 Cox-Ross-Rubinstein diffusion approximation

For mesh $h$, the risk-neutral up probability is $(e^{\rho h}-d)/(u-d)$. The log-price increment has mean $(\rho -{\sigma }^2/2)h+O(h^2)$ and variance ${\sigma }^{2}h+O(h^2)$. The Donsker invariance principle gives convergence of the log-price process to Brownian motion with this drift, hence of prices to the Black-Scholes model. Payoff-price convergence additionally uses continuity at almost every limiting path and uniform integrability for unbounded payoffs.