Short Stock Combined With Calls And Puts

Why Combine the Positions Before Evaluating Them?

A short stock position can be paired with options to change how the position responds to the stock price at expiration. The combined result is not found by looking at each contract in isolation: each position contributes its own signed payoff, and the total payoff is their sum. The useful method is to write those position functions first, mark every option strike, and then simplify the sum in each price region.

This tutorial considers positions held to a single expiration date. It focuses on contractual expiration payoffs, not on dividends, stock-borrow charges, or interim cash flows. Those costs can affect a short sale’s realized result, but they are outside the payoff calculation here.

Model and conventions: Let $S_0$ be the stock price when the position is opened and $S_T$ its price at expiration. A short position in one share has expiration trading payoff $S_0-S_T$, before financing, dividends, and borrow costs. A long call with strike $K$ pays $(S_T-K)^+=\max(S_T-K,0)$; a short call pays the negative of this amount. A long put pays $(K-S_T)^+=\max(K-S_T,0)$; a short put pays the negative of this amount. Unless a question says otherwise, the payoff is per position as specified, and option premiums are not included in it.

Write Each Position With Its Sign

For a short position in $q$ shares, where $q>0$, the trading payoff is $q(S_0-S_T)$. A position of $n$ long calls at strike $K_c$ contributes $n(S_T-K_c)^+$; a position of $m$ short calls contributes $-m(S_T-K_c)^+$. The same convention applies to puts: long puts contribute positively and short puts negatively. If one position contains several strikes, include one term for each strike.

The total expiration payoff is the algebraic sum of these signed terms. A compact expression is helpful for checking the setup, but a piecewise expression shows exactly how the payoff changes as $S_T$ moves across the strikes.

\text{Total payoff} = \text{short-stock payoff} +\text{long-option payoffs} -\text{short-option payoffs}.

In that expression, “short-option payoffs” means the positive contractual payoff the option buyer receives. The seller’s position contributes its negative. For example, a short call contributes $-(S_T-K)^+$, not $+(S_T-K)^+$.

Why the Payoff Is Piecewise Linear

Between any two consecutive strikes, each option is either in the money throughout that region or out of the money throughout it. An in-the-money call contributes $S_T-K$, while an out-of-the-money call contributes zero. An in-the-money put contributes $K-S_T$, while an out-of-the-money put contributes zero. The stock term is linear in $S_T$ everywhere. Adding these terms therefore gives a linear expression within each region.

Justification: For $S_T<K$, $(S_T-K)^+=0$ and $(K-S_T)^+=K-S_T$. For $S_T>K$, $(S_T-K)^+=S_T-K$ and $(K-S_T)^+=0$. At $S_T=K$, both option payoffs are zero. Thus the active option terms are fixed within each interval formed by the strikes, and their sum with the stock payoff is linear in each interval. The formulas on either side of a strike agree at the strike because the option payoff there is zero.

Write the signed positions.
Use $q(S_0-S_T)$ for $q$ short shares. Use a positive sign for a long option and a negative sign for a short option.

Order the distinct strikes.
The strikes divide nonnegative expiration prices into regions. If strikes coincide, treat them as one boundary and combine their terms.

Simplify in each region.
Replace every option payoff by either its active linear expression or zero, then collect the $S_T$ terms and constants.

Check the boundaries.
Evaluate neighboring formulas at each strike. They should agree, since an option has zero payoff exactly at its strike.

Worked Examples

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

An investor shorts one share when the stock price is $S_0=\$50$, buys one call with strike $K_c=\$55$, and buys one put with strike $K_p=\$45$. Find the total expiration payoff as a piecewise function. The call premium is $\$3$ and the put premium is $\$2$. Also find the payoff and the profit, excluding financing and short-sale costs, when $S_T=\$60$.

Method. Add the short-stock, long-call, and long-put position functions. The strikes, in order, are $\$45$ and $\$55$, so consider three regions.

