Proving Long-Short Payoff Symmetry

Why Long and Short Positions Are Symmetric

A purchased option gives its holder a right, while a written option creates the corresponding obligation. If the two positions have the same option type, strike, expiration date, and underlying, the writer’s expiration payoff is the negative of the holder’s payoff. The premium cash flows are also opposite: the purchaser pays the premium, and the writer receives it. Together, these facts make the two positions’ profits exact opposites when the same profit convention is used.

This symmetry is useful for checking calculations and interpreting profit diagrams. It is aligned with the SOA Exam FM (Financial Mathematics) syllabus. The discussion here concerns one unit of an option held to expiration; it does not analyze early exercise or scale positions by contract multipliers.

Definition: Let $S_T$ be the underlying asset’s price at expiration $T$, and let $K$ be the option’s strike price. For an option held through expiration, the call payoff to its purchaser is $\max(S_T-K,0)$, and the put payoff to its purchaser is $\max(K-S_T,0)$. A written option’s payoff is the negative of the corresponding purchased option’s payoff. These are expiration payoffs, before considering any premium.

Assume throughout that the options being compared have identical terms and that each position concerns one unit of the underlying. Ignore transaction costs, taxes, and counterparty default. An option payoff is determined by its exercise value at expiration: a call is exercised when $S_T>K$, while a put is exercised when $S_T<K$. At $S_T=K$, either payoff is zero.

Deriving the Payoff Symmetry

Write $x^+=\max(x,0)$. A purchased call’s payoff is $(S_T-K)^+$. The writer must deliver exactly the value the purchaser receives, so the writer’s payoff is its negative. A purchased put and its writer have the same relationship. The equations below also make the signs explicit in each price region.

\begin{aligned} \text{Long call payoff} &= (S_T-K)^+ \\ \text{Written call payoff} &= -(S_T-K)^+ =\begin{cases} 0, & S_T\leq K,\\ K-S_T, & S_T>K, \end{cases}\\[6pt] \text{Long put payoff} &= (K-S_T)^+ \\ \text{Written put payoff} &= -(K-S_T)^+ =\begin{cases} S_T-K, & S_T<K,\\ 0, & S_T\geq K. \end{cases} \end{aligned}

For example, when $S_T>K$, the call holder receives $S_T-K$. The writer pays that amount, so the writer’s payoff is $K-S_T=-(S_T-K)$. When $S_T\leq K$, neither side receives an exercise payment, and both payoffs are zero. For a put with $S_T<K$, the holder receives $K-S_T$, and the writer’s payoff is its negative, $S_T-K$. When $S_T\geq K$, both payoffs are zero. Thus, in either exercise region—and at the strike—the writer’s payoff is the negative of the purchaser’s payoff.

Theorem: For the same option terms and one unit of the underlying, the written payoff equals the negative of the purchased payoff at every expiration price: $$ \text{Written payoff}=-\text{Long payoff}. $$ This is a point-by-point identity, not merely a statement that the positions have equal maximum gains or losses.

Premium Cash Flows and Profit Symmetry

A payoff records the option-related cash flow at expiration. Profit also accounts for the premium. Under the standard expiration-profit convention used for option profit diagrams, the premium is subtracted from a purchased option’s payoff and added to a written option’s payoff. This convention treats the premium as a fixed amount and does not include interest earned or forgone between the trade date and expiration.

\begin{aligned} \text{Long profit} &= \text{Long payoff}-\text{premium},\\ \text{Written profit} &= \text{Written payoff}+\text{premium}. \end{aligned}

Suppose the premium is $q$. Since the written payoff is the negative of the long payoff, the writer’s profit is $ -\text{Long payoff}+q$. That is the negative of the purchaser’s profit, $ -(\text{Long payoff}-q)$. The premium sign matters: it is a cost to the purchaser and an inflow to the writer.

\text{Written profit} =-\text{Long payoff}+q =-(\text{Long payoff}-q) =-\text{Long profit}.

If profit is instead measured as a true terminal-value amount, the premium’s time value must be treated consistently. Suppose a premium $q$ is paid or received at time 0, the option expires after $T$ years, and a known annual effective interest rate $i$ is used to accumulate cash flows to expiration. The premium’s accumulated value is $q(1+i)^T$. The corresponding terminal profits are the option payoff less this accumulated cost for the purchaser, and the writer’s payoff plus the accumulated premium received. They are still exact opposites.

\begin{aligned} \text{Long terminal profit} &=\text{Long payoff}-q(1+i)^T,\\ \text{Written terminal profit} &=\text{Written payoff}+q(1+i)^T =-\text{Long terminal profit}. \end{aligned}

The choice of convention changes the numerical profit, but not the symmetry, provided both positions use the same convention and the premium flows are treated as opposites.

Worked Examples

Worked Example: Pairing a Call’s Payoffs and Profits

A hypothetical call has strike $K=\$50$ and premium $q=\$4$. At expiration, consider underlying prices of $\$42$, $\$50$, and $\$61$. Calculate the long and written payoffs and profits using the standard expiration-profit convention.

Method. First calculate the purchased call payoff, $\max(S_T-K,0)$, at each price. Negate it for the written payoff. Then subtract the premium for the long profit and add the premium for the written profit.

At $S_T=\$42$, the long payoff is $\max(42-50,0)=\$0$. The written payoff is $-\$0=\$0$. The long profit is $0-4=-\$4$, and the written profit is $0+4=\$4$.

At $S_T=\$50$, the long payoff is $\max(50-50,0)=\$0$, and the written payoff is $\$0$. The profits are $0-4=-\$4$ for the long position and $0+4=\$4$ for the written position.

