Short Sales Combined With Options

Short Sales and Terminal Cash Flows

A short sale creates a liability as well as an initial cash receipt. The short seller borrows shares, sells them, and later must buy shares to return to the lender. If the stock price rises, repurchasing the shares costs more; if it falls, repurchasing costs less. Calls and puts can change this exposure, but their payoffs must be combined with the short-sale cash flows rather than mistaken for the total profit.

For the calculations below, assume a short seller sells shares at time 0, invests the sale proceeds at a stated effective annual rate, and repays the borrowed shares at time $T$. Assume no dividends, stock-borrow fees, taxes, transaction costs, margin constraints, or early option exercise. Options are European and expire at $T$. These assumptions isolate the cash flows being studied; if a short seller owes dividends or incurs other costs, those must also be included.

Definition: For one share sold short at time 0 for $S_0$, the accumulated sale proceeds at time $T$ are $S_0(1+i)^T$, where $i$ is the effective annual rate. The cost to repurchase and return the share is $S_T$. Thus, before including options, the short position’s terminal dollar profit is $S_0(1+i)^T-S_T$.

The accumulated proceeds are not themselves the short seller’s profit. They must be compared with the cost of repurchasing the share. The formula includes interest earned on the sale proceeds. If the proceeds cannot be invested or the account earns a different rate, replace the accumulation assumption with the actual cash-flow treatment.

Combining a Short Sale with an Option

At expiration, a call with strike $K$ pays $\max(S_T-K,0)$, while a put pays $\max(K-S_T,0)$. These are option payoffs, not profits: the premium paid or received at time 0 also matters. To compare all cash flows at $T$, accumulate a time-0 premium using the same stated rate and term.

Result: Suppose $q$ shares are sold short at $S_0$, and $n_j$ options of type $j$ are held, where $n_j$ is positive for a long position and negative for a short position. If each option has premium $p_j$ per contract or share as specified, terminal profit is the accumulated short-sale proceeds less the repurchase cost, plus signed option payoffs, less the accumulated cost of signed option purchases: $$ \Pi_T=q\bigl(S_0(1+i)^T-S_T\bigr)+\sum_j n_j\left(\text{payoff}_j(S_T)-p_j(1+i)^T\right). $$ Here a negative $n_j$ represents a written option, so its payoff and premium cash flow both reverse sign. Keep quantities and premiums on a consistent per-share or per-contract basis.

The premium term follows from the time-0 cash flow. A long option costs $p_j$ at time 0, so that cost is $p_j(1+i)^T$ at the comparison date. A written option receives the premium, represented by a negative $n_j$, and owes the option payoff if exercised. This signed-quantity convention handles both cases with one formula.

Why the Short-Stock Formula Holds

The short seller receives $S_0$ when the share is sold. Under the stated investment assumption, that amount grows to $S_0(1+i)^T$. At expiration, returning the borrowed share requires spending $S_T$ to repurchase it. Subtracting that outflow from the accumulated receipt gives the short stock’s terminal cash-flow result:

\text{terminal short-stock result} = \text{accumulated sale proceeds} - \text{repurchase cost} = S_0(1+i)^T-S_T.

For $q$ shares, each cash flow scales by $q$. Adding an option means adding its signed expiration payoff and accounting for its signed premium at the same terminal date. This is a cash-flow derivation; it does not assume that the option premium is zero or that payoff equals profit.

Worked Example: A Short Sale Hedged with a Long Call

An investor sells one share short for $S_0=\$100$, buys a one-year call with strike $K=\$105$ for a premium of $\$6$, and invests the short-sale proceeds at an effective annual rate of $i=5\%$. There are no dividends. Find the terminal profit if $S_T=\$90$, and if $S_T=\$125$. Explain the hedge behavior.

Method and timing. Accumulate both the sale proceeds and the call premium for one year. The proceeds are $100(1.05)=\$105$, and the premium cost is $6(1.05)=\$6.30$.

For $S_T=\$90$, repurchasing the share costs $\$90$, and the call expires worthless. Therefore:

\Pi_T=105-90+\max(90-105,0)-6.30 =15+0-6.30 =\$8.70.

For $S_T=\$125$, repurchasing the share costs $\$125$, and the call pays $125-105=\$20$. Therefore:

\Pi_T=105-125+\max(125-105,0)-6.30 =-20+20-6.30 =-\$6.30.

The call offsets the short stock’s additional loss once the stock price exceeds $\$105$: above that strike, the stock-price part of the position is $-S_T+(S_T-105)=-105$. The terminal profit is then constant at $105-105-6.30=-\$6.30$. This call limits the position’s upward price exposure, but the premium remains a cost. The calculation avoids the mistake of reporting the call’s $\$20$ payoff as the combined position’s profit.

What a Put Does to a Short Position

A long put pays when the stock falls below its strike. A short stock position already benefits from a price decline, so a long put generally adds to that downside gain rather than protecting against a stock-price rise. This is the opposite direction from the long call hedge in the prior example. The outcome still depends on the put premium and the accumulated short-sale proceeds.

Worked Example: Two Short Shares with Two Long Puts

A trader sells two shares short at $\$80$ per share and buys two one-year puts with strike $K=\$75$, each costing $\$4$ per share. The proceeds and premiums are compared at the end of one year using $i=4\%$. There are no dividends. Calculate terminal profit when $S_T=\$60$ and when $S_T=\$90$.

