 Cox-Ross-Rubinstein diffusion approximation

For a time step $h>0$, choose the undiscounted stock multipliers and money-market multiplier asFor 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 ${\sigma }^{2}h+O(h^2)$. The Donsker invariance principle, in its triangular-array form, gives the risk-neutral log-price limitThus prices converge to the geometric Brownian motion used in the Black-Scholes model.^[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 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 payoffs are unbounded. A process limit alone does not justify taking expectations of every payoff.