For a binomial tree, everywhere in Hull and other literature, we have found the formulas for

$$u = \exp(\sigma \sqrt{h})$$

but for binomial trees based on forward prices, we get a different formula


Could anyone please provide an explanation of why there is this extra term of $\exp(r-\delta)$ multiplied here?

I understand that $\delta$ is for the constant dividend yield but why is there a difference in formulas for $u$ when binomial tress are constructed using forward prices?


1 Answer 1


there are many different trees. The first one, the CRR tree, used $$ u = e^{\sigma\sqrt{h}} $$ and $d = 1/u.$ However, you can take any real-world drift and still get the same prices in the limit so you can put $$ u = e^{\mu h +\sigma\sqrt{h}}, \text{ and } d = e^{\mu h -\sigma\sqrt{h}} $$ for any fixed $\mu.$

$\mu = 0$ is a poor choice for convergence. Better choices are $$ \mu = r - d - 0.5\sigma^2 $$ and $$ \mu = \frac{1}{T}(\log K - \log S_0). $$

There has been a huge amount of work on binomial trees in the last 40 years and there is now over 30 of them. More sophisticated trees achieve higher order convergence for European options.

I give a comprehensive survey in my book, More Mathematical Finance.

  • $\begingroup$ If it is useful to you, please upvote and accept it $\endgroup$
    – Gordon
    Commented Feb 19, 2017 at 14:03

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