Option Bounds in Discrete Time: Extensions and the Pricing of the American Put

Abstract

In a recent paper (Perrakis and Ryan 1984), upper and lower bounds on European option prices were derived in discrete time. These bounds were based on an extension of the discrete-time derivation of Black-Scholes pricing, first presented in the work of Black and Scholes (1973). The bounds are functions of the stock and exercise prices, the riskless rate of interest, the time to maturity, and the entire distribution of stock returns. In other words, they need more information than Black-Scholes for their computation, and the derived results are weaker.' On the other hand, they are significantly more general in their assumptions, insofar as they do not rely on the continuous-time riskless hedge of the original Black-Scholes derivations or on the assumption of a binomial distribution of stock returns used in Cox, Ross, and Rubinstein (1979) and Rendleman and Bartter (1979). For this reason they are likely to be particularly useful in valuing options on assets, for which the Black-Scholes methodology is clearly This paper generalizes and tightens the discrete-time bounds for European options and develops similar bounds for American puts. The stock return has a distribution with a finite range and is subject to the restriction that the stock be beta. The bounds are derived from arbitrage portfolios using the stock, the option, and the riskless asset. They converge to the previous ones when the range coincides with the nonnegative real line, and they both become equal to the binomial option model under a two-state stock return. Upper and lower bounds for the American put are derived recursively by the same method, with arbitrage portfolios involving the stock, the riskless asset, and American and European puts. An example is provided with a trinomial return distribution. * Part of this research was done while I was a visiting professor (1983-84), Ecole Superieure de Commerce et d'Administration des Entreprises, Reims, France. I wish to thank Peter Ryan, Albert Madansky, and a referee of this Journal for helpful advice and comment, while retaining all responsibility for any remaining errors or omissions. 1. Note, however, that this is also true of other BlackScholes generalizations, such as Merton (1976).

Cite this work

No public copy is stored for this record, so only the metadata is cited.

Stored only when both papers are in the corpus — 86 citations exist upstream.

Papers that cite the same references — the relational fallback while embeddings are still backfilling.