Details

Use the controls to select the Bermudan put parameters, and then monitor how the optimal temporal points are reached step by step, with Mathematica's built-in function FindMaximum (upper graph); you can use the last control to reset the starting time points. The lower graph shows the Bermudan optimal early-exercise temporal points and their corresponding critical prices (red dots), compared to the early-exercise boundary of the American put (blue line), according to Mathematica's built-in function FinancialDerivative.

The Geske–Johnson analytical formula [1] is applied to estimate the payoff function of the Bermudan put. The goal of the optimization process is to find the two early-exercise time points at which the payoff function of the option is maximized; these times are related to corresponding asset critical prices, as in the case of American options.

Geske and Johnson, in the "G–J" method [1], apply Richardson extrapolation on uniformly time-discretized Bermudan options to approximate the value of an American option. Bunch and Johnson, in the "two-point maximum" and "three-point maximum" approximation methods [2], suggest using optimally time-discretized Bermudan options instead, in order to get a better approximation for the American option.

References

[1] R. Geske and H. Johnson, "The American Put Option Valued Analytically," The Journal of Finance, 39(5), 1984 pp. 1511–1524. doi:10.1111/j.1540-6261.1984.tb04921.x.

[2] D. Bunch and H. Johnson, "A Simple and Numerically Efficient Valuation Method for American Puts Using a Modified Geske–Johnson Approach," The Journal of Finance, 47(2), 1992 pp. 809–816. doi:10.1111/j.1540-6261.1992.tb04412.x.