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Why Binomial Trees Use Reciprocal Up and Down Factors

Article Quant Q&A · Author: Morgan Lu
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Summary

The document explains why binomial option pricing models often set the up and down factors as reciprocals. That condition makes the price tree recombine: an up move followed by a down move reaches the same node as a down move followed by an up move, keeping the number of distinct nodes manageable.

The key clarification is that reciprocal factors are a convenience, not a requirement for pricing. A model can use other up and down factors if the option is priced from the underlying principles, though the resulting tree may not recombine and calculations can be more involved. The note gives a conceptual explanation rather than a worked valuation or comparison of numerical accuracy, and it points to an example outside the document without describing its details.

Key ideas

  • Reciprocal up and down factors make a binomial price tree recombine.
  • Recombination means that different move sequences can lead to the same asset price.
  • Reciprocal factors are a computational convenience, not a universal pricing requirement.
  • Non-recombining trees can still be priced from fundamental principles, with greater calculation complexity.

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# Binomial Pricing Model d and u


# Binomial Pricing Model d and u












In the binomial pricing model, why do the magnitude of the up factor $(u)$ and down factor $(d)$ have to be multiplicative inverses? I have read from multiple sources that the reason for this is that an up move followed by a down move $(ud)$ will have the same effect as a down move followed by an up move $(du)$, which simplifies calculations. However, doesn't this property still hold when $u$ and $d$ are not inverses? For example, $1.1*.8 = .8*1.1$

## Answer by Anirban Saha (score 1)

https://quant.stackexchange.com/a/60467

It does not need to be so always. You can always relax that assumption and come with the pricing by using the fundamental principles. As @Kermittfrog and @Dimitri Vulis commented it is just a matter of convenience for calculations and is called the recombining property. You can find an example in this link here which does not use this assumption to price the options.

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.