Integrated Short Sale And Option Position Problems

Why Combined Positions Need a Cash-Flow Table

A short stock position and an option position can offset one another in some expiration-price regions and reinforce one another in others. The direction of the result is not reliably identified by a label such as “hedge” or “spread.” It comes from adding every transaction with its correct sign, first at initiation and then at expiration.

The method in this tutorial applies to hypothetical positions held until expiration. Unless a problem specifies otherwise, assume no dividends, stock-borrow charges, transaction costs, margin interest, or interest earned on cash proceeds. Assume options are European-style or are otherwise held to expiration, and use a contract multiplier of 100 shares. Under these assumptions, initial and expiration cash flows can be added directly to calculate dollar profit; this is not an annualized investment return.

Definition: A short sale of $N$ shares at stock price $S_0$ produces initial cash proceeds of $NS_0$. If the position is closed at expiration when the stock price is $S_T$, the repurchase costs $NS_T$. The short-stock profit before other cash flows is therefore $N(S_0-S_T)$. For an option position, expiration profit contribution means its expiration payoff plus its initial premium cash flow: a premium paid is negative, and a premium received is positive.

For a call with strike $K$, the payoff per share at expiration is $\max(S_T-K,0)$. For a put it is $\max(K-S_T,0)$. A long option receives its payoff and pays its premium; a short option receives its premium and owes the option payoff. If a position has multiple option contracts, multiply each per-share amount by the number of contracts and by 100. The stock and option share quantities need not be the same, so track them separately.

A Reliable Calculation Sequence

Write down the position before evaluating any price scenario. Then sort all strikes from lowest to highest. Each strike can change which options have a nonzero payoff, so the strikes divide possible values of $S_T$ into regions. Within a region, each option payoff is either zero or a linear function of $S_T$; the total profit is obtained by adding those functions to the short-stock cash flows.

List initial transactions.
Record short-sale proceeds as positive cash, premiums paid as negative cash, and premiums received as positive cash.

Write expiration cash flows.
Include the cost of repurchasing shorted shares and the payoff owed or received on each option.

Partition by strike.
Use the ordered strikes to determine the active payoff formula in each region, including the regions below the lowest strike and above the highest strike.

Add and check.
Combine initial and expiration cash flows, test representative prices, and verify that adjacent profit formulas agree at each shared strike.

Why the Short-Stock Formula Holds

The short seller first receives $NS_0$ from selling $N$ shares. Closing the position requires buying those same shares for $NS_T$. The net cash from those two stock transactions is the sale proceeds minus the repurchase cost:

$$ NS_0-NS_T=N(S_0-S_T). $$

Thus a price decline creates a positive short-stock contribution, while a price increase creates a negative one. To obtain the total position profit, add the initial option premium cash flows and the option payoffs at expiration. The formula does not include interest, dividends, borrow costs, or other charges; if a problem supplies any of them, each must also be entered as a cash flow with the appropriate sign.

Worked Example: Short Stock With a Call Vertical Spread

A hypothetical investor shorts 100 shares at $S_0=\$60$, buys one call with strike $\$55$ for a premium of $\$8.00$ per share, and writes one call with strike $\$65$ for a premium of $\$3.00$ per share. Find the total profit as a function of $S_T$, and evaluate it at $S_T=\$50,\$60,$ and $\$70$. Assume the stated contract multiplier and no other cash flows.

Method. First total the initiation cash flows. The short sale produces $100(\$60)=\$6{,}000$. Buying the lower-strike call costs $100(\$8)=\$800$, and writing the higher-strike call brings in $100(\$3)=\$300$. The net initial cash flow is

\$6{,}000-\$800+\$300=\$5{,}500.

