This paper presents a theory of rational option pricing, focusing on the properties of option prices under various assumptions. The author, Robert C. Merton, discusses the development of a general theory of contingent-claims pricing, with options as a key example. The paper begins by examining the pricing of warrants and options, noting that while options are relatively simple financial instruments, their pricing can provide insights into broader financial theories. The paper outlines several theorems and assumptions that define the properties of rational option pricing. It discusses the differences between American and European options, noting that American options can be exercised at any time before expiration, while European options can only be exercised at expiration. The paper also addresses the effects of dividends and changes in exercise prices on option pricing. Key findings include theorems that show the value of an American warrant is at least as high as its European counterpart, and that the value of a perpetual warrant equals the value of the underlying stock. The paper also discusses the homogeneity property of option pricing, which implies that the price of an option should scale proportionally with the underlying asset's price. The paper further explores the implications of different market conditions, such as the presence of dividends and changes in exercise prices, on option pricing. It also compares the pricing of American and European options, showing that American options can command a premium due to the possibility of early exercise. The paper concludes by discussing the Black-Scholes model and its implications for option pricing, emphasizing the importance of assumptions about market conditions and investor behavior in deriving the pricing formula. The analysis highlights the need for rigorous assumptions and the importance of understanding the underlying economic principles that govern option pricing.This paper presents a theory of rational option pricing, focusing on the properties of option prices under various assumptions. The author, Robert C. Merton, discusses the development of a general theory of contingent-claims pricing, with options as a key example. The paper begins by examining the pricing of warrants and options, noting that while options are relatively simple financial instruments, their pricing can provide insights into broader financial theories. The paper outlines several theorems and assumptions that define the properties of rational option pricing. It discusses the differences between American and European options, noting that American options can be exercised at any time before expiration, while European options can only be exercised at expiration. The paper also addresses the effects of dividends and changes in exercise prices on option pricing. Key findings include theorems that show the value of an American warrant is at least as high as its European counterpart, and that the value of a perpetual warrant equals the value of the underlying stock. The paper also discusses the homogeneity property of option pricing, which implies that the price of an option should scale proportionally with the underlying asset's price. The paper further explores the implications of different market conditions, such as the presence of dividends and changes in exercise prices, on option pricing. It also compares the pricing of American and European options, showing that American options can command a premium due to the possibility of early exercise. The paper concludes by discussing the Black-Scholes model and its implications for option pricing, emphasizing the importance of assumptions about market conditions and investor behavior in deriving the pricing formula. The analysis highlights the need for rigorous assumptions and the importance of understanding the underlying economic principles that govern option pricing.