Options
Options give the buyer a right, not an obligation, to buy or sell at a fixed strike before or at expiry. The exam tests payoff construction, premium and breakeven arithmetic, put-call parity, the pricing inputs and the Greeks.
Rights, payoffs and value
A call buyer pays a premium for the right to buy; a put buyer pays for the right to sell. Writers receive the premium and carry the obligation. Long call: maximum loss the premium, gain unlimited. Long put: maximum loss the premium, maximum gain strike minus premium. Short call: maximum gain the premium, loss unlimited. Short put: maximum gain the premium, loss strike minus premium. Value splits into intrinsic value and time value; moneyness describes whether the option is in, at or out of the money.
Pricing and the Greeks
European options can be exercised only at expiry; American options at any time. Inputs are spot, strike, time, volatility, interest rate and dividends. Put-call parity for European options on a non-dividend stock is c + K·e^(−rT) = p + S₀. Delta measures change in premium per unit of spot, gamma the change in delta, theta time decay, vega sensitivity to volatility and rho sensitivity to rates. Black-Scholes assumes lognormal prices, constant volatility, no early exercise and frictionless markets.
Worked example
S₀ = 100, K = 100, r = 5%, T = 1, c = 8.00 and p = 3.50. PV(K) = 95.12, so parity requires c − p = 100 − 95.12 = 4.88, but the market shows 4.50. The call is cheap relative to the put: buy the call, sell the put, sell the stock and invest 95.12 at 5%. Net inflow = −8 + 3.50 + 100 − 95.12 = 0.38. At expiry the invested amount grows to 100 and the option and stock legs offset in every scenario, locking in 0.38 before costs.
Marks-losing traps
- Quoting breakeven as the strike; it is strike plus premium for a call and strike minus premium for a put.
- Forgetting the premium in profit and loss.
- Applying parity to American or dividend-paying options without adjustment.
- Assuming a short call loss is limited to the premium.