Notations: Given a binomial tree with $N$ periods and time to maturity $T,$ let $\Delta t = T / N.$
It is well-known that CRR uses the up and down multipliers as $$u = e^{\sigma\sqrt{\Delta t}} \quad \text{and} \quad d = e^{\sigma\sqrt{\Delta t}} = \frac{1}{u}.$$
In another post, Mark Joshi suggested that one can take any real-world drift and still get the same prices in the limit so you can put $$ u = e^{\mu \Delta t +\sigma\sqrt{\Delta t}}\quad \text{ and }\quad d = e^{\mu \Delta t -\sigma\sqrt{\Delta t}} $$ for any fixed $\mu.$ $\mu =0 $ is a bad choice. Better choices are
$$ \mu = r - d - 0.5\sigma^2 $$ and $$ \mu = \frac{1}{T}(\log K - \log S_0). $$
I notice that $\mu = r - d - 0.5\sigma^2$ is derived from the discrete version of the solution of Geometric Brownian motion, that is, $$\log S_{j\Delta t} = \log S_{(j-1)\Delta t} + \left( r - d - \frac{1}{2} \sigma^2 \right)\Delta t + \sigma \sqrt{\Delta t} Z_j \quad \text{for all } j=1,2,...,N$$ where $Z_j$ is a Bernoulli random variable on $\{-1,1\}$ with $\mathbb{P}(Z_j = -1) = \mathbb{P}(Z_j = 1) = \frac{1}{2}.$
However, I do not see the motivation of $\mu = \frac{1}{T}(\log K - \log S_0).$ Can someone give a reference on where this $\mu$ is used?
Remark: I coded binomial trees using both CRR and discrete Geometric Brownian Motion multipliers. Some simulations show that they indeed converge to the same price as $N$ tends to infinity.
If you are interested, you can find the codes at my Github page.
The source codes for binomial tress can be found at the script https://github.com/hongwai1920/Implement-Option-Pricing-Model-using-Python/blob/master/scripts/Binomial_tree.py.
The simulation can be found at jupyter notebook https://nbviewer.jupyter.org/github/hongwai1920/Implement-Option-Pricing-Model-using-Python/blob/master/4.%20Recombining_Trees.ipynb (under CRR trees section)
