# Stock price modelling under binomial tree model?

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?

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.

• Note that S is assumed to be lognormal ( log S is assumed normal ) which is how the exponential creeps into the result. Obvious to many but I figured it was worth throwing that info in. Commented Dec 4, 2023 at 4:42
• Thank you! It really helps! Commented Dec 4, 2023 at 7:14
• Just to add a remark: the answer has no drift in it (which you need to add). The derivation of d and u can be found in a footnote in Hull (2021).
– T123
Commented Dec 4, 2023 at 12:12