# Jabbour-Kramin-Young ABMC Binomial Parameterization

The JKY ABMC Model (taken from Jabbour, et al. 2001) parameterizes the binomial model (in a risk-neutral world) such that,

$u = e^{r\Delta t} + e^{r\Delta t}\sqrt{e^{\sigma^2\Delta t} - 1}$

$d = e^{r\Delta t} - e^{r\Delta t}\sqrt{e^{\sigma^2\Delta t} - 1}$

JKY continue and say that this is equivalent to,

$u = 1 + \sigma\sqrt{\Delta t} + R\Delta t + \mathcal O(\Delta t^\frac{3}{2})$

$d = 1 - \sigma\sqrt{\Delta t} + R\Delta t + \mathcal O(\Delta t^\frac{3}{2})$

I'm having trouble seeing this rigourously. Specifically, I can find that the first term $e^{r\Delta t} = 1 + R\Delta t + \mathcal O(\Delta t^2)$ from the Taylor expansion of $e^x$, but I'm having troubling seeing how the second term contributes to the $\pm\sigma\sqrt{\Delta t}$ and how it leads to the restriction of the error to $\mathcal O(\Delta t^\frac{3}{2})$

Thank you

• I've got a detailed comparison of nearly all binomial trees and their features in "more mathematical finance" – Mark Joshi Sep 24 '15 at 21:23