= Cox-Ross-Rubinstein diffusion approximation
{c}
{title2=$U_h=e^{\sigma\sqrt h},\quad D_h=e^{-\sigma\sqrt h},\quad R_h=e^{\rho h}$}
= Binomial diffusion approximation
{synonym}
For a time step $h>0$, choose the undiscounted stock multipliers and money-market multiplier as
$$
U_h=e^{\sigma\sqrt h},\qquad D_h=e^{-\sigma\sqrt h},\qquad R_h=e^{\rho h},\qquad \sigma>0.
$$
For sufficiently small $h$, $D_h<R_h<U_h$ and the risk-neutral up <probability> is $q_h=(R_h-D_h)/(U_h-D_h)$. A one-step log-stock increment takes values $\pm\sigma\sqrt h$; expanding $q_h$ gives <mean> $(\rho-\sigma^2/2)h+O(h^2)$ and <variance> $\sigma^2h+O(h^2)$. The <Donsker invariance principle>, in its triangular-array form, gives the risk-neutral log-price limit
$$
\log S_t=\log S_0+(\rho-\sigma^2/2)t+\sigma W_t.
$$
Thus prices converge to the <geometric Brownian motion> used in the <Black-Scholes model>.<Option Pricing: A Simplified Approach by John C. Cox and others>[Section5, manuscript pp.17 onward, small-step limits]{ref}
The parent model's discounted multipliers are instead $u_h=U_h/R_h$ and $d_h=D_h/R_h$. Their log-price <drift coefficient>[drift] loses the $\rho$ term, as required for the discounted price to be a <martingale>. Convergence of claim prices also needs the <payoff> functional to be continuous at almost every limiting path, together with <Uniform integrability> when <payoff>[payoffs] are unbounded. A process limit alone does not justify taking <expected values>[expectations] of every <payoff>.
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