The American Put Option Valued Analytically

Authors

  • ROBERT GESKE,

  • H. E. JOHNSON

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    • University of California-Los Angeles and Louisiana State University, respectively. This paper was presented and benefitted from comments at the Western Finance Association in Portland and the European Finance Association in Fontainbleau, France, and at seminars at the University of Utah, Washington State University, and McGill University. The authors are grateful for individual comments from Warren Bailey, Cliff Ball, Ed Blomeyer, Nai-fu Chen, Ken French, Chi-Cheng Hsia, David Mayers, Richard Roll, Mark Rubinstein, and Walt Torous, and the programming assistance of Ho Yang and B. F. Wexler.

ABSTRACT

An analytic solution to the American put problem is derived herein. The hedge ratio and other derivatives of the solution are presented. The formula derived implies an exact duplicating portfolio for the American put consisting of discount bonds and stock sold short. The formula is extended to consider put options on stocks paying cash dividends. A polynomial expression is developed for evaluating these formulae. Values and hedge ratios for puts on both dividend and nondividend paying stocks are calculated, tabulated, and compared with values derived by numerical integration and binomial approximation. As with European options, evaluating an analytic formula is more efficient than approximating the stock price process or the partial differential equation by binomial or finite difference methods. Finally, applications of this American put solution are discussed.

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