For $S_T<45$, the call is out of the money and the put is in the money. The payoff is $50-S_T+0+(45-S_T)=95-2S_T$.

For $45\leq S_T\leq55$, both options have zero payoff, including at their respective strikes. The payoff is $50-S_T$.

For $S_T>55$, the call is in the money and the put is out of the money. The payoff is $50-S_T+(S_T-55)+0=-5$.

P(S_T)= \begin{cases} 95-2S_T, & 0\leq S_T<45,\\ 50-S_T, & 45\leq S_T\leq55,\\ -5, & S_T>55. \end{cases}

The boundary checks give $95-2(45)=5=50-45$ and $50-55=-5$, matching the value in the upper region. At $S_T=\$60$, the short stock contributes $50-60=-\$10$, the call contributes $60-55=\$5$, and the put contributes $\$0$. The total payoff is $-10+5+0=-\$5$.

The option premiums paid total $\$3+\$2=\$5$. Under the stated convention of excluding financing and other short-sale costs, profit is payoff minus premiums: $-\$5-\$5=-\$10$. The method avoids treating the option premiums as part of the contractual expiration payoff.

Worked Example: Two Short Shares, a Long Call, and Short Puts

A trader shorts two shares at $S_0=\$42$, buys one call with strike $\$40$ for a premium of $\$4$, and writes two puts with strike $\$35$, receiving $\$1.50$ for each put. Find the total payoff function and the profit at $S_T=\$37$, ignoring financing and short-sale costs.

Method. The two-share short-stock payoff is $2(42-S_T)=84-2S_T$. The long call contributes $(S_T-40)^+$, and the two short puts contribute $-2(35-S_T)^+$. The ordered strikes are $\$35$ and $\$40$.

If $0\leq S_T<35$, the call is out of the money and both puts are in the money: $84-2S_T-2(35-S_T)=84-2S_T-70+2S_T=14$.

If $35\leq S_T\leq40$, both option terms are zero, so the payoff is $84-2S_T$.

If $S_T>40$, the call is in the money and the puts are out of the money: $84-2S_T+(S_T-40)=44-S_T$.

P(S_T)= \begin{cases} 14, & 0\leq S_T<35,\\ 84-2S_T, & 35\leq S_T\leq40,\\ 44-S_T, & S_T>40. \end{cases}

At $S_T=35$, the middle expression is $84-70=14$, matching the lower region. At $S_T=40$, it is $84-80=4$, matching $44-40=4$. At $S_T=\$37$, the short shares contribute $84-74=\$10$, and both options expire with zero payoff. The total payoff is $\$10$.

The opening option cash flow is $-\$4+2(\$1.50)=-\$1$: the call purchase is an outflow and the put sales are inflows. Profit, without carrying those cash flows forward, is payoff plus the net opening option cash flow: $\$10-\$1=\$9$. The short-sale proceeds are not added again to this trading payoff.

Worked Example: Multiple Strikes and a Short Call

A position consists of one short share opened at $S_0=\$60$, one short call with strike $\$65$, two long puts with strike $\$50$, and one short put with strike $\$45$. Find the payoff function and the payoff at $S_T=\$47$ and $S_T=\$70$. Option premiums are excluded.

Method. The signed payoff is $60-S_T-(S_T-65)^+ +2(50-S_T)^+ -(45-S_T)^+$. The ordered strikes are $\$45,\$50,\$65$, producing four regions.

For $0\leq S_T<45$, both put strikes are above the stock price, and the call is out of the money: $60-S_T+2(50-S_T)-(45-S_T)=115-2S_T$.

For $45\leq S_T<50$, the long puts are in the money but the short put at $\$45$ and the call are out of the money: $60-S_T+2(50-S_T)=160-3S_T$.

For $50\leq S_T\leq65$, all options have zero payoff, so the total is $60-S_T$. For $S_T>65$, the short call contributes $-(S_T-65)$, giving $60-S_T-(S_T-65)=125-2S_T$.

