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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?

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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.

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  • $\begingroup$ 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. $\endgroup$
    – mark leeds
    Dec 4, 2023 at 4:42
  • $\begingroup$ Thank you! It really helps! $\endgroup$
    – Clay ZHAI
    Dec 4, 2023 at 7:14
  • $\begingroup$ 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). $\endgroup$
    – T123
    Dec 4, 2023 at 12:12
  • $\begingroup$ See also here: quant.stackexchange.com/questions/25390/… $\endgroup$
    – T123
    Dec 4, 2023 at 12:13

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