Options: pricing, strategies and risk
What is tested
This domain tests option payoffs, put-call parity, Black-Scholes inputs and the Greeks, and common strategies. You must distinguish intrinsic and time value, identify moneyness, and explain how delta, gamma, theta, vega and rho change with market moves.
Core rules
- Call payoff at expiry = max(S − K, 0); put = max(K − S, 0). Premium is paid upfront, so profit = payoff − premium.
- European put-call parity: c + Ke^{−rT} = p + S0 for non-dividend stock. Rearranged, c − p = S0 − Ke^{−rT}.
- Black-Scholes inputs: spot, strike, volatility, time, risk-free rate, dividends. Volatility is the only unobservable input; higher volatility raises both call and put values.
- Greeks: delta (call 0 to 1, put −1 to 0), gamma (rate of change of delta), theta (time decay, usually negative for long options), vega (sensitivity to volatility), rho (sensitivity to rates).
- Strategies: covered call (income, capped upside), protective put (insurance), straddle/strangle (long volatility), bull/bear spreads (directional, limited risk), collar (hedge funded by call sale).
Worked example
S0 = 100, K = 100, r = 4%, T = 1 year, call = 8.00. Fair put from parity: p = c − S0 + Ke^{−rT} = 8 − 100 + 100e^{−0.04} = 8 − 100 + 96.08 = 4.08. If the put trades at 3.50, it is cheap. Buy the put and the stock, sell the call: cash outflow = 3.50 + 100 − 8 = 95.50. At expiry the payoff is K = 100 regardless, so profit = 100 − 95.50 × 1.04 = 100 − 99.32 = 0.68 per share.
Traps that cost marks
- Applying parity to American options or ignoring dividends.
- Confusing payoff with profit: premiums and financing must be included.
- Reversing Greek signs, especially put delta and theta.
- Assuming higher volatility always helps a strategy; it hurts short option positions.
- Forgetting early exercise and assignment risk on American options.