P(S_T)= \begin{cases} 115-2S_T, & 0\leq S_T<45,\\ 160-3S_T, & 45\leq S_T<50,\\ 60-S_T, & 50\leq S_T\leq65,\\ 125-2S_T, & S_T>65. \end{cases}

Check the boundaries: at $\$45$, the first expression gives $115-90=25$, and the second gives $160-135=25$. At $\$50$, the second gives $160-150=10$, and the third gives $60-50=10$. At $\$65$, the third gives $-\$5$, and the fourth gives $125-130=-\$5$.

At $S_T=\$47$, use the second region: $160-3(47)=160-141=\$19$. At $S_T=\$70$, use the last region: $125-2(70)=125-140=-\$15$. The method avoids applying the short-call payoff when the stock price is below its strike.

Payoff, Profit, and Common Traps

The expiration payoff records the value of the positions at expiration under the stated convention. It does not include premiums paid or received at opening. To calculate profit in the simple, undiscounted convention used above, add the opening option cash flow to the expiration payoff: subtract premiums paid for long options and add premiums received for short options. If a problem specifies an investment return, financing, or other cash flows, those must be handled according to that problem’s timing and assumptions.

In particular, the initial proceeds from shorting stock are not an extra expiration payoff. The short-stock trading payoff $S_0-S_T$ already compares the sale price with the later repurchase price. Adding the sale proceeds a second time would count the opening transaction twice.

A frequent error is to give a short call a positive payoff. For example, at $S_T=\$70$, a short call with strike $\$65$ contributes $-(\$70-\$65)=-\$5$, not $+\$5$. If the position also has one short share opened at $\$60$, its stock payoff is $60-70=-\$10$; the combined payoff is $-10-5=-\$15$. The incorrect sign would produce $-\$10+\$5=-\$5$, overstating the result by $\$10$.

Check the position direction. A short stock payoff falls as $S_T$ rises. A short call is a liability when the stock finishes above its strike; a short put is a liability when it finishes below its strike.
Use the correct region. For each option, compare $S_T$ with that option’s own strike. Do not treat every option as active merely because another option is in the money.
Check every strike. The formulas immediately to the left and right of a strike must give the same payoff at that strike.
Keep premiums separate. A payoff calculation and a profit calculation answer different questions unless the problem explicitly defines them otherwise.

Key takeaway: Write the short-stock term and every signed option term before simplifying. Divide the expiration-price axis at the distinct strikes, simplify each region, and verify continuity at each strike. Add premiums only when the requested quantity is profit rather than expiration payoff.

Check Your Understanding

For each problem, use the stated payoff convention and keep premiums separate unless profit is requested.

A trader shorts one share at $\$30$ and buys one put with strike $\$28$. Write the payoff function and find the payoff if $S_T=\$25$. (Answer: $P(S_T)=30-S_T+(28-S_T)^+$; payoff at $\$25$ is $\$8$.)
A trader shorts two shares at $\$24$, buys one call with strike $\$26$, and writes one put with strike $\$20$. Find the total payoff at $S_T=\$18$. (Answer: $\$10$.)
A trader shorts one share at $\$75$, buys a call with strike $\$70$ for $\$6$, and buys a put with strike $\$65$ for $\$3$. At $S_T=\$80$, find the expiration payoff and the profit, ignoring financing and short-sale costs. (Answer: payoff $=\$5$; profit $=-\$4$.)
A position shorts three shares at $\$40$, buys two calls with strike $\$42$, and writes one put with strike $\$35$. Find the expiration payoff at $S_T=\$50$. (Answer: $-\$14$.)
A trader shorts one share at $\$52$, writes two calls with strike $\$55$, and buys three puts with strike $\$48$. Give the piecewise payoff and evaluate it at $S_T=\$46$ and $S_T=\$60$. (Answer: $P(S_T)=52-S_T+3(48-S_T)^+-2(S_T-55)^+$; at $\$46$, payoff $=\$12$; at $\$60$, payoff $=-\$18$.)

What You'll Learn