

An American option is an option that entitles the holder to buy or sell an asset at a pre-determined price at any time within the period of the option contract. A perpetual American option does not have an expiration date. In this study, we solve the optimal stopping problem of a perpetual American stock option from optimization point of view using linear programming duality under the assumption that underlying’s price follows a discrete time and discrete state Markov process. We formulate the problem with an infinite dimensional linear program and obtain an optimal stopping strategy showing the set of stock-prices for which the option should be exercised. We show that the optimal strategy is to exercise the option when the stock price hits a special critical value. We consider the problem under the following stock price movement scenario: We use a Markov chain model with absorption at zero, where at each step the stock price moves up by ∆x with probability p, and moves down by ∆x with probability q and does not change with probability 1 − (p + q). We examine two special type of exotic options. In the first case, we propose a closed form formula when the option is collar type. In the second case we study multiple type options, that are written on multiple assets, and explore the exercise region for different multiple type options.