For $S_T\leq \$55$, both calls expire without value. For $\$55<S_T<\$65$, only the long call pays, by $100(S_T-\$55)$. For $S_T\geq \$65$, the long call pays $100(S_T-\$55)$ and the written call costs $100(S_T-\$65)$. Including the cost of repurchasing the short stock gives:

\text{Profit}(S_T)= \begin{cases} 5500-100S_T, & S_T\leq 55,\\ 5500-100S_T+100(S_T-55)=0, & 55<S_T<65,\\ 5500-100S_T+100(S_T-55)-100(S_T-65)=6500-100S_T, & S_T\geq 65. \end{cases}

At $S_T=\$50$, profit is $\$5{,}500-100(\$50)=\$500$. At $S_T=\$60$, the short-stock close-out is $-\$6{,}000$ and the long call payoff is $\$500$, so total profit is $\$5{,}500-\$6{,}000+\$500=\$0$. At $S_T=\$70$, the stock close-out is $-\$7{,}000$, the long-call payoff is $\$1{,}500$, and the written-call payoff is $-\$500$. Total profit is $\$5{,}500-\$7{,}000+\$1{,}500-\$500=-\$500$.

The middle-region profit is zero throughout because the long call’s slope offsets the short stock’s slope, while the initial cash flow offsets the remaining constant amount. As a boundary check, both neighboring formulas give zero at $\$55$ and at $\$65$. Common mistake avoided: treating the short call as an additional gain when it is in the money. Its premium is received initially, but its expiration payoff is a cash outflow.

Short Stock With a Call and Put

A combined position may have option strikes on opposite sides of the initial stock price. For instance, a long call can limit some losses from a rising stock, while a written put changes the payoff when the stock falls below its strike. Do not infer the total position’s risk from either option alone: calculate the combined profit in each strike region.

Worked Example: Short Stock With a Long Call and Written Put

A hypothetical investor shorts 100 shares at $S_0=\$40$, buys one call with strike $\$45$ for $\$2.50$ per share, and writes one put with strike $\$35$ for $\$1.00$ per share. Calculate the profit function, the profit at $S_T=\$30,\$40,$ and $\$50$, and any breakeven price.

The initial cash flow is $\$4{,}000-\$250+\$100=\$3{,}850$: the short sale generates $\$4{,}000$, the call costs $\$250$, and the written put brings in $\$100$.

If $S_T\leq \$35$, the call expires worthless and the written put costs $100(\$35-S_T)$. Adding the short-stock close-out gives profit $3850-100S_T-100(35-S_T)=350$. Between the strikes, neither option has a payoff, so profit is $3850-100S_T$. At or above $\$45$, the long call pays $100(S_T-\$45)$, so profit becomes $3850-100S_T+100(S_T-45)=-650$.

\text{Profit}(S_T)= \begin{cases} 350, & S_T\leq 35,\\ 3850-100S_T, & 35<S_T<45,\\ -650, & S_T\geq 45. \end{cases}

At $S_T=\$30$, profit is $\$350$. At $S_T=\$40$, it is $\$3{,}850-\$4{,}000=-\$150$. At $S_T=\$50$, it is $-\$650$. In the middle region, the breakeven equation is $3850-100S_T=0$, so $S_T=\$38.50$, which lies between $\$35$ and $\$45$. The adjacent formulas also agree at both strikes: they give $\$350$ at $\$35$ and $-\$650$ at $\$45$.

Common mistake avoided: omitting the written put’s loss when $S_T<\$35$. In that region its payoff owed is $100(\$35-S_T)$, not zero; the short-stock gain and written-put loss together produce the constant $\$350$ profit.

Short Stock With a Put Vertical Spread

A put spread contributes a payoff that changes at two strikes. A long put at the higher strike gives a positive payoff below that strike; a written put at the lower strike offsets part of that payoff below the lower strike. When combined with short stock, the profit may have a different slope in each region. The net profit, rather than the spread payoff alone, determines the breakeven.

Worked Example: Short Stock With a Put Spread

A hypothetical investor shorts 100 shares at $S_0=\$80$, buys one put with strike $\$75$ for $\$2.40$ per share, and writes one put with strike $\$60$ for $\$0.80$ per share. Find the profit function, evaluate it at $S_T=\$50,\$68,$ and $\$82$, and determine the breakeven.

The initial cash flow is $\$8{,}000-\$240+\$80=\$7{,}840$. The put spread pays $\$1{,}500$ when $S_T\leq \$60$, because its net payoff per share is $(75-S_T)-(60-S_T)=\$15$. For $\$60<S_T<\$75$, only the long put pays, giving $100(75-S_T)$. Above $\$75$, both puts expire without value.

