MC 与二叉树

经验法则：

• 树/格点：低维、需要提前行权（美式）或障碍边界清晰时更合适。
• Monte Carlo：高维、多因子、强路径依赖（亚式、篮子、路径事件）更合适。

缺点互补：MC 处理美式较难（需 LSM 等）；树在高维会遭遇维度灾难。

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英文解析

The great virtue of a binomial tree is that it is easy to do early exercise. Its rate of convergence for continuous payoffs including American options is $1 / n$ ,

where $n$ is the number of steps. Since the number of computations increases as $n^2$ , this means that the rate of convergence is $t^{- 1 / 2}$ where $t$ is computational time. (Your interviewer may not know this...) For higher dimensional trees, convergence will be slower. Path- dependence is not natural in trees but can be dealt with by using an auxiliary variable which increases the dimension by 1. The great virtues of Monte Carlo are:

The great virtues of Monte Carlo are:

Convergence is order $t^{- 1 / 2}$ in all dimensions. Path- dependence is easy.

The great downside is that early exercise is hard (see Question 2.57). The other downside is slowness of convergence in low dimensions, although it is no slower than a binomial tree, there are other faster lattice methods in low dimensions.

The convergence speed and early exercise can be coped with, but it requires much work. As a rule of thumb, use binomial trees for low- dimensional problems involving early exercise, and Monte Carlo for high- dimensional problems involving path- dependence.