At $S_T=\$61$, the long payoff is $\max(61-50,0)=\$11$. The written payoff is $-\$11$. The long profit is $11-4=\$7$, and the written profit is $-11+4=-\$7$.

Check. At each price, the written payoff is the negative of the long payoff, and the written profit is the negative of the long profit. The strike price gives zero payoff for both positions, not zero profit, because the premium remains.

Common mistake avoided: Treating the call’s $\$11$ payoff as the long profit. The premium must be subtracted to find profit.

Worked Example: Pairing a Put’s Payoffs and Profits

A hypothetical put has strike $K=\$80$ and premium $q=\$7$. At expiration, the underlying price is either $\$65$ or $\$88$. Find each position’s payoff and profit using the standard expiration-profit convention.

Method. For a purchased put, calculate $\max(K-S_T,0)$. The written payoff is its negative. Apply the premium after calculating the payoff.

At $S_T=\$65$, the long put payoff is $\max(80-65,0)=\$15$. The written put payoff is $-\$15$. The long profit is $15-7=\$8$; the written profit is $-15+7=-\$8$.

At $S_T=\$88$, the long put payoff is $\max(80-88,0)=\$0$, and the written payoff is $\$0$. The long profit is $0-7=-\$7$, and the written profit is $0+7=\$7$.

Check. The paired profits sum to zero in both cases: $8+(-8)=0$ and $-7+7=0$. The written payoff is not the amount the put holder would receive; it has the opposite sign.

Common mistake avoided: Using $S_T-K$ as the purchased put payoff. A put’s purchaser benefits when the underlying finishes below the strike, so the correct payoff is $\max(K-S_T,0)$.

Worked Example: Including the Premium’s Time Value

A hypothetical call has strike $K=\$120$, premium $q=\$9$, and expires in one year. The annual effective interest rate is $i=5\%$. Compare the purchaser’s and writer’s terminal profits when the underlying finishes at $\$136$ and $\$115$, accumulating the premium to expiration.

Method. Calculate the call payoff first. Accumulate the premium using the one-year factor $1+i=1.05$. Subtract that accumulated cost from the long payoff; add the accumulated premium to the written payoff.

The premium’s terminal value is $9(1.05)^1=\$9.45$.

At $S_T=\$136$, the long payoff is $\max(136-120,0)=\$16$, so the written payoff is $-\$16$. The long terminal profit is $16-9.45=\$6.55$. The written terminal profit is $-16+9.45=-\$6.55$.

At $S_T=\$115$, the long payoff is $\max(115-120,0)=\$0$, and the written payoff is $\$0$. The long terminal profit is $0-9.45=-\$9.45$, and the written terminal profit is $0+9.45=\$9.45$.

Check. The terminal profits sum to zero at both prices. If the standard expiration-profit convention were used instead, the long profits would be $\$7$ and $-\$9$, with written profits of $-\$7$ and $\$9$. Those figures differ because they do not accumulate the premium, but the symmetry remains.

Common mistake avoided: Comparing a terminal payoff with an unaccumulated premium while claiming to calculate terminal-value profit. Both cash flows must be expressed on the same time basis.

Common Sign Errors

Long-short symmetry is an effective check, but it does not prevent a sign error unless the payoff and premium are handled separately. For a call with strike $\$60$, premium $\$5$, and expiration price $\$68$, the call payoff to the purchaser is $\max(68-60,0)=\$8$. The written payoff is $-\$8$. Under the standard expiration-profit convention, the writer’s profit is $-8+5=-\$3$.

A wrong calculation might report the writer’s profit as $8+5=\$13$, by using the purchaser’s positive payoff and then adding the premium. This gives the wrong sign for the option obligation. The correction is to negate the purchased payoff first, then add the premium: $-8+5=-\$3$. The purchaser’s profit is $8-5=\$3$, confirming that the two profits are opposites.

Key takeaway: Calculate the purchased payoff from the exercise rule, negate that payoff for the writer, and then apply opposite premium cash flows. Under matching assumptions and a consistent time basis, the written payoff and profit are each the negative of the corresponding long amount.

Check Your Understanding

For each question, calculate the requested payoff and profit for both positions, using the stated convention.

A hypothetical call has strike $\$44$ and premium $\$3$. At expiration, the underlying is $\$55$. Find the long and written payoffs and profits under the standard expiration-profit convention. (Answer: Payoffs $+\$11$ and $-\$11$; profits $+\$8$ and $-\$8$, respectively.)
A hypothetical put has strike $\$72$ and premium $\$5$. At expiration, the underlying is $\$61$. Find the long and written payoffs and profits under the standard expiration-profit convention. (Answer: Payoffs $+\$11$ and $-\$11$; profits $+\$6$ and $-\$6$, respectively.)
A hypothetical call has strike $\$90$ and premium $\$8$. The underlying finishes exactly at the strike. Find both payoffs and profits, and state why the profits are not zero. (Answer: Both payoffs are $\$0$; long profit is $-\$8$, written profit is $+\$8$, because the premium remains.)
A hypothetical put has strike $\$105$ and premium $\$12$. The underlying finishes at $\$87$. Find the paired payoffs and profits under the standard expiration-profit convention. (Answer: Payoffs $+\$18$ and $-\$18$; profits $+\$6$ and $-\$6$, respectively.)
A hypothetical put has strike $\$70$, premium $\$4$, and expires in two years. The annual effective interest rate is $10\%$, and the underlying finishes at $\$62$. Accumulate the premium to expiration and find both terminal profits. (Answer: Accumulated premium $4(1.10)^2=\$4.84$; payoffs $+\$8$ and $-\$8$; terminal profits $+\$3.16$ and $-\$3.16$, respectively.)

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