CRR Binomial Trees and the No-Arbitrage Step-Size Condition

Summary

The document explains a no-arbitrage condition for a binomial asset-price model: the risk-free growth factor for a time step must lie between the down and up growth factors. Under the Cox-Ross-Rubinstein (CRR) choice, the up factor is set by volatility and the square root of the step length, while the down factor is its reciprocal. If the step is too long relative to the rate and volatility, those assumptions can violate the condition.

The accepted response gives a practical remedy: shorten the time step by adding more steps, so the CRR factors can satisfy the no-arbitrage bound. It also comments that CRR trees are outdated, but does not explain or substantiate that assessment. The exchange provides no worked example, alternative tree construction, or detailed criteria for choosing a step size. Its answer is therefore a concise modeling pointer rather than a comprehensive comparison of binomial pricing methods.

Key ideas

• A binomial model requires the risk-free growth factor to fall between the down and up asset growth factors.
• The CRR choice links the up factor to volatility and the square root of the time step.
• A sufficiently long step can make the CRR assumption inconsistent with the no-arbitrage condition.
• The response recommends using more, shorter time steps to avoid that violation.

Tags

# Binomial pricing model: When the Cox-Ross-Rubinstein assumption is not arbitrage-free


# Binomial pricing model: When the Cox-Ross-Rubinstein assumption is not arbitrage-free












I understand that in an arbitrage-free Binomial model, we assume that $S_{t+1} = S_t \cdot u$ in the event of an up-jump and $S_{t+1} = S_t \cdot d$ in the event of a down-jump. We call $u$ and $d$ the growth factors. A neccessary condition for no-arbitrage is that $d < e^{r \delta t} < u$, where $r$ is the risk-free lending rate and $\delta t$ is the time elapsed during each step. A common assumption, due to Cox, Ross and Rubinstein, is to let $u = 1/d = e^{\sigma \sqrt{\delta t}}$.

It seems that this assumption is inconsistent with no-arbitrage in the case where $e^{r \delta t} \geq e^{\sigma \sqrt{\delta t}}$ or, equivalently, when $r \geq \sigma / \sqrt{\delta t}$. What is the typical recourse in this situation?

## Answer by Mark Joshi (score 3, accepted)

https://quant.stackexchange.com/a/18107

if you let $\delta t$ be small enough, this won't happen. So the solution is to take more steps.

The CRR tree is very out dated in any case.

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