On the analytical-numerical valuation of the Bermudan and American options

The paper further develops, both from the theoretical and numerical points of view the analytical valuation of the American options, initiated by Geske and Johnson (1984) for the American put with no dividend. We present and prove closed form formulas for the value of the Bermudan put and call, with dividend, paid continuously at a constant rate, where a general number and not necessarily equal length intervals subdivide the time. Based on the obtained formulas and recent, efficient numerical integration techniques, to obtain values of the multivariate normal c.d.f., the Bermudan put and call option values are calculated for up to twenty subdividing intervals. The sequences of option values are smoothed by sums of exponential functions and the latters are used to predict the values of the American options. Numerical results are presented and compared with those, published in the literature. It is shown that the binomial method systematically overestimates the option price, and, according to our numerical results, so do many other methods. Some properties of Richardson extrapolation are explored.