Graphing Put Payoffs and Profits

What a Put Graph Shows

A put’s expiration graph relates the underlying asset price $S_T$ to the option position’s value at expiration. A payoff graph shows the option’s value from exercise alone. A profit graph also accounts for the premium paid or received when the position was opened. The two graphs therefore have related shapes but different vertical positions.

In this tutorial, $K$ is the strike price and $p$ is the put premium per unit. Assume a standard option held to expiration, with $S_T\geq 0$, $K>0$, and $p>0$. Graphs are per unit of the underlying, and profit is payoff adjusted by the premium without accumulating or discounting that premium. This is the usual expiration-graph convention; if a problem specifies financing or interest on the initial premium, that cash flow must also be included.

Definition: A long put is a purchased put. Its expiration payoff is $\max(K-S_T,0)$, and its profit under the stated convention is its payoff minus the premium $p$. A written put is a sold put. Its payoff is the negative of the long put’s payoff, and its profit is its payoff plus the premium received.

The graph’s horizontal axis is the expiration price $S_T$, and the vertical axis is payoff or profit per unit. Mark the strike $K$ on the horizontal axis. For profit graphs, also mark the break-even price, where profit equals zero. That price is not generally where payoff equals zero.

Equations and Graph Shapes

A put gives its holder the right to sell at $K$. At expiration, exercising is worthwhile when $S_T<K$: selling at $K$ rather than at the market price produces $K-S_T$ per unit. When $S_T\geq K$, exercise has no positive value. Thus the long put’s payoff is positive below the strike and zero at or above it.

\begin{aligned} \text{Long put payoff} &= \max(K-S_T,0) =\begin{cases} K-S_T,&0\leq S_T<K,\\ 0,&S_T\geq K, \end{cases}\\ \text{Long put profit} &= \max(K-S_T,0)-p =\begin{cases} K-S_T-p,&0\leq S_T<K,\\ -p,&S_T\geq K. \end{cases} \end{aligned}

The long payoff is a descending straight line below the strike, with slope $-1$, and a horizontal line at zero above the strike. Subtracting the constant premium shifts the entire payoff graph downward by $p$, without changing its slopes. The long profit graph therefore has slope $-1$ below $K$ and is horizontal at $-p$ above $K$.

The writer has the opposite exercise-based payoff: the writer loses the amount the holder gains from exercise. The writer also receives the premium. Consequently, the writer’s profit is the negative payoff plus $p$.

\begin{aligned} \text{Written put payoff} &= -\max(K-S_T,0) =\begin{cases} S_T-K,&0\leq S_T<K,\\ 0,&S_T\geq K, \end{cases}\\ \text{Written put profit} &= p-\max(K-S_T,0) =\begin{cases} S_T-K+p,&0\leq S_T<K,\\ p,&S_T\geq K. \end{cases} \end{aligned}

The written payoff rises with slope $+1$ below the strike and is zero above it. Adding the premium shifts that payoff graph upward by $p$. The written profit therefore rises with slope $+1$ below $K$ and is horizontal at $p$ above $K$.

Break-even: For either a long or written put, the profit is zero below the strike when $S_T=K-p$, provided $0\leq p\leq K$. The long put’s profit is positive below that price and negative above it; the written put’s profit is negative below it and positive above it. The payoff itself is zero at $S_T\geq K$, not at the break-even price.

The break-even follows by setting the relevant profit equation below $K$ equal to zero. For the long put, $K-S_T-p=0$, giving $S_T=K-p$. For the written put, $S_T-K+p=0$, giving the same price. If $p>K$, then $K-p<0$, outside the possible range $S_T\geq0$; neither position has a zero-profit price at a nonnegative expiration price under this convention.

A Reliable Plotting Procedure

A clear graph should identify which position is being shown and whether the vertical coordinate is payoff or profit. Then use the formula to find anchor points and label the slopes. The following steps keep the payoff and profit calculations separate.

Set the axes and units.
Use $S_T$ on the horizontal axis and dollars per unit of payoff or profit on the vertical axis. Keep the same price units for $K$, $p$, and $S_T$.

Mark the strike.
At $S_T=K$, switch from the linear exercise region to the flat no-exercise region. Use the relevant piecewise equation on each side.

Find the break-even for a profit graph.
Set profit equal to zero and solve. For a put with $p\leq K$, mark $S_T=K-p$; do not label that point as zero payoff.

Calculate and connect anchor points.
Evaluate the graph’s own equation at $S_T=0$, the break-even if applicable, $S_T=K$, and a price above $K$. Connect the points with the stated straight-line segments.

