Convergence in the CRR model

Question

Under certain conditions, the option price of the CRR (Cox-Ross-Rubinstein) Binomial model converges to the Black-Scholes price as the maximal step size of the partition converges to zero (i.e. a smoother partition of $\left[0,T\right]$ is taken).

What are these conditions? Does the same convergence can be observed in case of Black formula (the lognormal pricing model of options on forwards/futures), or is it only applicable to Black-Scholes pricing?

Answer 1

In general, in lattice models, you are approximating the true strike $K$ or barrier $H$ of an option with $\hat{K}$ or $\hat{H}$, a terminal node in the tree. As you take finer partitions, $\hat{K} \to {K}$. So, intuitively, the sawtooth pattern of convergence must apply to most lattice models.