Margin and Short Positions¶

Picture two traders on opposite sides of a single naked call: strike $100, premium $5, one contract on 100 shares. The buyer hands over $500 and goes home; no matter what the stock does, that $500 is the most she can ever lose. The writer collects the $500 and goes home with an obligation. If the stock closes at $150, he owes $5,000; at $300, he owes $20,000; at $1,000, he owes $90,000. Same trade, same contract — but one side's exposure is bounded and the other's is not. That single asymmetry, repeated millions of times across the market every day, is the entire reason exchanges require option writers to post collateral before the trade is allowed to print.

When an investor buys an option, the maximum loss is the premium paid — a known, finite amount settled at the time of purchase. The writer of that same option, however, accepts an obligation whose eventual cost depends on the future path of the underlying. This fundamental asymmetry between bounded and potentially unbounded loss is the reason margin requirements exist: they ensure that option writers can meet their obligations when called upon.

Risk Asymmetry Between Buyer and Writer¶

Key idea: Margin exists because losses are asymmetric — the buyer's loss is bounded by the premium, but the writer's loss can be far larger.

Recall (see § The Four Elementary Positions): the max-loss column of the four-position table makes the asymmetry explicit — long calls and long puts lose at most the premium, a naked call writer faces unbounded loss as $S_T \to \infty$, and a naked put writer faces a bounded but potentially large loss of up to $K - P_0$ (attained when $S_T = 0$).

Concretely, the writer's loss profiles are

\[ L_{\text{call}} = (S_T - K)^+ - C_0, \qquad L_{\text{put}} = (K - S_T)^+ - P_0 \leq K - P_0. \]

The first has no finite upper bound; the second, although finite, can still be very large relative to the premium received when $K$ is large. This is the risk profile that margin requirements are designed to cover.

Margin as a Performance Guarantee¶

Because the exchange stands between buyer and writer as the central counterparty, it must ensure that every writer can fulfill the contract. In essence, the exchange is asking: can you pay if things go wrong? Margin is the collateral — typically cash or eligible securities — that the writer deposits with the broker to guarantee performance.

The initial margin is set at the time the position is opened. For a naked call, the required margin is larger than for a naked put, reflecting the unbounded loss profile. Covered positions (e.g., a call written while holding the underlying stock) require substantially less margin because the writer's obligation is offset by the asset already held.

Premium Scale: Why Margin Matters

At-the-money SPX options command premiums of 150–200 index points, each worth $100 per point. Writing a single naked ATM call therefore creates potential exposure on the order of tens of thousands of dollars per contract. Even a naked put, whose loss is bounded, can require the writer to pay up to $K \times 100$ in the extreme case — for an SPX put with $K = 6{,}600$, that is $660,000. These magnitudes explain why exchanges demand substantial margin collateral from option writers and why margin calls can be triggered by seemingly modest index movements.

Margin Calls and Maintenance¶

Why does this matter for pricing theory? Margin requirements constrain the positions that market participants can take and affect the supply of options. From a modeling perspective, the existence of margin reflects the fact that credit risk is a real-world friction that the idealized Black-Scholes framework abstracts away.

As the underlying price moves against the writer's position, the margin account may fall below a prescribed maintenance margin level. When this happens, the broker issues a margin call, requiring the writer to deposit additional funds promptly — often within one trading day. Failure to meet the margin call allows the broker to close the position, crystallizing the loss.

This mechanism prevents the accumulation of unrealized losses that the writer cannot cover, protecting both the counterparty and the integrity of the exchange.

Exercises¶

Exercise 1. An investor writes a naked call with strike $K = 80$ for a premium of $5. Compute the writer's profit or loss at maturity when (a) $S_T = 70$, (b) $S_T = 90$, and (c) $S_T = 120$.

Solution to Exercise 1

The writer's profit at maturity is $C_0 - (S_T - K)^+$.

(a) $S_T = 70 < K = 80$: the call expires worthless. Profit $= 5 - 0 = \$5$.

(b) $S_T = 90 > K = 80$: the writer pays the payoff. Profit $= 5 - (90 - 80) = 5 - 10 = -\$5$.

(c) $S_T = 120 > K = 80$: Profit $= 5 - (120 - 80) = 5 - 40 = -\$35$.

As $S_T$ increases, the writer's loss grows without bound.

Exercise 2. An investor writes a naked put with strike $K = 60$ for a premium of $4. Determine the maximum possible loss and the value of $S_T$ at which it occurs.

Solution to Exercise 2

The writer's loss at maturity is $(K - S_T)^+ - P_0$. The worst case occurs when $S_T = 0$:

\[ L_{\max} = (60 - 0) - 4 = \$56 \]

Unlike the naked call writer, the naked put writer's loss is bounded. The maximum loss is $K - P_0 = 60 - 4 = \$56$, occurring when the underlying asset becomes worthless.

Exercise 3. Explain why a covered call (writing a call while holding the underlying stock) requires less margin than a naked call. Describe what happens to the writer's overall position when $S_T > K$.

Solution to Exercise 3

A naked call writer must deliver a stock worth $S_T$ in exchange for $K$ when $S_T > K$. Since $S_T$ is unbounded, the potential outlay is unlimited, and the margin must reflect this open-ended risk.

A covered call writer already holds the stock. If $S_T > K$, the writer delivers the stock (already owned) and receives $K$. The writer forgoes the upside above $K$ but does not need to purchase the stock at the market price. The worst outcome is opportunity cost, not an out-of-pocket loss.

Because the obligation is fully offset by the asset held, the covered call has a bounded risk profile, and exchanges accordingly require far less margin collateral.

Exercise 4. A trader writes a naked call at strike $K = 100$ and posts an initial margin of $15 per share. Suppose the maintenance margin is $10 per share and the premium received is $6. If the stock price rises from $95 to $108 after one day, determine whether a margin call is triggered by comparing the margin account balance to the maintenance requirement.

Solution to Exercise 4

At inception the margin account holds the initial margin plus the premium received: $15 + 6 = \$21$ per share.

After the stock rises to $108, the call is in the money with intrinsic value $(108 - 100)^+ = 8$. The unrealized loss to the writer is $8 per share, reducing the margin account balance to $21 - 8 = \$13$ per share.

Since $13 exceeds the maintenance margin of $10, no margin call is triggered. However, if the stock were to rise further to, say, $112 (unrealized loss of $12, balance $= 21 - 12 = \$9 < \$10$), a margin call would be issued.