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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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Contrary to economic theory and common sense, stock returns are negatively related to both expected and unexpected inflation. We argue that this puzzling empirical phenomenon does not indicate causality.Instead, stock returns are negatively related to contemporaneous changes in expected inflation because they signal a chain of events which results in a higher rate of monetary expansion. Exogenous shocks in real output, signalled by the stock market, induce changes in tax revenue, in the deficit, in Treasury borrowing and in Federal Reserve "monetization" of the increased debt. Rational bond and stock market investors realize this will happen. They adjust prices (and interest rates) accordingly and without delay.Although expected inflation seems to have a negative effect on subsequent stock returns, this could be an empirical illusion, since a spurious causality is induced by a combination of: (a) a reversed adaptive inflation expectations model and (b) a reversed money growth/stock returns model.If the real interest rate is not a constant, using nominal interest proxies for expected inflation is dangerous, since small changes in real rates can cause large and opposite percentage changes in stock prices. THERE ISA WELL-documented but puzzling empirical relation between stock returns and inflation. Expected inflation, unexpected inflation, and changes in expected inflation are all negatively related to stock returns. See Fama and Schwert [12] and the less comprehensive but consistent work by Lintner [26], Jaffe and Mandelker [23], and Nelson [32]. We offer here an explanation for this phenomenon and present evidence supporting the explanation.The empirical results merit attention because they appear to be in conflict with both economic theory and common sense, according to which stock returns should be positively related to both expected and unexpected inflation. A positive * Graduate School of Management, University of California, Los Angeles. We are grateful for many helpful conversations with Brad Cornell, for the assistance of Ted Lloyd of the UCLA macroeconomic forecasting group in assembling data, for the assistance of Winston Cheong in computations, and for comments and suggestions from Eugene Fama, Kenneth French, David Mayers, Bruno Solnik, Sheridan Titman, and from workshop participants at Claremont Graduate School, Tennessee, Toronto, and UCLA (Economics). We owe a particular debt to Michael Brennan for many helpful editorial and substantive suggestions. However, no one named above necessarily agrees with or is responsible for the resulting contents of this paper. 2The Journal of Finance stock price reaction to unexpected inflation is suggested by the traditional idea that equities are "hedges" against (unanticipated) inflation because they represent claims to real assets. Stock returns should be positively related to expected inflation according to the Fisherian theory of interest; the nominal expected return on any asset equals real interest and a real risk premium (if appropriate), plus expected infla...