The simplest American and Real Option approximations: Geske–Johnson interpolation in maturity and yield

Abstract

The American early exercise feature of the Real Option to invest in a new project is important in capital budgeting and project valuation. Closed form solutions for American, and therefore Real, Options are known for two special cases; an infinite horizon generates the Merton (Bell Journal of Economics, 4, 141–83, 1973) solution while a zero dividend yield on the project generates Black-Scholes (Journal of Political Economy, 81, 637–59, 1973) prices since early exercise is never optimal. Geske–Johnson (Journal of Finance, 39, 1511–24, 1984) approximation is extended to a bivariate case by assuming various forms of separability for option prices as a function of time to maturity and yield to produce fully explicit and asymptotically correct approximations. These methods are compared with another simple approximation method due to Barone-Adesi and Whaley (Journal of Finance, 42, 301–20, 1987) and MacMillan (Advances in Futures Options and Research, 2, 117–42, 1987) and the estimated error these expressions contain compared to an accurate numerical benchmark technique.