Boundary Values and Kinks at the Strike

Why the Strike Is a Special Point

An option payoff changes formula when the stock price at expiration crosses the strike. At the strike itself, the option is at the boundary between the two regions: exercising produces no intrinsic gain, while not exercising also produces no payoff. The payoff is therefore easy to misread, especially when a problem asks for profit rather than payoff. Profit also reflects the premium paid or received.

This tutorial considers a European call or put at its expiration date. Let $S_T$ denote the stock price at expiration, $K$ the strike price, and $c$ or $p$ the call or put premium per share. Assume one option on one share, no transaction costs, and the simple expiration-profit convention: payoff at expiration minus the premium originally paid. The premium is not accumulated with interest. Written positions have the opposite payoff and profit to the corresponding long positions.

Definition: A long call’s expiration payoff is $\max(S_T-K,0)$, and a long put’s expiration payoff is $\max(K-S_T,0)$. A long position’s profit is its payoff minus the premium. A written position’s payoff and profit are the negatives of the corresponding long position’s payoff and profit.

At $S_T=K$, both expressions inside the maximum equal zero. Thus, neither the call nor the put has a positive payoff at the strike. A long option’s profit at that price is not zero unless its premium is zero: the long call’s profit is $-c$, and the long put’s profit is $-p$. The written positions have profits $c$ and $p$, respectively.

Piecewise Values at and Around the Strike

Writing the payoffs in pieces makes the boundary behavior explicit. At equality, either adjacent branch gives the same payoff. The profit branches follow by subtracting the premium for a long position, or equivalently by taking the opposite profit for the writer.

\begin{aligned} \text{Long call payoff:}\quad C_T &= \begin{cases} 0, & S_T \le K,\\ S_T-K, & S_T > K, \end{cases} & \text{Long call profit:}\quad \Pi_C &= \begin{cases} -c, & S_T \le K,\\ S_T-K-c, & S_T > K; \end{cases} \\[6pt] \text{Long put payoff:}\quad P_T &= \begin{cases} K-S_T, & S_T < K,\\ 0, & S_T \ge K, \end{cases} & \text{Long put profit:}\quad \Pi_P &= \begin{cases} K-S_T-p, & S_T < K,\\ -p, & S_T \ge K. \end{cases} \end{aligned}

The choice of which branch includes equality is a matter of presentation here, not a difference in value: both branches meet at the same payoff at $S_T=K$. For instance, the call’s out-of-the-money branch gives payoff zero at equality, while its other branch gives $K-K=0$. The put has the same agreement at the boundary.

Worked Example: Long Call and Long Put at the Same Strike

A hypothetical European call has strike $K=\$50$ and premium $c=\$4$ per share. A hypothetical European put on the same stock has strike $K=\$50$ and premium $p=\$3$ per share. Find each long position’s payoff and profit at $S_T=\$49$, $\$50$, and $\$51$, then determine the slopes immediately to the left and right of the strike.

Method. Apply the maximum-payoff formulas at each stock price, then subtract the relevant premium to obtain profit.

At $S_T=\$49$, the call payoff is $\max(49-50,0)=\max(-1,0)=\$0$, so call profit is $0-4=-\$4$. The put payoff is $\max(50-49,0)=\$1$, so put profit is $1-3=-\$2$.

At $S_T=\$50$, the call payoff is $\max(50-50,0)=\$0$, so call profit is $0-4=-\$4$. The put payoff is also $\max(50-50,0)=\$0$, so put profit is $0-3=-\$3$.

At $S_T=\$51$, the call payoff is $\max(51-50,0)=\$1$, so call profit is $1-4=-\$3$. The put payoff is $\max(50-51,0)=\$0$, so put profit is $0-3=-\$3$.

For the call, profit changes from $-\$4$ at $\$49$ to $-\$4$ at $\$50$: the left-hand slope is $({-4}-(-4))/(50-49)=0$. From $\$50$ to $\$51$, it changes from $-\$4$ to $-\$3$: the right-hand slope is $({-3}-(-4))/(51-50)=1$. For the put, the left-hand slope is $({-3}-(-2))/(50-49)=-1$, and the right-hand slope is $({-3}-(-3))/(51-50)=0$.

