Stock price modelling under binomial tree model?

Question

In binomial tree model, the stock price is modelled in the form of $S_{k\delta}=S_{(k-1)\delta}\exp(\mu\delta+\sigma\sqrt\delta Z_k)$, where $\delta$ is time invertal between two observations $S_{k\delta},S_{(k-1)\delta}$, $Z_k=1,-1$ for upward and downward scenarios of the stock price change.

I noted some illustrations of variance and mean to explain why the model is set in the form, but I cannot find more explicit explanation. Could someone help?

Answer 1

Typically the binomial tree is written as:

$$S_{t+\Delta t} = uS_t$$ $$S_{t+\Delta t} = dS_t$$

where $u=e^{\sigma\sqrt{\Delta t}}$ and $d=e^{-\sigma\sqrt{\Delta t}}$.

There is $Z$ instead to eloquently write the $u$ and $d$ in the same line. Then you now have to include discounting, $$e^{-\mu\Delta t}S_{t+\Delta t} = uS_t$$ $$e^{-\mu\Delta t}S_{t+\Delta t} = dS_t$$ which is the same as what you have - there is no difference.