An approximation to the American Put due to Geske and Johnson

Define to be the price of the put option with dates of exercise
left. is then the price of an european option, with the one exercise
date the expiry date. is the price of an option that can be exercised
twice, once halfway between now and expiry, the other at expiry.
Geske-Johnson shows how these options may be priced, and then develops a
sequence of approximate prices that converges to the correct price.
An approximation involving 3 option evaluations is

is the ordinary (european) Black Scholes value

where

The evaluations of and are easy, the problem is the evalutation of , since it involves the evaluation of a trivariate normal cumulative distribution. For this see the notes on the normal distribution, but it is a nontrivial problem, involving some numerical integration.