Cash Flows and Characteristics of Puts and Calls

A portfolio of equities backing variable annuity guarantees drops 15% in a quarter. The guarantees are deep in the money, billions more owed than expected. A colleague with put options sleeps soundly. The difference: understanding exactly when options pay off. Every pricing model in Topics 6b-6d builds on payoff mechanics.

HIGH-FREQUENCY: Payoff formulas for calls and puts, and long vs. short positions, appear on nearly every FAM exam.

$$\text{Long call} = \max(S_T - K, 0), \quad \text{Short call} = -\max(S_T - K, 0)$$

$$\text{Long put} = \max(K - S_T, 0), \quad \text{Short put} = -\max(K - S_T, 0)$$

KEY: Profit accounts for the premium accumulated to expiration, always use $e^{rT}$, not the raw premium.

Profit accounts for the premium accumulated to expiration:

$$\text{Long call profit} = \max(S_T - K, 0) - C \cdot e^{rT}$$

$$\text{Long put profit} = \max(K - S_T, 0) - P \cdot e^{rT}$$

European options exercise only at expiration. American options permit exercise before expiration. FAM pricing calculations use simple European-style derivatives on one non-dividend-paying asset.

Common mistakes

• Confusing payoff with profit. Payoff ignores the premium. Profit subtracts it (long) or adds it (short). Trap: payoff = 7 reported as profit when the accumulated premium of 3.28 should be deducted.
• Forgetting to accumulate the premium. Premium is paid at time 0; payoff is at time $T$. Comparing them without $e^{rT}$ introduces a small error. Trap: profit off by $C(e^{rT} - 1)$.
• Reversing call and put formulas. Call: $S_T - K$. Put: $K - S_T$. Swapping gives a negative payoff incorrectly set to zero. Trap: zero payoff when the option is ITM.

Bottom line

• Long call payoff: $\max(S_T - K, 0)$. Long put payoff: $\max(K - S_T, 0)$. Short positions negate the long payoff.
• Profit = payoff $\pm$ accumulated premium, where the premium is carried forward at $e^{rT}$.
• European options exercise at expiration; FAM pricing uses a single non-dividend-paying asset.
• Call ITM when $S > K$. Put ITM when $S < K$.

Exam shortcut

Write the max expression first, then deal with premium accumulation separately. This two-step approach prevents sign errors. For moneyness, draw a quick number line with $K$ (right of $K$ is ITM for calls, left is ITM for puts. Remember: "Call up, Put down") calls profit when the stock goes up, puts profit when it goes down.

The full lesson (about 1,684 words, 11 min read) adds 2 worked examples, all 5 common mistakes, a self-check, free in the app.

Learning objectives

• 6a

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