In this paper, novel singular perturbation techniques are applied to price European, American, and barrier options. Employment of these methods leads to a significant simplification of the problem in all cases, by reducing the number of parameters. For American options, the valuation problem is reduced to a procedure that may be performed on a rudimentary handheld calculator. The method also sheds light on the evolution of option prices for all of the cases considered, the results being particularly illuminating for American and barrier options.
Widdicks, M., Duck, P. W., Andricopoulos, A. D., & Newton, D. (2005). The Black-Scholes Equation Revisited: Asymptotic Expressions and Singular Perturbations. Mathematical Finance, 15(2), 373-391. https://doi.org/10.1111/j.0960-1627.2005.00224.x