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

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.