Derivation — From Binomial Trees to Black-Scholes in the Continuous Limit

Multi-period binomial trees showed numerically that as the number of steps $n \to \infty$, the binomial European call price converges to the Black-Scholes formula $c = S_0 N(d_1) - Ke^{-rT}N(d_2)$. This note derives that convergence analytically.

The proof has a clean structure: write the $n$-step binomial price as a discounted expected payoff under the risk-neutral measure $Q$, split the sum into two parts ($U_1$ and $U_2$), apply the central limit theorem to each, and read off $d_1$ and $d_2$. The key step is recognising that $U_2$ is a binomial tail probability (giving $N(d_2)$) and $U_1$ involves a different binomial distribution with a shifted probability (giving $N(d_1)$). This shifted probability is the embryonic form of the Girsanov change of measure that underlies continuous-time risk-neutral pricing.