三叉树不唯一

用于数值计算时，树是对给定风险中性扩散（如 GBM）的离散近似，而不是“树本身必须无套利地唯一决定价格”。

三叉树常见优势：

• 收敛更平滑、振荡比二叉树小
• 节点/概率自由度更高，便于把节点放在障碍/关键执行价上
• 对某些 PDE/格点实现更稳定

因此银行用三叉树作为数值近似器来逼近目标风险中性测度下的价格。

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英文解析

The crucial point here is that when using a model as a computational device, we are not doing no arbitrage arguments. Instead, we are approximating a given risk- neutral measure. In this case, the risk- neutral measure we are approximating is geometric Brownian motion with a growth rate of $r$ .

We can think of four operations:

• Passing to the risk-neutral measure.- Discretization with a trinomial tree.- Pricing.- Passing to the limit.

The crucial point here is that the first two of these do not commute.

One reason why a trinomial tree is a useful computational device is that it does not naturally oscillate in the same way as a binomial tree. That is if the number of time steps in the tree changes by one, the price will not change as much as in a binomial tree. Trinomial trees also have much flexibility in terms of node placement since one can play with 3 node locations and two probabilities, whereas with a binomial tree there are two node locations and one probability. We have to match two quantities (at least asymptotically): mean and variance. This means that one can adapt node placement to the product, so, for example, nodes lie on barriers.