The Journal of Finance

THE PRICING OF OPTIONS WITH STOCHASTIC DIVIDEND YIELD

Published: 05/01/1978   |   DOI: 10.1111/j.1540-6261.1978.tb04871.x

Robert Geske

A formula is derived in discrete time for pricing options when the underlying stock has a stochastic dividend yield. The result implies that regarding the dividend yield as certain when it is not results in misestimation of the variance of the underlying stock. Comparative statics indicate that this adjustment could diminish a bias of the Black‐Scholes model. This model systematically underprices deep‐out‐of‐the‐money options. A numerical example demonstrates that this stochastic adjustment may be more important for longer‐lived options and warrants.

On Valuing American Call Options with the Black‐Scholes European Formula

Published: 06/01/1984   |   DOI: 10.1111/j.1540-6261.1984.tb02319.x

ROBERT GESKE, RICHARD ROLL

Empirical papers on option pricing have uncovered systematic differences between market prices and values produced by the Black‐Scholes European formula. Such “biases” have been found related to the exercise price, the time to maturity, and the variance. We argue here that the American option variant of the Black‐Scholes formula has the potential to explain the first two biases and may partly explain the third. It can also be used to understand the empirical finding that the striking price bias reverses itself in different sample periods. The expected form of the striking price bias is explained in detail and is shown to be closely related to past empirical findings.

The Fiscal and Monetary Linkage between Stock Returns and Inflation

Published: 03/01/1983   |   DOI: 10.1111/j.1540-6261.1983.tb03623.x

ROBERT GESKE, RICHARD ROLL

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.

The American Put Option Valued Analytically

Published: 12/01/1984   |   DOI: 10.1111/j.1540-6261.1984.tb04921.x

ROBERT GESKE, H. E. JOHNSON

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.

Over‐the‐Counter Option Market Dividend Protection and “Biases” in the Black‐Scholes Model: A Note

Published: 09/01/1983   |   DOI: 10.1111/j.1540-6261.1983.tb02295.x

ROBERT GESKE, RICHARD ROLL, KULDEEP SHASTRI

Most options are traded over‐the‐counter (OTC) and are dividend “protected;” the exercise price decreases on the ex date by an amount equal to the dividend. This protection completely inhibits the early exercise of American call options. Nevertheless, OTC‐protected options have market values which differ systematically from Black‐Scholes values for European options on non‐dividend paying stocks. The pricing difference is related to both the variance of the underlying stock return and to time until expiration of the option, but it is quite small in dollar amount.