Research on the American Call Options on the Stocks Paying Multiple Dividends

Cassimon et al. (2007) propose a pricing formula of American call options under the multiple dividends by extending Roll (1977). However, because these studies investigate the option pricing formula under the escrow model, there is inconsistency for the assumption of the stock prices. This paper proposes pricing formulas of American call options under the multiple dividends and piecewise geometric Brownian motion. For the formulas, I approximate the log prices of ex-dividend dates to follow a multivariate normal distribution, and decompose the option price as a function of payoffs and exercise boundaries. Then, I obtain an upper bound of the American call options by substituting approximated log prices into the both of the payoffs and the exercise boundaries. Besides, I obtain a lower bound of the price by substituting approximated price only into the exercise boundaries. These upper and lower bounds are exact prices when the amounts of dividends are linear to the stock prices. According to the numerical study, the lower bound produces relatively small errors. Especially, it produces small errors when the dividends are more sensitive to the stock price changes.