Label slopes and flat levels.
For a long position, the slope below $K$ is $-1$; for a written position it is $+1$. Above $K$, the graph is horizontal at its applicable payoff or profit level.

Worked Graphs

Worked Example: Plotting a Long Put and Its Profit

A purchased put has strike $K=\$75$ and premium $p=\$9$ per unit. Construct its payoff and profit graphs by finding the values at $S_T=0$, the break-even price, the strike, and $S_T=\$90$. Label the slopes and identify which graph has zero value at the break-even.

Method. Use the long-put piecewise equations, first for payoff and then for profit. The premium is subtracted only when calculating profit.

Break-even. Here $K-p=75-9=66$, so the break-even price is $S_T=\$66$. Check that this lies below the strike: $66<75$.

Payoff coordinates. At $S_T=0$, payoff is $\max(75-0,0)=\$75$. At $S_T=66$, payoff is $\max(75-66,0)=\$9$. At $S_T=75$, payoff is $\max(75-75,0)=\$0$. At $S_T=90$, payoff is $\max(75-90,0)=\$0$. The payoff graph therefore passes through $(0,75)$, $(66,9)$, $(75,0)$, and $(90,0)$, with coordinates in dollars.

Profit coordinates. Subtract the premium of $\$9$ from each payoff. The corresponding profits are $\$66$, $\$0$, $-\$9$, and $-\$9$. Thus the profit graph passes through $(0,66)$, $(66,0)$, $(75,-9)$, and $(90,-9)$.

Slopes and labels. Both graphs have slope $-1$ for $0\leq S_T<75$ and slope $0$ for $S_T\geq75$. Label $K=\$75$ on both horizontal axes and mark the profit break-even at $\$66$. The payoff at the break-even is $\$9$, whereas the profit there is zero. This distinction prevents the common error of placing $(66,0)$ on the payoff graph.

Check. At $S_T=\$75$, the put is worth zero and the long position’s profit is $-\$9$, exactly the premium paid. Above the strike, both values remain at those levels.

Worked Example: Plotting a Written Put

An investor writes a put with $K=\$120$ and receives $p=\$8$ per unit. Construct the payoff and profit graphs at $S_T=0$, the break-even, $S_T=120$, and $S_T=150$. State the region of loss and the slopes.

Method. Apply the written-put equations: the payoff is the negative of the holder’s exercise value, and the premium received is added to obtain profit.

Break-even. Setting the below-strike profit equal to zero gives $S_T-K+p=0$. Hence $S_T=120-8=\$112$, which is below $K=\$120$.

Payoff coordinates. At $S_T=0$, payoff is $0-120=-\$120$. At $S_T=112$, it is $112-120=-\$8$. At $S_T=120$, payoff is zero, and at $S_T=150$, it remains zero. The graph passes through $(0,-120)$, $(112,-8)$, $(120,0)$, and $(150,0)$.

Profit coordinates. Add the received premium of $\$8$ to each payoff. The profits are $-\$112$, $\$0$, $\$8$, and $\$8$, respectively. The profit graph passes through $(0,-112)$, $(112,0)$, $(120,8)$, and $(150,8)$.

Slopes and interpretation. Below $K$, both graphs rise with slope $+1$; above $K$, payoff is horizontal at zero and profit is horizontal at $+\$8$. The written put loses money for $S_T<\$112$, breaks even at $\$112$, and earns a positive profit above $\$112$. At the break-even, the payoff is $-\$8$, not zero; the premium received offsets that negative payoff.

Check. At $S_T=0$, the writer owes the full strike amount of $\$120$ through exercise, but has received $\$8$; net profit is $-120+8=-\$112$, consistent with the graph.

Worked Example: Comparing Long and Written Puts

A put has $K=\$50$ and $p=\$7$. One investor buys the put and another writes an otherwise identical put. Find payoff and profit coordinates for both positions at $S_T=\$20$, the break-even, $S_T=\$50$, and $S_T=\$65$. Explain why the positions share a break-even price.

Break-even. For both positions, $K-p=50-7=\$43$. Since $43<50$, this is a valid nonnegative break-even in the linear region.

Long put calculations. At $S_T=20$, payoff is $50-20=\$30$, so profit is $30-7=\$23$. At $S_T=43$, payoff is $50-43=\$7$, so profit is $7-7=\$0$. At $S_T=50$, payoff is $\$0$, so profit is $-\$7$. At $S_T=65$, payoff remains $\$0$, and profit remains $-\$7$.

