= Cox-Ross-Rubinstein diffusion approximation
{c}
{title2=$u=e^{\sigma\sqrt h},\quad d=e^{-\sigma\sqrt h},\quad B_{k+1}/B_k=e^{\rho h}$}
= Binomial diffusion approximation
{synonym}
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 $\sigma^2h+O(h^2)$. 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.
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