Option pricing — binomial and Black-Scholes

**One-period binomial**: Up factor u, down factor d. Stock S → uS or dS. Risk-neutral probability: π = (1+r-d)/(u-d). Call value: C = [π × Cu + (1-π) × Cd] / (1+r). **Black-Scholes-Merton (European call, no dividend)**: C = S×N(d1) – K×e^(-rT)×N(d2) d1 = [ln(S/K) + (r + σ²/2)T] / (σ√T) d2 = d1 – σ√T Key assumptions: lognormal stock, constant volatility, constant rate, no dividends, continuous trading, no friction. **Put-call parity**: C – P = S – K×e^(-rT). Allows synthetic creation: long call + short put = long forward.

**Greeks** (sensitivity of option price): - **Delta** (∂C/∂S): N(d1) for call. Hedge ratio. Approaches 1 as deep ITM, 0 as deep OTM. - **Gamma** (∂²C/∂S²): rate of change of delta. Highest near-ATM. - **Vega** (∂C/∂σ): sensitivity to volatility. Always positive for long options. Highest near-ATM. - **Theta** (∂C/∂T): time decay. Negative for long options. - **Rho** (∂C/∂r): sensitivity to rates. American vs European: - American puts can be optimal to exercise early (deep ITM, no dividend). - American calls on non-dividend stock = European call (never optimal to exercise early). - Use binomial with early-exercise check at each node.

L2 vignette tricks: - BSM doesn't directly handle dividends — use forward stock or modify. - Implied volatility ≠ realised volatility. IV depends on supply/demand for options. - Volatility smile: OTM puts trade above ATM IV in equity markets (post-1987 crash skew). - Vega is highest at ATM and longer-dated; gamma highest at ATM and shorter-dated. Replication intuition: long call = long Δ shares + short risk-free borrowing. Allows risk-neutral pricing. Delta hedging: as Δ changes (gamma), portfolio rebalances continuously. In practice, discrete rebalancing → P&L sensitivity to gamma.

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