Interpreting Cumulative Returns in an Implied Binomial Tree
Summary
The document clarifies the meaning of the cumulative return variable used in Rubinstein’s implied binomial tree method. The questioner has derived terminal path probabilities and recursively aggregated them to earlier nodes, then asks how to obtain the up and down values in the recursion for cumulative returns. They wonder whether those values should come from the original Cox-Ross-Rubinstein tree.
The response says that each node’s R represents implied risk-neutral cumulative growth along the tree, expressed as one plus the cumulative risk-neutral return to that node. It illustrates the interpretation with an underlying that has grown by 20 percent, corresponding to an R value of 1.20. The answer distinguishes these values conceptually from merely reusing CRR prices, but offers no worked numerical tree, implementation procedure, or discussion of assumptions and calibration accuracy. The source question also notes that its spreadsheet example does not implement this step.
Key ideas
- In the implied binomial tree, R represents cumulative risk-neutral growth to a node.
- The value is expressed as one plus the cumulative risk-neutral return.
- The up and down R values refer to growth along the implied tree rather than automatically to CRR prices.
- The explanation is conceptual and does not provide a full worked calculation.
Tags
Full text
# Rubinsteins Implied Binomial Tree - how to calculate the cumulative returns # Rubinsteins Implied Binomial Tree - how to calculate the cumulative returns I am working on Rubinsteins IBT and use the following paper to implement this into excel: http://papers.ssrn.com/sol3/papers.cfm?abstract_id=541744 the original paper can be found here: http://www.haas.berkeley.edu/groups/finance/WP/rpf232.pdf I am stuck in the last step: I calculated the path probabilities, denoted by "Q". In the paper "Implied Binomial Trees in Excel without VBA" page 7 and 17. So I have now the results of Panel B on page 17. These are the Q at the last node, now I calculate the Q at the nodes before with $Q=Q^+ + Q^-$. So I have the resulting path probabilities Q shown in the uploaded and attached picture (the numbers are a bit different from the paper, because my excel solver was not that accurate, but the numbers should be the same). So now I want to calculate the "R". These are the cumulative returns. In the paper it says on page 7: $R=(qR^+ + (1-q)R^-)/r$. I know that the small qs are the up probabilities, calculated by $q=Q^+/Q$. Ok, the small r is a discounting factor, ok. But what are the $R^+$ and $R^-$. Where do I get them? Are these just the original prices form the CRR binomial tree? Thanks a lot! (in the excel file which can be downloaded, this step is not implemented) edit: > It should be in the paper, though, for example "a 20% growth in the underlying gives R = 1.20" My underlying, values u and d calculated by CRR? So the values of the underlying are: @ Freddy Could you please use my example and do an example calculation? The comment is to general and not specific enough.... ## Answer by Matt Wolf (score 1) https://quant.stackexchange.com/a/4921 R denotes the implied risk-neutral cumulative growth which is different for each note. You get them through: 1 + cumulative risk-neutral return through the tree to a particular node. It should be in the paper, though, for example "a 20% growth in the underlying gives R = 1.20"
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.