Check. At the strike, both payoffs are zero, but the long positions’ profits are negative their respective premiums. The computed slopes agree with the piecewise profit formulas. The common mistake this method avoids is reporting zero profit merely because the payoff is zero.

Why the Graph Has a Kink

A slope describes how much a graph’s vertical value changes for a small change in its horizontal value. Here the horizontal variable is $S_T$, and the vertical variable is payoff or profit. The relevant slopes are one-sided: a left-hand slope uses stock prices below $K$, while a right-hand slope uses stock prices above $K$.

Theorem: At the strike, the long call payoff has left-hand slope $0$ and right-hand slope $1$. The long put payoff has left-hand slope $-1$ and right-hand slope $0$. Subtracting a fixed premium shifts a long position’s graph vertically and leaves these slopes unchanged. The two slopes differ at the strike, so the graph has a kink there and no single derivative at that point.

The result follows directly by comparing nearby values. Take a positive amount $h$. For a call, at $K-h$ and $K$, the payoff is zero at both prices; the change in payoff divided by the change in stock price is therefore zero. From $K$ to $K+h$, the call payoff rises from zero to $h$, so the corresponding slope is $h/h=1$. For a put, the payoff falls from $h$ at $K-h$ to zero at $K$, giving slope $-h/h=-1$; it stays zero from $K$ to $K+h$, giving slope zero.

At the boundary, the one-sided values agree: each payoff is zero at $K$. This is why the payoff graph is continuous there. But continuity does not require the slopes to agree. The call changes from flat to rising, while the put changes from falling to flat. Their different one-sided slopes create the kink. A fixed premium does not alter that argument: subtracting the same constant from every long-position payoff value changes the vertical level, not the change between nearby values.

For a written call or put, the payoff and profit are the negatives of the corresponding long position’s values. Negating a graph reverses the signs of both one-sided slopes. Thus, the written call’s slopes are $0$ on the left and $-1$ on the right; the written put’s are $1$ on the left and $0$ on the right. These are slopes with respect to the stock price and are measured in dollars of payoff or profit per dollar of stock-price change.

Evaluate the boundary payoff.
Set $S_T=K$ in the option’s maximum-payoff formula. For either a call or put, the result is zero.

Convert payoff to profit.
Subtract the premium for a long position; reverse the long-position result for a written position.

Find the side-specific slopes.
Use the branch below $K$ for the left-hand slope and the branch above $K$ for the right-hand slope.

Compare the slopes.
If the one-sided slopes differ, the graph is continuous at the strike but has a kink there; its ordinary derivative at the strike does not exist.

Worked Example: Written Positions at the Boundary

A hypothetical written call has strike $K=\$80$ and premium received $c=\$2.50$ per share. A hypothetical written put has the same strike and premium received $p=\$1.50$ per share. For each written position, calculate payoff and profit at $S_T=\$79$, $\$80$, and $\$81$, and find its one-sided slopes.

Method. First calculate the long option’s payoff; the written payoff is its negative. The written profit is the premium received plus the written payoff.

For the written call, at $\$79$ and $\$80$ the long call payoff is zero, so the written payoff is $\$0$ and written profit is $2.50+0=\$2.50$ at each price. At $\$81$, the long call payoff is $81-80=\$1$; the written payoff is $-\$1$, and written profit is $2.50-1=\$1.50$. Its left-hand slope is $(2.50-2.50)/(80-79)=0$; its right-hand slope is $(1.50-2.50)/(81-80)=-1$.

For the written put, at $\$79$ the long put payoff is $80-79=\$1$, so the written payoff is $-\$1$ and written profit is $1.50-1=\$0.50$. At $\$80$ and $\$81$, the long put payoff is zero, so the written payoff is zero and written profit is $\$1.50$ at each price. Its left-hand slope is $(1.50-0.50)/(80-79)=1$; its right-hand slope is $(1.50-1.50)/(81-80)=0$.

