Robust Pricing of the American Put Option: A Note on Richardson Extrapolation and the Early Exercise Premium

Abstract

Aunque existe una formula que revolucionó la valoración de derivados para opciones Europeas, la famosa fórmula de Black-Scholes-Merton, no existe tal formula cerrada para opciones Americanas. Estas últimas requiren métodos numéricos. En este paper muestro como la descomposición de la opción Americana en el early-exercise-premium más la contraparte equivalente Europea puede ser utilizada para valorar numéricamente la opción americana. El método es muy eficiente y exacto, dominando de hecho a la mayoría de los métodos en la literatura. El método utiliza también la extrapolación de Richardson para mejorar la eficiencia.

This paper presents a detailed analysis of the numerical implementation of the American put option decomposition into an equivalent European option plus an early exercise premium (Kim 1990, Jacka 1991, Carr et al. 1992). It subsequently introduces a new algorithm based upon this decomposition and Richardson extrapolation. This new algorithm is based upon (a) the derivation of the correct order for the error term when applying Richardson extrapolation, which is used to control the error of the extrapolated prices, (b) an innovative adjustment of Kim's (1990) discrete-time early exercise premium, so that these premiums monotonically converge and, therefore, it is appropriate to use them in extrapolation, and (c) the optimal exercise frontier can be quickly computed through Newton's method, permitting the efficient implementation of the decomposition formula in practice. Numerical experiments show that this new algorithm is accurate, efficient, easy to implement, and competitive in comparison with other methods. Finally, it can also be applied to other American exotic securities.