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

.

[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.