Weak Convergence of Tree Models for Option Pricing
Summary
The document introduces a question about when discrete trees approximate an underlying stochastic process closely enough to price options. For terminal-value payoffs under a lognormal model, it describes matching the mean and variance of each tree step and invoking the central limit theorem to motivate convergence. For path-dependent payoffs, it points to functional weak convergence, such as the type established by Donsker’s theorem, as the relevant idea because the payoff depends on the trajectory rather than only the endpoint.
The text asks how broadly stepwise moment matching can approximate processes, whether it is necessary as well as sufficient, and where to learn more. It does not provide answers, proof, or empirical pricing results. Its claims are explicitly tentative and assume sufficiently well-behaved payoffs; the suggested convergence intuition should not be treated as a general result for arbitrary processes or payoffs.
Key ideas
- Terminal-value payoffs can be studied through convergence of the tree’s terminal distribution.
- Matching stepwise means and variances is presented as an intuition for lognormal tree convergence.
- Path-dependent payoffs require convergence of the process paths, not just the terminal value.
- The document leaves the generality and necessity of moment matching unresolved.
Tags
Full text
# Theory of the convergence of option prices using trees
# Theory of the convergence of option prices using trees
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.
- 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.
- 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):
- What is the most general class of processes we can approximate weakly using this "mean/variance matching of each step of the tree" construction?
- 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?
- Can you suggest some material regarding this part of the theory?
Any help is appreciated! Thank you.Shown in full with attribution under the source's licence. Licence: CC BY-SA 4.0 (Stack Exchange)
This summary was written by Stratmill's research agent from the original; it is not a copy of the source.