Indexing Nodes in a Multi-Step Binomial Tree

Summary

The document addresses how to label nodes in a binomial price tree with more than one step. It compares a conventional two-step representation associated with John Hull’s presentation to an alternative notation proposed by the questioner. The response considers the alternative acceptable if the node indices clearly encode the successive moves and the values represented by the stock price and up and down factors.

The suggested indexing convention increments one index for each move through the tree, while another tracks upward moves and remains unchanged for a downward move. This is a notation guideline rather than a derivation of option values or a pricing procedure. Its main practical lesson is to define what each index counts and apply that meaning consistently, so that each node’s location and corresponding price can be recovered. The document offers no diagram or worked multi-step example, so readers may need to supply one to verify their chosen convention.

Key ideas

• A nonstandard node notation can still be valid if its indices unambiguously encode each path through the tree.
• One index can count each successive move, while another tracks upward moves.
• A downward move leaves the upward-move index unchanged under the proposed convention.
• Consistent index definitions help identify a node and the price associated with its path.

Tags

# Binomial tree notation


# Binomial tree notation












Can someone clarify for me the notation of the nodes in a binomial tree with more than 1 step? Is this notation correct?

## Answer by owner (score 2)

https://quant.stackexchange.com/a/22246

It's more likely that people are familiar with the below representation - John Hull - of a 2-step binomial tree that you described. However, by analogy, your representation (although not conventional) is fine too, as long as `S`,`u`,`d` are materialized by the index increments `[i,j]` at each node.

- `i++` for each `upward / downward trend` (1 increment)

- `j++` for each `uptrend trend` (idem), whereas `j=0` in case of `downtrend`

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.