美式看跌期权的价值和期数关系

为什么美式看跌期权的价值和期数是呈现振动收敛的关系？如图？

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。

CateFul 发表于 2015-2-25 05:17
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。

no3621 发表于 2015-2-28 06:30
According to your graph shown before, I think m u refer to is the steps used in binomial tree. Thus, ...

maybe ur right but actually i do not know why in each sector, the graphic has a lowest point. Basically, it means the points fluctuate and takes this pattern to converge to BS model price. i never studied physics before, so could u make some explanation on the "odd-even" effect? thanks a lot:)

昨天我问了教授这个问题。

教授给我的答复是，我以前在网络上看到的是bull*****。囧，whatever。

根据教授的说法，是由于树的结构原因，当然没有给出我详细的证明。

比如二叉树，one－step是2个ending－nodes，two－step是3个ending－nodes。
如果你连接one－step，three－step，。。。odd－step，那么convergence就是光滑的逼近，even－step同理。

更进一步，比如三叉树，every step都是odd ending nodes，所以也是一条光滑逼近曲线。

虽然我说了我自己对continuous和discrete的看法，教授说，those guys are bull*****。。。。。。好吧，问不下去了，也不给我为什么。。。

no3621 发表于 2015-3-4 23:16
昨天我问了教授这个问题。

教授给我的答复是，我以前在网络上看到的是bull*****。囧，whatever。

太谢谢你了。。。我觉得这个问题确实是我太吹毛求疵了，因为本来蒙特卡洛方法模拟出来的 就是数值方法。。这个很难解释的，我之前也只在网上找到了一篇相关的资料，你可以看看 是量化金融的大神李洋老师（faruto）的一篇论文