We provide a comprehensive treatment of option pricing with particular emphasis on the valuation of American options on dividend-paying assets. We begin by reviewing principles for European contingent claims in a financial market in which the underlying asset price follows an Ito process and the interest rate is stochastic. Then this analysis is extended to the valuation of American contingent claims. In particular, the early exercise premium and the delayed exercise premium representations of the American option price are presented. These results are specialized in the case of the standard market model, i.e., when the underlying asset price follows a geometric Brownian motion process and the interest rate is constant. American capped options with constant and growing caps are then analyzed. Valuation formulas are first provided for capped options on dividend-paying assets in the context of the standard market model. Previously unpublished results are then presented for capped options on nondividend-paying assets when the underlying asset price follows an Ito process with stochastic volatility and the cap's growth rate is an adapted stochastic process.