Check. At the strike, each written option has zero payoff and profit equal to the premium received. The slopes are the negatives of the corresponding long-option slopes. Confusing the writer’s premium receipt with a positive payoff would give the wrong boundary value.

Worked Example: Recovering Boundary Profit and Verifying the Kink

A hypothetical long call has strike $K=\$64$ and premium $c=\$2.40$ per share. A quoted expiration profit is needed at the strike and at stock prices $\$3$ below and $\$3$ above it. Calculate all three profits and use the changes to verify the one-sided slopes.

Method. Evaluate the long call’s payoff at each price, subtract the premium, then calculate each change in profit per dollar of stock-price change.

At $S_T=\$61$, the payoff is $\max(61-64,0)=\$0$, and profit is $0-2.40=-\$2.40$. At $S_T=\$64$, the payoff is $\max(64-64,0)=\$0$, and profit is also $-\$2.40$. At $S_T=\$67$, the payoff is $\max(67-64,0)=\$3$, and profit is $3-2.40=\$0.60$.

The left-hand slope from $\$61$ to $\$64$ is $(-2.40-(-2.40))/(64-61)=0/3=0$. The right-hand slope from $\$64$ to $\$67$ is $(0.60-(-2.40))/(67-64)=3/3=1$. The graph is at the same profit on the left side through the boundary, then rises on the right side.

Check. The boundary value is $-\$2.40$, not zero, because the premium remains a cost even though the expiration payoff is zero. The slopes verify the long-call branches on both sides of the strike.

Common Boundary Mistakes

Using the stock price as the payoff. A call payoff is not $S_T$. At $S_T=K$, exercise would require paying $K$ for a share worth $K$, so the intrinsic payoff is $K-K=0$, not $K$.
Reporting zero profit because payoff is zero. For a long call at the strike, profit is $0-c=-c$; for a long put, it is $0-p=-p$. A written position has profit equal to its premium received under the stated convention.
Assigning one slope at the kink. The call’s left and right slopes differ, as do the put’s. State both one-sided slopes; the ordinary derivative at the strike does not exist.
Reversing the written-position sign. A writer’s payoff is the negative of the long holder’s payoff. The premium receipt affects profit, but it does not change the writer’s payoff formula.

Careful boundary evaluation is useful whenever a payoff or profit is specified piecewise. For questions aligned with the SOA Exam FM (Financial Mathematics) syllabus, write down the value at equality separately, then check the behavior on each side. That prevents a branch-boundary convention from being mistaken for a change in the option’s value.

Check Your Understanding

For each problem, evaluate the boundary value before using nearby prices to check the slopes.

A hypothetical long call has strike $\$42$ and premium $\$3$. Find its payoff and profit at $S_T=\$42$, and give its left- and right-hand profit slopes. (Answer: payoff $\$0$, profit $-\$3$, slopes $0$ and $1$.)
A hypothetical long put has strike $\$95$ and premium $\$4$. Find its profit at $S_T=\$95$, and calculate the profit at $\$94$ and $\$96$. (Answer: at $\$95$, $-\$4$; at $\$94$, $-\$3$; at $\$96$, $-\$4$.)
A hypothetical written call has strike $\$110$ and premium received $\$6$. Find its payoff and profit at the strike and its two one-sided profit slopes. (Answer: payoff $\$0$, profit $\$6$, slopes $0$ and $-1$.)
A hypothetical written put has strike $\$72$ and premium received $\$2.50$. Calculate its profit at $S_T=\$70$, $\$72$, and $\$74$, then state its left- and right-hand slopes. (Answer: $\$0.50,\ \$2.50,\ \$2.50$; slopes $1$ and $0$.)
A hypothetical long call has strike $\$125$ and premium $\$7$. A candidate claims its profit at $S_T=\$125$ is zero because the payoff is zero. State the correct boundary profit and explain which amount the candidate omitted. (Answer: profit $-\$7$; the candidate omitted the premium paid.)

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