Actually I am not familiar with this, but I will try my best.

As we know, Normal distribution is approximated by small rectangles, that is, area under curve can be calculated with sum of those rectangles. If the number of those rectangles is infinite, then we will find that two areas are almost the same. However, if u use less rectangles, the result will have big mistakes from the real one.

Basically, the future price of the underlying follows normal distribution (just an example, I know in BS we assume it follows lognormal), so if we use BS model, the potential price range is continuous rather than discrete that is under Tree model. To this point, the gap between two methods can be regarded, in some extent, to the performance of approximating continuous distribution with discrete method.

Odd-Even effect is found in the behavior of approximating normal with rectangles. Say, the real area under curve is 1. U use 3 rectangles to approximate it, u get value of 1.1. Then, u think it is a big mistake, so u wanna increase rectangles to have a better result. When u use 4 rectangles, the result, say, 0.95. With this procedure, finally, u will find the approximating value is fluctuating around 1 and typically, one is above 1, and next is below 1. That is Odd-Even effect.

For this reason, ppl wanna find some more sophisticated way to do it, which can be applied to extrapolation.

If u wanna dig more on this thesis, u can search google, find some courseware to figure it out.

Finally, Hope this will help you.

因为美式put可以在到期前exercise, 那么派息次数就不确定。所以会摇摆。

这个是没有派息的美式期权。。。

期数是什么？别用简称啊！

According to your graph shown before, I think m u refer to is the steps used in binomial tree. Thus, ur question can be translated as why the type of convergence of binomial tree is sawtooth.

And the error is between Binomial-Method and BS-Method.

The reason behind sawtooth is "odd-even" effect, that is, the move from value of step odd to value of step even (one difference) is relatively large, therefore u get another value is on the other side of the true value.

The "odd-even" effect is about the relationship between binomial and normal. We will see that in the end nodes of binomial tree, the points represent the discrete potential prices of underlying asset. However, under BS model, it suggests the price is continuous. That is the difference.

And, u wanna learn more, search odd-even effect on Google.

Thx.

楼主有些东西不知道中文怎么翻译，所以用英文，英文书写也不是很好，见谅看看。