From the Binomial Tree to Black-Scholes

Write $C_0 = \frac{1}{r^n}\sum_{k=0}^{n}\binom{n}{k}p^{*k}(1-p^*)^{n-k}\max[0,\ u^k d^{n-k}S_0 - K]$ and price one two-period example both by backward induction through the tree and by the formula.

Record that a period can be a year, a minute or a millisecond, and count the paths and distinct final prices of an eight-step recombining tree as the example.

Write $\frac{1}{2}\sigma^2S^2\frac{\partial^2 C}{\partial S^2} + rS\frac{\partial C}{\partial S} + \frac{\partial C}{\partial t} = rC$ with $C(S,T) = \max[S-K,\ 0]$ and $C(0,t) = 0$, and state it as the arbitrage condition with infinitely many steps.

Record the two practical advantages of trees, numerical solution for complex securities and easy parallelization, and state that the model assumes IID coin tosses, with correlated returns requiring Markov pricing.