A Quasi-Closed-Form Solution for the Valuation of American Put Options

Abstract

This study develops a quasi-closed-form solution for the valuation of an American put option and the critical price of the underlying asset. This is an important area of research both because of a large number of transactions for American put options on different underlying assets (stocks, currencies, commodities, etc.) and because this type of evaluation plays a role in determining the value of other financial assets such as mortgages, convertible bonds or life insurance policies. The procedure used is commonly known as the method of lines, which is considered to be a formulation in which time is discrete rather than continuous. To improve the quality of the results obtained, the Richardson extrapolation is applied, which allows the convergence of the outputs to be accelerated to values close to reality. The model developed in this paper derives an explicit formula of the finite-maturity American put option. The results obtained, besides allowing us to quickly determine the option value and the critical price, enable the graphical representation—in two and three dimensions—of the option value as a function of the other components of the model.