American options on assets with dividends near expiry

Explicit expressions valid near expiry are derived for the values and the optimal exercise boundaries of American put and call options on assets with dividends. The results depend sensitively on the ratio of the dividend yield rate D to the interest rate r. For D>r the put boundary near expiry tends parabolically to the value rK/D where K is the strike price, while for D≤r the boundary tends to K in the parabolic-logarithmic form found for the case D=0 by Barles et al. (1995) and by Kuske and Keller (1998). For the call, these two behaviors are interchanged: parabolic and tending to rK/D for D<r, as was shown by Wilmott, Dewynne, and Howison (1993), and parabolic-logarithmic and tending to K for D≥r. The results are derived twice: once by solving an integral equation, and again by constructing matched asymptotic expansions.