An asymptotic computational method for the nonlinear weakly singular integral models in option pricing

The integral representation of the optimal exercise boundary problem for generating the continuous-time early exercise boundary for the American put option is a well-known topic in the mathematical finance community. The main focus of this paper is to provide  an efficient asymptotically computational method to improve the accuracy of American put options and their optimal exercise boundary. Initially, we reformulate the nonlinear singular integral  model of the early exercise premium problem given in [Kim et al.,  A simple iterative method for the valuation of American options,  Quant. Finance. 13  (2013) 885--895] to an equivalent form which is more tractable from a numerical  point of view. We then obtain the existence and uniqueness results with verifiable conditions on the functions and parameters in the resulting operator  equation.  The asymptotic behavior for the early exercise boundary  is also analyzed which is mostly compatible with some realistic financial models.