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# Pricing American Options with the Two- and Three-Point Maximum Methods

This Demonstration shows the application of the "two-point maximum" and "three-point maximum" methods [2] in order to approximate the value of an American put. The Demonstration uses the trinomial method [3] and the fact that Bermudan options approximate American options to locate the optimal early exercise temporal points and estimate the values and . Use the controllers to set the time discretization for the trinomial tree and the American option parameters.
• The table shows the maximized and values of the Bermudan put options and their optimal early exercise temporal points, respectively. The table also displays the American put approximations after the application of Richardson extrapolation once and twice, respectively.
• The upper graph shows the value of a Bermudan option , depending on the choice of its early exercise temporal point . The red dot represents the optimal early exercise temporal point, which maximizes the value of the Bermudan option .
• The lower 3D graph shows the value of a Bermudan option , depending on the choice of its early exercise temporal points and . The red dot represents the optimal early exercise temporal points, which maximize the value of the Bermudan option .

### DETAILS

The "two-point maximum" and "three-point maximum" methods [2] derive from the analytical method [1], which uses the values of
• a European put option , which can only be exercised at its maturity ,
• a Bermudan put option , which can be exercised at or , and
• a Bermudan put option , which can be exercised at , , or.
Then, according to the method [1], Richardson extrapolation is applied twice to approximate the American put value as , with an error term of the form: .
Instead, the modified "three-point maximum" method [2] uses optimally time-discretized Bermudan options with maximized values and , respectively. When Richardson extrapolation is applied twice, the American put approximation is . The error term is again of the form , but in most cases its absolute value is smaller compared to the method [1]. The disadvantage of the modified method [2] compared to the method [1] is that it requires more computational time and effort. The "two-point maximum" method refers to the application of Richardson extrapolation once; the American put approximation is then , with an error term of the form .
References
[1] R. Geske and H. Johnson, "The American Put Option Valued Analytically," The Journal of Finance, 39(5), 1984 pp. 1511–1524.
[2] D. Bunch and H. Johnson, "A Simple Numerically Efficient Valuation Method for American Puts Using a Modified Geske–Johnson Approach," The Journal of Finance, 47(2), 1992 pp. 809–816.
[3] P. Boyle, "Option Valuation Using a Three-Jump Process," International Options Journal, 3, 1986 pp. 7–12.

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