A call option gives the buyer the right to purchase 100 shares of the underlying stock at the strike price before expiration. If AAPL trades at $180 and you buy a $190 strike call for a $5 premium, you pay $500 (100 shares × $5). If AAPL rises to $210 before expiration, your call is worth at least $20 intrinsic value ($210 - $190). If AAPL stays below $190, the call expires worthless and you lose the $500 premium. Maximum loss for the buyer is the premium paid; maximum gain is theoretically unlimited. The seller of the call (writer) has the opposite payoff: pockets the premium but faces unlimited upside risk.

Intrinsic value = max(Stock Price - Strike, 0) for calls. Time value is the premium above intrinsic value that reflects the probability of future price movement before expiration. An option far out-of-the-money (OTM) has zero intrinsic value but positive time value — it has a chance of becoming profitable if the stock moves enough. At expiration, time value reaches zero — all that remains is intrinsic value or nothing. This time decay (theta decay) is the primary mechanism by which sold options lose value as expiration approaches, and the primary way options sellers make money.

A put option gives the buyer the right to sell 100 shares at the strike price. If AAPL trades at $180 and you buy a $170 put for $4, you pay $400. If AAPL falls to $150, your put is worth $20 intrinsic value ($170 - $150). If AAPL stays above $170, the put expires worthless. Put buyers profit from price declines; they use puts for speculation on falling prices or as portfolio insurance (protective puts). Put sellers (who are short the put) have the obligation to buy shares at the strike price if assigned — they profit from time decay and modest price stability.

Put-call parity is a fundamental no-arbitrage relationship: Call - Put = Stock Price - Present Value of Strike. This relationship ensures that European calls and puts with the same strike and expiration are priced consistently with the underlying stock. Violations create arbitrage opportunities that sophisticated traders immediately exploit. In practice, put-call parity holds very tightly for liquid, exchange-listed options — any apparent deviation is typically explained by dividends, borrow costs for short sellers, or early exercise considerations.

Option premium is determined by six variables: current stock price, strike price, time to expiration, implied volatility, risk-free interest rate, and dividends. Of these, implied volatility (IV) is the most important non-obvious driver. Two options with identical structure can have radically different premiums based solely on IV differences — a stock with high IV (high expected future volatility) has more expensive options because there is more probability that the stock will move enough to make the option profitable. IV is the market's collective forecast of future price volatility, derived by inverting the Black-Scholes model with the observed market price.

Options provide non-linear payoffs — the relationship between stock price change and option P&L is not linear. A small stock move can cause a large percentage change in an option's value (leverage). This non-linearity is both the attraction (leverage without margin debt) and the risk (leverage multiplies losses as well as gains). Delta measures the linear approximation of this relationship; gamma measures how delta changes with price — the non-linearity term. Understanding these sensitivities is the entry point to sophisticated options risk management.