The accumulated short-sale proceeds are $2(80)(1.04)=\$166.40$. The two put premiums total $\$8$ at time 0, or $8(1.04)=\$8.32$ at expiration. The combined profit is:

\Pi_T=166.40-2S_T+2\max(75-S_T,0)-8.32.

At $S_T=\$60$, returning the two shares costs $2(60)=\$120$, and the two puts pay $2(75-60)=\$30$. Hence:

\Pi_T=166.40-120+30-8.32=\$68.08.

At $S_T=\$90$, returning the shares costs $2(90)=\$180$, and the puts expire worthless. Hence:

\Pi_T=166.40-180+0-8.32=-\$21.92.

The puts add a payoff when the stock falls, increasing the gain in the first case after accounting for their premium. They do not offset the loss from a rising stock price in the second case. A common mistake is to treat a long put as a general hedge without checking which direction of the stock movement produces its payoff.

Option Quantity and Residual Exposure

An option hedge can offset all or only part of a short share position’s price movement. To see the effect, consider the position above the strike of a long call: each call then pays one dollar more for each additional dollar in the stock price. If there are fewer calls than short shares, some upward exposure remains. Below the strike, those calls pay zero, so they do not offset the short stock’s price movement there.

Worked Example: A Partial Call Hedge

A trader sells three shares short at $\$50$ per share and buys two one-year calls with strike $K=\$55$, each costing $\$3$ per share. The short-sale proceeds and option premiums are accumulated for two years at an effective annual rate of $i=3\%$. There are no dividends. Calculate terminal profit for $S_T=\$40$ and $S_T=\$70$, and identify the residual exposure above the strike.

The two-year accumulation factor is $1.03^2=1.0609$. Accumulated short-sale proceeds are $3(50)(1.0609)=\$159.135$. The total time-0 call premium is $2(3)=\$6$, which accumulates to $6(1.0609)=\$6.3654$. Thus:

\Pi_T=159.135-3S_T+2\max(S_T-55,0)-6.3654.

At $S_T=\$40$, both calls expire worthless and repurchasing three shares costs $3(40)=\$120$:

\Pi_T=159.135-120+0-6.3654=\$32.7696.

At $S_T=\$70$, the three-share repurchase costs $3(70)=\$210$, and the two calls pay $2(70-55)=\$30$:

\Pi_T=159.135-210+30-6.3654=-\$27.2304.

Above $\$55$, the three short shares lose one dollar per share for each one-dollar increase in $S_T$, while the two calls gain one dollar each. The net price sensitivity is therefore that of one short share: $-3+2=-1$. The calls reduce, but do not eliminate, upward exposure. The calculation avoids assuming that owning any call automatically makes a short position fully hedged.

Common Traps and Practical Checks

Before finalizing a short-sale-and-option calculation, check that the cash flows are assigned the correct signs and dates. A reliable setup separates the accumulated proceeds, the repurchase liability, option payoffs, and option premiums. If dividends, borrow costs, or other payments are specified, include them as additional cash flows; the examples’ no-dividend assumption would not apply unchanged.

Confusing proceeds with profit. The sale proceeds must be compared with the repurchase cost. A stock sale receipt alone is not a gain.
Using payoff as profit. An option payoff excludes its premium. Accumulate the premium to the same date as the other cash flows when using terminal profit.
Reversing the short-stock sign. A short position benefits when the stock price falls and loses when it rises, before other cash flows are considered.
Ignoring option quantity. Two calls hedge only two shares’ worth of upward exposure above the strike, not an arbitrary number of short shares.
Forgetting the model assumptions. A dividend owed to a share lender is an additional short-sale cost; it is not included in $S_0(1+i)^T-S_T$.

Key takeaway: First write the short-sale proceeds and share-repurchase liability. Then add each option’s signed expiration payoff and account for its premium at the same comparison date. Only after those cash flows are combined can the position’s profit and hedge behavior be judged.

Check Your Understanding

For each problem, calculate terminal profit under the stated assumptions. Treat all options as European, assume no dividends or transaction costs, and use the stated annual effective rate.

A trader sells one share short for $\$72$, invests the proceeds for one year at $5\%$, and has no options. Find the profit if $S_T=\$65$. (Answer: $\$10.60$)
A trader sells one share short for $\$40$ and buys a one-year call with strike $\$44$ for $\$2$. The effective annual rate is $5\%$. Find profit if $S_T=\$50$. (Answer: $-\$4.10$)
A trader sells two shares short at $\$30$ per share and buys two one-year puts with strike $\$28$ for $\$1$ per share. The effective annual rate is $3\%$. Find profit if $S_T=\$20$. (Answer: $\$35.74$)
A trader sells four shares short at $\$25$ per share and buys one one-year call with strike $\$27$ for $\$1.50$ per share. The effective annual rate is $4\%$. Find profit if $S_T=\$35$. (Answer: $-\$29.56$)
A trader sells one share short at $\$90$ and writes one one-year call with strike $\$95$, receiving a premium of $\$4$. The effective annual rate is $5\%$. Find profit if $S_T=\$110$. (Answer: $-\$26.30$)

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