Binomial Tree Option Pricer
Implement a recombining binomial tree to price options under the Cox-Ross-Rubinstein (CRR) model with $N$ time steps.
Your implementation must:
1. Price a **European call** option.
2. Price an **American put** option with early exercise.
3. Run in $O(N^2)$ time and $O(N)$ memory.
4. Clearly state the risk-neutral up/down factors $u$, $d$ and the risk-neutral probability $q$.
5. Explain how you would verify convergence to the Black-Scholes price as $N \to \infty$.
**Inputs:** Stock price $S_0$, strike $K$, risk-free rate $r$, volatility $\sigma$, time to expiry $T$, number of steps $N$.
**Constraints:**
-
\leq N \leq 10{,}000$
- $S_0, K, r, \sigma, T > 0$
**Example:**
- Input: $S_0 = 100$, $K = 100$, $r = 0.05$, $\sigma = 0.20$, $T = 1.0$, $N = 200$
- Output: European call price $\approx 10.45$, American put price $\approx 6.08$
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