American Closed Form Solution

American Option Pricing

American Options has been object of investigation for at least 50 years. Research in this field continues today. New insights emerge all the time and novel applications of American Option theory reverberate all the time with new manifestations the area of investment decisions, portfolio management and pollution rights. Yet valuation can be tortuous and each proposed technique seems to present trade-offs between speed of computation and accuracy. According to Barone-Adesi (2005) "On some level attempts to model the American put resemble the search for elementary particles in physics, which is continuing long after the original questions about the main constituents of the atom have been answered. The beauty of our search stems from its purely intellectual formulation, abstracting from the anomalous empirical evidence that often fuels research in the physical sciences."

Early research dates back to Samuelson (1967) and McKean (1967) and Van Moerbeke (1974). The binomial method of Cox, Ross and Rubinstein (1979) became rapidly the favored method in the classroom. While it appealed at an intuitive level however static implementations, devoid of optimization required substantial computer time to yield accurate results. Barone-Adesi and Whaley (1987), Ju and Zhong (1999) and Bjerksund and Stensland (1993) (2002) developed ever more accurate analytical approximations. Below, I post three snippets of C++ code originally made available by Fabrice Rouah to estimate the value of American Options using these 4 well known techniques. Also, please check out J.R.M Roman VBA code.