How to Determine Parameters in a Non-recombining Binomial Tree for Option Pricing

External question — Quantitative Finance Stack Exchange Author: Gull23 Original post: https://quant.stackexchange.com/questions/76625 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. For a CRR recombining Binomial Tree, let the underlying stock price be $S_0$ at $t=0$ and the time interval be $\Delta t$ . The nodes at $t=\Delta t$ and probabilities reaching them can be written as: $ \left\{ \begin{array}{**lr**} S_u = S_0e^{\sigma \Delta T},\ p_u=\frac{e^{r \Delta t}-d}{u-d}\\ S_d = S_0e^{-\sigma \Delta T}, \ p_d=1-p_u \end{array} \right. $ . And we will have $S_{ud}=S_{du}$ at $t=2\Delta t$ because $ud=1$ . Now, if I want to construct a $N$ step non-recombining Binomial Tree which is only limited to $d<e^{r \Delta t}<u$ . How should I derive $u$ , $d$ and $p$ under risk-neutral condition, except using real option prices to calibrate? I've been going through literature reviews about numerous Binomial Trees proposed until now but failed to find a general method. Any textbook or paper link is welcomed! Thanks!

External answer — Quantitative Finance Stack Exchange Author: Hritabrata Das Original post: https://quant.stackexchange.com/a/85752 License: CC BY-SA 4.0 — https://creativecommons.org/licenses/by-sa/4.0/ Adaptation: HTML converted to plain text; contact email addresses removed. A non-recombining binomial tree basically drops the assumption of up-down movement equals down-up movement. We would need to specify two path dependent volatilities so that at step 2 your stock prices are different and they do not convergence. Once done , use standard risk neutral formula and your p's should be reflected.