Options Markets and Properties of Options

A risk manager pays $4foraone-yearputona\$100stockwithstrike\$95.That$4 is insurance, protection against the bottom falling out. The exam tests whether you can decompose that premium into intrinsic and time value, bound it by no-arbitrage, and recognize when put-call parity reveals a mispricing the floor traders already arbitraged away.

The premium on a vanilla option moves with six inputs. Memorize the directions before you memorize the formulas.

European calls and puts on dividend-paying stocks can violate the time-to-maturity rule: a longer-dated European call can be worth less than a shorter-dated one if a fat dividend falls between them. American options always benefit from more time because you can always exercise sooner if it pays.

KEY: Higher volatility raises both call and put values. Options have asymmetric payoffs (capped on one side), so wider distributions of S(T) increase expected payoff while leaving the downside floor at zero.

Intrinsic value is the immediate exercise payoff. Call intrinsic = max(S - K, 0); put intrinsic = max(K - S, 0).

Common mistakes

• Forgetting to discount the strike in the lower bound. Call lower bound is $S_0-K{e}^{-rT}$, not $S_0-K$. Trap: a candidate sees S = 100, K = 95, r = 5%, T = 1 and computes "intrinsic = $5." The actual lower bound is $100-95\times e^{-0.05}=\$9.63$, so a call at $7 is below the no-arbitrage floor.
• Applying European put-call parity to American options. The clean equation $c-p=S_0-K{e}^{-rT}$ holds only for Europeans. For Americans the spread is bounded but not equal. Trap: a problem says "compute the American put given the American call price" and offers the European parity result; that answer is too high (American puts are worth more).
• Treating dividends and yield as interchangeable. Discrete dividends use $S_0-I$, where I is the present value of cash dividends. Continuous yield uses $S_0{e}^{-qT}$. They behave similarly but the formulas differ. Plugging an annual dividend dollar amount into the q slot inflates the discount and understates the parity-implied put.

Bottom line

• Moneyness: ITM = positive intrinsic, OTM = zero intrinsic, ATM = strike near spot; intrinsic + time value = premium, and ATM options carry the largest time value.
• Six drivers (S, K, T, σ, r, D) move calls and puts oppositely on most inputs, but higher volatility raises both.
• Lower bounds (no dividends): call $c\ge S_0-K{e}^{-rT}$; put $p\ge K{e}^{-rT}-S_0$; both bounded below by zero.
• Put-call parity (European, no dividend): $c-p=S_0-K{e}^{-rT}$; with dividend yield q, $c-p=S_0{e}^{-qT}-K{e}^{-rT}$.

Exam shortcut

When a parity question lands, write $c-p$ on one side and $S_0-K{e}^{-rT}$ (or its dividend-adjusted form) on the other. If they don't match, the option that makes the equation work too high is the one to sell. The trap is forgetting to discount K, which gives you intrinsic value instead of the lower bound.

The full lesson (about 2,657 words, 18 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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