美式期权可能行权的概率如何计算？？？

在经典BS模型的框架下，美式看涨（跌）期权可能行权的概率如何计算？最好提供解析形式的解，尽量避免依赖MC和二叉树

This is a classic problem in American option. Some discussion can be found in Shreve's book Ch 8 (8.4.1)

The problem can be summarized as find an exercise boundary, i.e. For example for a put option, we need to find a exercise boundary L(T-t), whenever the stock price S(t) falls bellow L(T-t), we should exercise. For your problem, it is the probability that lognormal distribution goes below L(T-t).

This problem is a little hard because it needs simultaneously solve PDE with several boundary conditions. So the common practice is to use finite difference (I know you don't like :-( ) to numerically find the boundary. Than you can calculate the probability.

More details, see Shreve's book. I think this is a popular issue. There should be many research papers talking about how to get the exercise boundary more efficiently (perhaps some analytical solutions :-) ).

best,

看看

问题不太专业，影响因素太多了，比如假设的标的资产的变化形式，是否是完全市场等等，你得具体问题具体分析

菜园老头 发表于 2013-12-18 14:33
问题不太专业，影响因素太多了，比如假设的标的资产的变化形式，是否是完全市场等等，你得具体问题具体分析

已经改正了，请不吝赐教

Chemist_MZ 发表于 2013-12-18 23:35
This is a classic problem in American option. Some discussion can be found in Shreve's book Ch 8 (8. ...

在二叉树的框架下可以得到近似的最优执行边界，并且可以知道那些路径穿过了边界，执行概率也就算出来了。不过这种方法不够“高大上”。
在连续情境下，执行概率问题可以抽象为一个“首达时”问题，不过时变的边界条件比教科书上的反射原理复杂多了。可以推荐一些关于执行概率的论文吗？
执行概率对定价没太大帮助，但是对交易来说很有参考价值，特别是那些期望卖出裸期权的投机者。
把执行概率和期望收益结合起来编制一个效用函数

xuruilong100 发表于 2013-12-19 17:50
在二叉树的框架下可以得到近似的最优执行边界，并且可以知道那些路径穿过了边界，执行概率也就算出来了。 ...

OK, I know you don't like binomial. But actually it is the most efficient one. And actuate enough.

PS: actually binomial modal catches the character of the the path dependent derivatives. I highly recommend it.

Stopping time can solve infinite maturity (perpetual) put. But not finite maturity (bad)

If you are just consider the trading issue, then numerical and approximation is ok (Including binomial and all kinds of ways, just google it).

This is a paper talking about something you may interested in (very attractive abstract :-)  ). You can take a look if you can drag it down.

http://www.tandfonline.com/doi/a ... 80600699811#preview

best,