A Laplace Space Approach to American Options

In this paper, we extend the lower-upper bound approximation (LUBA) idea of Broadie and Detemple [Broadie, M., Detemple, J., (1996) American option valuation: New bounds, approximations, and comparison of existing methods. Review of Financial Studies. 9(4): 1211-1250] to the Laplace space. We construct tight lower and upper bounds for the price of a finite-maturity American option when the underlying stock is modeled by a large class of stochastic processes, for which there exist closed-form expressions for the Laplace transforms of the corresponding “capped (barrier) option prices”. The method is applied to a time-homogeneous diffusion process and a jump diffusion process. The novelty of the method is to first take the Laplace transform of the price of the “capped (barrier) option” with respect to the time to maturity, and then carry out optimization procedures similar as Broadie and Detemple in the Laplace space. Finally we numerically invert the Laplace transforms to obtain the lower bound of the price of the American option, and further utilize the early exercise premium (EEP) representation in the Laplace space to obtain the upper bound. We obtain explicit expressions in the case of the constant elasticity of variance (CEV) model (Wong and Zhao) and the double-exponential jump diffusion (DEJD) model (Leippold and Vasiljevic). Numerical examples show that our lower and upper bounds are accurate and efficient compared to results in the literature. To the best of authors' knowledge, it is the first time that the LUBA idea of Broadie and Detemple is applied to a model with jumps, and this solves an open question stated on page 1181 of Kou and Wang