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)

  • $\begingroup$ @noob2 I think Joshi means that for any $\mu,$ if $N \to\infty,$ then the binomial tree will give the same price as Black-Scholes analytical pricing. $\endgroup$ – Idonknow Jun 6 at 6:41
  • $\begingroup$ Something I don't understand: If $K$ stays the same and we increase $\mu$, the number of cases where the Call has a positive payoff increases. Doesn't that affect the discounted PV of the option? $\endgroup$ – noob2 Jun 6 at 7:15
  • $\begingroup$ @noob2 your arguments seem to make sense. However, I am not sure about the answer. $\endgroup$ – Idonknow Jun 6 at 7:18
  • $\begingroup$ dm63 is right of course, the probabilities are adjusted accordingly. You may be interested in this post on "CRR with drift" that I found. goddardconsulting.ca/matlab-binomial-crrdrift.html $\endgroup$ – noob2 Jun 6 at 17:26
  • $\begingroup$ Nice! Thanks for the link. $\endgroup$ – Idonknow Jun 7 at 0:48

It appears that the motivation for $\mu = (\log K - \log S_0)/T$ may be that K is in the middle of the tree at $T$. I could see how this may improve accuracy since K is where the ‘action’ is.

@noob2 I think that in the case of various choices of $\mu$, the up/down probabilities in the tree may be adjusted to give the correct risk neutral expectation for the stock.

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