This thesis presents two numerical procedures for valuing American call and put options written on individual stocks, stock indices, foreign currencies, and futures contracts for which early exercise may be optimal. The valuation problem of American options do not admit simple closed form solutions and for such reasons numerical solution is the only means of obtaining quantitative results. Chapters 2 and 3 present a general study on the properties of optimal exercise boundaries in binomial option pricing model and apply them to a variety of American options. In these chapters, the properties associated with the optimal exercise boundary in the binomial option pricing model are examined and an efficient recursive valuation method which incorporates these properties is presented. This valuation method represents a substantial improvement over the conventional binomial model in terms of computational efficiency with exactly the same accuracy.
The early exercise premium representation of the American option value in terms of the optimal exercise boundary was derived in a collection of papers, working from McKean's formulation of the free boundary problem. As a result, the determination of the optimal exercise boundary has become an important element in the valuation of American options. Chapter 4 presents a formula that relates the optimal exercise boundaries of American call and put options on futures contract. It is shown that the geometric mean of the optimal exercise boundaries for call and put on the same futures contract with the same exercise price is equal to the exercise price. This chapter also investigates the properties of American calls and puts written the same futures contract. Chapter 5 develops a computationally efficient technique for the valuation of American options. The approximation method utilizes the early exercise premium representation for the American option price. This chapter pay particular attention to the form and behavior of the optimal exercise boundary. This chapter also reports the results of a number of tests for the method's efficiency. The results of the thesis will be of interest to financial analysts and practitioners who have an active need to acquire useful techniques for valuing American type contingent claims.
