Difference in formulas for u & d in Binomial trees

Question

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

$$u=\exp((r−\delta)h+\sigma\sqrt{h})$$

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?

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.