# Derivation of $u=e^{\sigma\sqrt{dt}}$ and $d=e^{-\sigma\sqrt{dt}}$

Anyone could provide me a proof of how, starting from $$\frac{dS_T}{S_t}\sim \operatorname{N}(\mu dt,\sigma^2 dt)$$ with $$p:=\frac{e^{rdt}-d}{u-d}$$, we can obtain the parameters $$u$$ and $$d$$ as from title? Proof on Hull-White is unclear.

Thanks for any help or links.

• You have to choose the three numbers, $p,u$ and $d$ such that the average log return has mean $\mu dt$ and variance $\sigma^2 dt$. You can verify that this particular way of choosing them has this mean and this variance (simply compute the mean and variance and check), but is is not the only solution there are other choices, after all you have 3 numbers to choose and only two conditions to satisfy, so the problem is under-determined. Oct 21 '20 at 17:07
• The mean log return is $p \ln u + (1-p) \ln d$. Now substitute the above choices for $p,u,d$ and see if you get $\mu dt$ or not. Similarly for the variance. Oct 21 '20 at 17:14
• @noob2 Thanks for your answers! Have you any links that you can attach about the complete proof? It's for a thesis. Oct 21 '20 at 19:31