Options Markets and Properties of Options

A risk manager pays $4 for a one-year put on a \$100 stock with strike \$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.

An American option lets its holder exercise on permitted dates through expiration; a European option permits exercise only at expiration. These labels describe exercise timing, not where the option trades. A holder can often sell either style before expiration without exercising it. An assigned writer must perform the contract's cash or physical settlement obligations. Assignment follows exercise procedures; the writer cannot choose to refuse it because the trade is unprofitable. The additional American exercise right cannot reduce value relative to an otherwise identical European option.

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

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 equality does not determine the American put price.
• 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 $c \geq S_0 - K e^{-rT}$; put $p \geq K e^{-rT} - S_0$; 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 5,145 words, 34 min read) adds 2 worked examples, all 6 common mistakes, a self-check, free in the app.

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