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A call option gives the right (not obligation) to buy at a strike price; a put gives the right to sell. European options exercise only at maturity; American options exercise anytime. Payoffs are call = max(S − K, 0) and put = max(K − S, 0), where S is stock price and K is strike.
Draw payoff diagrams for long/short calls and puts at various strikes. Calculate payoffs at different stock prices and understand when exercise is optimal.
From your study of options basics and payoff diagrams, you know that options are contracts giving the holder a right without an obligation. Let's sharpen exactly what that right looks like for calls and puts, when you would use it, and how the payoff formulas encode those decisions.
A call option gives you the right to buy an asset at a predetermined strike price K. Suppose you hold a call on a stock with K = $50. If the stock price S at expiration is $70, you exercise: you pay $50 for something worth $70, pocketing a $20 gain per share. If S = $40, you do nothing — you would not pay $50 for something worth $40 when you can simply buy it in the market for $40. This is the max(S − K, 0) payoff formula in action: exercise when S > K, walk away when S ≤ K. The right, not obligation, is what caps your downside at zero.
A put option is the mirror image: the right to sell at strike K. If S = $30 and K = $50, you exercise by selling something worth $30 for $50 — a $20 gain. If S = $60, there is no point selling at $50 when you could sell in the market for $60, so you let the put expire. Payoff: max(K − S, 0). Puts increase in value when the underlying falls; calls increase in value when the underlying rises. This asymmetry is the defining feature of options: unlimited upside (for calls) or large downside protection (for puts), with losses capped at the premium paid.
The European versus American distinction matters for when exercise can happen. European options can only be exercised at expiration; American options can be exercised at any point before expiration. For a non-dividend-paying stock, it is almost never optimal to exercise an American call early — the option has time value that you sacrifice by exercising before maturity. Early exercise of American puts can be optimal, however: if a stock collapses near zero, you might prefer the certainty of the K − S payoff now rather than waiting while time value decays.
The payoff diagrams you drew in your prerequisite course encode a crucial point about who bears risk. The long call buyer has a limited loss (premium paid) and theoretically unlimited gain. The short call writer — the person on the other side — has limited gain (premium received) and unlimited potential loss. Long and short positions in the same option are perfect opposites in their risk profiles. This zero-sum structure at expiration is why options are central to hedging: for every risk-taker who wants exposure to a price move, there is a hedger who wants to transfer that exact risk away.