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...

Various strategies for sequencing heterogeneous azeotropic distillations have been explored. The design and optimization of these systems by simulation-based methods is generally acknowledged to be a very difficult problem (Kingsley and Lucia, 1988). In this article, a simple and accurate design and optimization procedure has been developed. The method does not involve any recycle convergence loops and has been applied successfully to several test problems. Unlike the simulation-based methods, our method is robust to changes in the system optimization variables (e.g., reflux ratio in the azeotropic column).In order to demonstrate the method, w e present design and optimization results for the dehydration of ethanol using a benzene entrainer. Our optimal designs consume significantly less energy than state-of-the-art designs reported in the literature.

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A simple design technique for heterogeneous distillation columns has been developed. The underlying geometric conditions which govern minimum reflux are shown to share certain common features with homogeneous distillations, but differ in some important respects. It is shown that heterogeneous columns display some unusual sensitivities to the design parameters. The most notable being the drastic swings in the composition and temperature profiles with very small changes in the level of trace impurities in the product stream.

Background: A distinctive feature of patients suffering from Crohn's disease is a predisposition to develop a variety of anal complications. The aetiology of such conditions is unclear, and the reported incidence of anal involvement in Crohn's disease varies party due to the various criteria used for classification. This study aims to review the management of patients with symptomatic anal pathology associated with Crohn's disease at St Vincent's Hospital, Melbourne. Methods: A database of 306 patients with Crohn's disease referred to the department between January 1978 and October 1994 was reviewed to identify those patients with symptomatic anal disease. The anal pathology was recorded and classified. Demographic data and the clinical and surgical history of the patient were recorded. Results: Of the 306 patients with Crohn's disease, 129 (42.4%) were identified as having symptomatic anal pathology. Patients were likely to present with anal symptoms after they had been diagnosed as having intestinal Crohn's disease (46.1 %). The commonest presentations were perianal abscess (29.5%), anal fissure (27.6%), and low anal fistula (26.7%). A minority of patients presented with highkomplex anal fistulae (3.8%), or recto-vaginal fistulae (5.2%). Five per cent of patients had Crohn's disease localized to the anal area. The pattern of intestinal disease in the remaining patients was small bowel 21.1%. small bowel and colon 31.9%, and colon 43.0%. A total of 244 local anal surgical procedures were performed on these patients; the commonest of these were drainage of an abscess (38.5%), examination under anaesthetic (29.1%). and laying open of a low anal fistula (22.5%). Following surgical treatment, the recurrence rate for perianal abscesses was 13%, and for low anal fistulae 6%. Conclusions: The majority of patients with Crohn's disease who develop anal pathology have an excellent prognosis. A minority of patients develop complex anal fistulae and these remain a therapeutic challenge.