“Closed-form Solutions for American Options”

Wai Man Tse, Chu Hai College of Higher Education 

American options unlike European options allow early exercise prior to its maturity. They are the second most actively traded options with transaction values of astronomical figures. Exact closed-form solutions exist only for American call options with single dividend payout and American call options with no dividend yield. Thus far, the American options problem is solved by numerical techniques: lattice methods, Monte Carlo simulation or by approximate closed-form solutions. While the approximate closed-form formula of Barone-Adesi and Whaley (1987) depends on a single spot boundary, Kim (1990) solved the American option problem by adding an early exercise premium to its European option value. In this study, we derived an exact closed-form formula for American options when stock returns follow normal distribution and an exact single integral solution when the returns follow Lévy processes. Closed-form formulas could exist under Lévy processes if their European options could be valued in closed-form. Our specific way of repesentation of optimal stopping time forms the key to our formulas derivation. Our constant-dividend-yield solution could be easily modified to solve in closed-form American put or call options with dividend payouts at discrete time points, and Bermudan options without curse of dimensionality.