An Approximate Formula for Pricing American Options

An approximate formula for pricing American options along the lines of MacMillan [1986] and Barone-Adesi and Whaley [1987] is presented. This analytical approximation is as efficient as the existing ones, but it is remarkably more accurate. In particular, it yields good results for long maturity options for which the existing analytical ones fare poorly. It is also demonstrated that this approximation is more accurate than the less efficient methods such as the four-point extrapolation schemes of Geske and Johnson [1984] and Huang, Subrahmanyam and Yu [1996]. There have been many attempts at pricing American options. Numerical methods such as the finite difference method of Brennan and Schwartz [1977] and the binomial tree model of Cox, Ross and Rubinstein [1979] are among the earliest and still widely used ones. Even though these methods are quite flexible, they are also among the most time consuming ones. A rare exception among the numerical methods is a recent paper by Figlewski and Gao [1999]. They show that efficiency and accuracy of the binomial method can be improved tremendously by fine tuning the tree in the regions where discretization induces the most serious pricing errors. The second group of methods includes approximate schemes based on exact representations of the free boundary problem of the American options or the partial differential equation satisfied by the option prices. This group includes Geske and Johnson [1984], Bunch and Johnson [1992], Huang, Subrahmanyam and Yu [1996], Carr [1998] and Ju [1998]. These methods are essentially analytic approximations and they are convergent in the sense that as more and more terms are included, they become more and more accurate. However these methods become inefficient very rapidly. Another category of methods uses regression techniques to fit an analytical approximation based on a lower bound and an upper bound of an American option. These methods include Johnson [1983], and Broadie and Detemple [1996]. These methods can be quite fast, but they all need regression coefficients which in turn require computing a large number of options accurately. Another drawback is that these methods are not convergent. A fourth category of potential methods includes analytical approximations. MacMillan [1986] and Barone-Adesi and Whaley [1987] are among these methods. A common feature of these methods is that they are many times faster than most of the aforementioned ones. A drawback is that they are not very accurate, especially for long maturity options, such as