Parameter Choices for Non-Recombining Binomial Option Trees

Summary

The document asks how to choose up and down moves and risk-neutral probabilities for a non-recombining binomial tree without calibrating to observed option prices. It contrasts this setup with the Cox–Ross–Rubinstein tree, where reciprocal up and down factors make paths recombine. In a non-recombining tree, the order of moves matters, so the price after an up-then-down path can differ from the price after a down-then-up path.

The response says that path dependence requires specifying distinct volatilities for the two paths and then applying the usual risk-neutral probability condition. This is a brief conceptual suggestion, not a general derivation: it does not specify how to select those path-dependent volatilities or show that the resulting parameters are unique. Thus the document highlights the extra modeling choices introduced by abandoning recombination but leaves the calibration or estimation problem unresolved.

Key ideas

• A recombining binomial tree assumes the up and down factors make opposite-order paths meet at the same node.
• A non-recombining tree allows the underlying price to depend on the order of moves.
• Path-dependent volatility assumptions are needed to describe distinct outcomes after different move sequences.
• Risk-neutral probabilities can be imposed after the tree’s move sizes are specified.
• The response does not provide a general rule for choosing path-dependent volatilities or uniquely determining the tree parameters.

Tags

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


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












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!

## Answer by Hritabrata Das (score 0)

https://quant.stackexchange.com/a/85752

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.

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This summary was written by Stratmill's research agent from the original; it is not a copy of the source.