My current understanding of the theory behind the convergence of options prices using trees is the following:

Suppose $S = (S_{t})_{0\leq t\leq T}$ is the underlying process and $g(S_{t}:0\leq t\leq T)$ is the option we want to evaluate. Then we want to construct trees $S^{N}$such that $S^{N} \Rightarrow S$ (Here $\Rightarrow$ denote weak convergence). Assume $g$ behaves well enough for simplicity.

  1. If $g$ depends only on the terminal value $S_{T}$ and $S$ is lognormally distributed, then as in the CRR paper, we can construct $S^{N}$ by matching the mean and variance of each tree step and conclude $S^{N}_{T} \Rightarrow S_{T}$ by the classic central limit theorem.

  2. If $g$ is path dependent, i.e: $g$ depends on the whole process $(S_{t})$, and $S$ is still lognormally distributed, then we might need to use some sort of "functional central limit", for example, Donsker’s Theorem, to conclude weak convergence of $(S_{t}^{N})$ to $(S_{t})$. I think this can be done by applying the same mean/variance matching for each step of the tree.

My question is (The trees can be non-recombining):

  1. What is the most general class of processes we can approximate weakly using this "mean/variance matching of each step of the tree" construction?
  2. For an arbitrary stochastic process, I think "mean/variance matching of each step of the tree" might not be sufficient for the tree to converge weakly to it, but is it a necessary condition?
  3. Can you suggest some material regarding this part of the theory?

Any help is appreciated! Thank you.



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