Expanding the JKY ABMC Binomial Up and Down Factors

Summary

The document asks how the Jabbour-Kramin-Young ABMC parameterization of a risk-neutral binomial model yields its small-time approximations. It gives the up and down factors in terms of the risk-free rate, volatility, and time step, then cites expansions with leading movements of plus or minus volatility times the square root of the time step and a common drift term proportional to the time step.

The specific difficulty is understanding how the square-root contribution arises from the exponential volatility expression and why the remaining error is of order time step to the three-halves power. The author identifies the ordinary Taylor expansion of the rate exponential but does not provide a derivation, numerical example, or discussion of conditions on the time step. The material is a focused question about asymptotic expansion in a binomial pricing model; readers should treat the stated approximation as the claim under examination rather than as a demonstrated result.

Key ideas

• The JKY ABMC formulation specifies risk-neutral up and down factors using rate, volatility, and time step.
• The cited approximation has opposite square-root volatility terms and a shared first-order drift term.
• The document asks how expanding the volatility expression produces the square-root terms.
• It also asks why the residual terms are bounded at order time step to the three-halves power.

Tags

# Jabbour-Kramin-Young ABMC Binomial Parameterization


# 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

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