Upper and lower bounds are derived for call options traded at discrete intervals. These bounds are independent of assumptions on the stock price distribution other than a restriction satisfied by the stock being "non-negative beta." The development of the bounds relies on the single-price law and arbitrage arguments. Both single-period and multiperiod results are produced, and put option bounds follow by extension. The bounds exist as equilibrium values given a consensus on stock price distribution; they are also valid for empirical studies, being adjustable for dividends and commissions. FOLLOWING THE IMPETUS of theBlack-Scholes pricing model, the majority of option analyses have been in continuous time. Such approaches have relied on the formation of perfect, riskless hedges involving the option, its underlying security, and a riskless asset; since the hedge must be continuously revised and maintained, while actual trading opportunities are discrete, this crucial assumption limits the accuracy and applicability of the Black-Scholes and related formulae. In fact, the Black-Scholes formula can only be considered as "a valid approximation to the discrete-time solution" (Merton [8, p. 663]), while the closeness of that approximation has not been ascertained in general.1 Earlier discrete models such as Boness [3] and Samuelson [12] have been succeeded by those of Brennan [5], Cox et al. [6], and Rubinstein [11], in which distributional restrictions are imposed on share returns; usually, the discrete formula converges to the Black-Scholes expression under appropriate limit conditions. Using the Rubinstein [11] approach, we derive upper and lower bounds for option prices with both a general price distribution and discrete trading opportunities. The bounds are functions of the share and exercise prices, the riskless rate of interest, the distribution of the share price change per period, and the number of periods of expiration of the option. The lower bound is tighter than that of Merton [7], derived from more general arbitrage considerations, while being less exact than the results already mentioned for restricted distributions. Thus the bounds form a range of prices in equilibrium derivable for any general distribution; they have the further advantage of being adjustable for dividends and commissions. The generality of assumptions and adjustable range provide important advantages for empirical studies. * Professor and Associate Professor, Faculty of Administration, University of Ottawa. A preliminary version of this paper was presented under a different title at the 4th World Congress of the Econometric Society at Aix-en-Provence, in August 1980. The authors wish to thank a referee of this Journal for helpful advice and comment. 1 See [6], [9], and [10] for pertinent comments on this point. 519 520 The Journal of Finance I. Call Option Bounds We adopt the following notation: S the initial price of the underlying asset, X the exercise price of the option, Y the random price change (AS) per period, F(Y) the probability...

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We derive equilibrium restrictions on the range of the transaction prices of American options on the stock market index and index futures. Trading over the lifetime of the options is accounted for, in contrast to earlier single-period results. The bounds on the reservation purchase price of American puts and the reservation write price of American calls are tight. We allow the market to be incomplete and imperfect due to the presence of proportional transaction costs in trading the underlying security and due to bid-ask spreads in option prices. The bounds may be derived for any given probability distribution of the return of the underlying security and admit price jumps and stochastic volatility. We assume that at least some of the traders maximize a time-separable utility function. The bounds are derived by applying the weak notion of stochastic dominance and are independent of a trader's particular utility function and initial portfolio position.