Exercise: American put via binomial tree

Tasks

1. Implement the CRR binomial tree pricer for an American put. Parameters: $S_0 = 100, K = 100, T = 1, r = 0.05, \sigma = 0.2$, $N = 1000$ time steps.

2. Verify convergence: compute prices for $N \in \{50, 100, 500, 1000, 5000\}$. The price should converge to \sim \6.09$ but with non-monotone oscillations. Why?

3. European put comparison. Modify your code to also compute the European put price (no early-exercise check). What's the early-exercise premium?
4. Deep ITM check. For $S_0 = 50, K = 100$ (deep ITM), the American put value should equal the immediate exercise value $K - S_0 = 50$. Verify.