Weak Solutions for the Valuation of American Options, I: Uniqueness and Equivalence

This article proposes a new formula for the valuation of American options by using a weak variational form for the corresponding free boundary value problem derived by Merton. It is shown that the weak form has a unique solution, and the classical solution of the original free boundary value problem satisses the weak form. Therefore, solutions to both weak and strong forms are unique. The weak variational formula for the American option pricing is advantageous over the classical one in that the identiication of the optimal exercise curve (free boundary) is separated from the determination of the valuation function. It is, therefore, possible to design eecient numerical schemes for computing the price of options of American type. 1. Introduction We are concerned with new mathematics and computational techniques for the valuation of American options. Merton 17] and McKean 18] showed that the valuation of American options can be determined as the solution of a free boundary value problem of parabolic type. The unknown function in the model equation corresponds to the valuation function, and the free boundary represents the time path of critical stock prices beyond which early exercise is warranted. The free boundary is also known as an optimal exercise curve in the pricing of American options. The main diiculty associated with the valuation of American options in theory and application is the fact that the optimal exercise curve must be identiied as part of the solution for the modeling equation. Research on this problem has been directed at either analytical methods or numerical methods. In the analytical approach, attempt has been made in the search for a simple characterization of the optimal exercise curve. For example, closed form solutions in terms of exogenous variables and the optimal exercise boundary are presented by Carr, Jarrow, and Myneni 9], Jacka 11], and Kim 13]. The method of variational inequalities was employed by Bensoussan and Lions 4] and Jaillet, Lamberton, and Lapeyre 12] for a study in the solution uniqueness and existence. In the numerical methods, implicit nite diierence methods of Brennan and Schwartz 6, 7] and Brenner, Courtadon, Subrahmanyam 8] and quadratic approximation techniques of Barone-Adesi