\text{Profit}(S_T)= \begin{cases} 7840-100S_T+1500=9340-100S_T, & S_T\leq 60,\\ 7840-100S_T+100(75-S_T)=15340-200S_T, & 60<S_T<75,\\ 7840-100S_T, & S_T\geq 75. \end{cases}

At $S_T=\$50$, profit is $\$9{,}340-\$5{,}000=\$4{,}340$. At $S_T=\$68$, it is $\$15{,}340-\$13{,}600=\$1{,}740$. At $S_T=\$82$, it is $\$7{,}840-\$8{,}200=-\$360$. In the middle region, the algebraic root is $S_T=15340/200=\$76.70$, which is outside that region and is not a valid breakeven there. In the upper region, $7840-100S_T=0$ gives $S_T=\$78.40$, which is in the region $S_T\geq \$75$. Thus $\$78.40$ is the breakeven. The formulas agree at $\$60$ (profit $\$3{,}340$) and $\$75$ (profit $\$340$).

Common mistake avoided: accepting a root without checking its region. A linear expression derived under a particular option-payoff condition is valid only when the price satisfies that condition.

Checks That Catch Sign and Region Errors

A correct piecewise function should give the same profit from either neighboring formula at each strike. At a strike, an option’s payoff is zero exactly at the boundary, so including the boundary in either adjoining region does not change the result. This continuity check is useful, but it does not replace checking the signs of the initial transactions or the option quantities.

Track proceeds separately from profit. Short-sale proceeds are an initial inflow, not the short position’s profit. The repurchase cost must also be included.
Scale every option amount. A quoted premium or payoff per share must be multiplied by the contract multiplier and contract count.
Use the correct sign for written options. A written option receives its premium initially and pays its payoff at expiration.
Check roots against their regions. Solve for zero profit within each piece, then reject any root that does not satisfy that piece’s strike conditions.
Keep the model assumptions visible. Dividends, borrow costs, fees, or interest on cash proceeds change the total cash-flow calculation when they are specified.

Key takeaway: For an integrated short-stock and option position, calculate initial net cash flow, write the expiration cash flows with signs, divide the price range at the strikes, and add the cash flows within each region. Verify boundary values and test every proposed breakeven in its stated region.

Check Your Understanding

For each hypothetical position, use a 100-share option contract multiplier and ignore dividends, borrow costs, fees, and interest. Check both the initial cash flow and the expiration payoffs.

An investor shorts 100 shares at $\$24$ and buys one call with strike $\$27$ for a premium of $\$1.50$ per share. Find profit at $S_T=\$20$ and $S_T=\$30$. (Answer: $\$250$ at $\$20$; $-\$450$ at $\$30$.)
An investor shorts 100 shares at $\$52$, buys one call with strike $\$50$ for $\$4.00$ per share, and writes one call with strike $\$60$ for $\$1.50$ per share. Find profit at $S_T=\$56$ and $S_T=\$65$. (Answer: $-\$50$ at $\$56$; $-\$550$ at $\$65$.)
An investor shorts 100 shares at $\$40$, buys one put with strike $\$42$ for $\$3.00$ per share, and writes one put with strike $\$32$ for $\$1.20$ per share. Find profit at $S_T=\$28$ and $S_T=\$37$. (Answer: $\$2{,}020$ at $\$28$; $\$620$ at $\$37$.)
An investor shorts 200 shares at $\$70$, buys two put contracts with strike $\$75$ for $\$2.20$ per share, and writes two put contracts with strike $\$65$ for $\$0.70$ per share. Find profit at $S_T=\$60$ and $S_T=\$80$. (Answer: $\$3{,}700$ at $\$60$; $-\$2{,}300$ at $\$80$.)
An investor shorts 100 shares at $\$48$, buys a call at strike $\$50$ for $\$2.80$, writes a call at strike $\$58$ for $\$1.10$, buys a put at strike $\$44$ for $\$1.60$, and writes a put at strike $\$38$ for $\$0.60$, all per share. Find profit at $S_T=\$40$ and $S_T=\$55$, and identify every breakeven price. (Answer: $\$930$ at $\$40$; $-\$470$ at $\$55$; the only breakeven is $S_T=\$45.30$.)

What You'll Learn