Adaptive mesh relocation refinement on Kim's method: enhanced approximations and upper bounds for American options

Abstract

This study demonstrates the efficiency of an adaptive mesh relocation refinement (AMrR) to Kim's (1990) American options pricing method. Given the suboptimal uniform time discretisation proposed by Kim, we test an r-adaptive strategy that controls the overall error using an adjoint objective function. Next, we build an a posteriori goal-oriented error estimator as a measure of the global error incurred by the time mesh used. The analytics show the AMrR with n = 30 time steps improves upon Kim's (1990) method root mean squared relative error (RMSRE) by 85.74% for short and by 73.70% for long-term American options. Furthermore, we use the AMrR to obtain tighter upper bounds for the option's theoretical value, built directly from Broadie and Detemple (1996) and Chung et al. (2010) upper bound methods. In respect to both studies, we find substantial improvements in the RMSRE for both short and long American options.