Written put calculations. At $S_T=20$, payoff is $20-50=-\$30$, and profit is $-30+7=-\$23$. At $S_T=43$, payoff is $43-50=-\$7$, and profit is $-7+7=\$0$. At $S_T=50$, payoff is $\$0$, and profit is $\$7$. At $S_T=65$, payoff is still zero and profit is still $\$7$.

Coordinates and graph labels. In order of $S_T=20,43,50,65$, the long payoff values are $30,7,0,0$, and the long profit values are $23,0,-7,-7$. The written payoff values are $-30,-7,0,0$, and the written profit values are $-23,0,7,7$. Mark the strike at $\$50$ and the profit break-even at $\$43$. The long graphs have slope $-1$ below the strike, while the written graphs have slope $+1$; each is flat above the strike.

Why the break-even matches. The written position’s payoff is the negative of the long position’s payoff, and its profit adds the premium instead of subtracting it. At $S_T=43$, the long receives $\$7$ from exercise value and pays $\$7$ premium; the writer has a $\$7$ exercise loss offset by the $\$7$ premium received. Both profits are zero, although their payoffs are $+\$7$ and $-\$7$, respectively.

Check. At the strike, neither position has exercise value. The long loses the premium and the writer keeps it, giving profits $-\$7$ and $+\$7$.

Common Graphing Traps

A graph is often easiest to check by substituting an anchor price into the exact equation for the quantity being plotted. In particular, do not use the payoff equation to label a profit graph or the profit equation to label a payoff graph. Their intercepts and flat levels differ by the premium.

Putting the break-even on the payoff graph as a zero. Break-even means profit is zero. For a long put with $K=75$ and $p=9$, at $S_T=66$ the payoff is $75-66=9$, and profit is $9-9=0$. The payoff point is $(66,9)$; the profit point is $(66,0)$.
Subtracting a premium from a written position. A writer receives the premium, so profit is payoff plus $p$. For example, a written put with payoff $-\$8$ and premium received $\$3$ has profit $-8+3=-\$5$, not $-\$11$.
Drawing the wrong slope. Below the strike, the long put’s value falls by one dollar for each one-dollar increase in $S_T$, giving slope $-1$. The written put has the opposite payoff and profit movement, with slope $+1$.
Confusing the strike with the break-even. The strike is where the payoff begins its flat region. The profit break-even is $K-p$, usually below the strike for a positive premium.
Mixing units. If $K$, $p$, and $S_T$ are quoted per share, graph values are per share too. A contract multiplier changes total contract profit but does not change these per-unit break-even and slope calculations.

These graphs are useful because a quick visual check exposes sign errors: a long put should benefit as the underlying price falls, while a written put’s profit should move in the opposite direction below the strike. They also separate the exercise-based payoff from the premium-adjusted profit, a distinction needed in multi-step option problems aligned with the SOA Exam FM (Financial Mathematics) syllabus.

Check Your Understanding

For each problem, identify payoff and profit separately. Unless stated otherwise, use dollars per unit and the expiration-profit convention in this tutorial.

A long put has $K=\$64$ and $p=\$6$. Find the break-even price, payoff and profit at $S_T=\$40$, and the profit at $S_T=\$64$. (Answer: Break-even $\$58$; at $\$40$, payoff $\$24$ and profit $\$18$; at $\$64$, profit $-\$6$.)
A written put has $K=\$90$ and $p=\$12$. Find its break-even and its payoff and profit at $S_T=\$60$ and $S_T=\$90$. (Answer: Break-even $\$78$; at $\$60$, payoff $-\$30$ and profit $-\$18$; at $\$90$, payoff $\$0$ and profit $\$12$.)
A long put has $K=\$48$ and $p=\$5$. Give its payoff and profit at $S_T=0,\$43,\$48,$ and $\$60$, and state the slopes below and above the strike. (Answer: Payoffs $\$48,\$5,\$0,\$0$; profits $\$43,\$0,-\$5,-\$5$; slopes $-1$ below the strike and $0$ above it.)
A written put has $K=\$150$ and $p=\$11$. Give its payoff and profit at $S_T=0,\$139,\$150,$ and $\$180$, and identify its loss region. (Answer: Payoffs $-\$150,-\$11,\$0,\$0$; profits $-\$139,\$0,\$11,\$11$; loss when $S_T<\$139$.)
A long put’s profit graph is horizontal at $-\$6$ for every $S_T\geq K$, and its break-even is $S_T=\$84$. Recover the premium and strike. (Answer: Premium $p=\$6$; since $K-p=84$, strike $K=\$90$.)

What You'll Learn