Asymptotic Approximations for Asian, European, and American Options with Discrete Averaging or Discrete Dividend/Coupon Payments

We develop approximations to the pricing of options on an asset which makes discrete dividend payments, focusing on the case of frequent payments. The principal mathematical tool is the method of multiple timescales, allied to matched asymptotic expansions. We first analyze European style options, deriving the continuously paid dividend equation as a limiting case of the relevant discrete problem, and we analyze the range accrual note to compute the relevant “continuity correction.” We also carry out the same analysis for Asian options with discrete averaging. We then give a detailed description of the intricate exercise policies that arise for American put (and, to a lesser extent, call) options when dividends are paid discretely, for the cases of proportionate and fixed-amount dividends.