Reading a Call Graph
A call graph shows how a call option’s value at expiration, or the profit associated with the position, changes as the underlying asset’s expiration price changes. The graph is piecewise linear: the strike price separates the region where the call expires without value from the region where its exercise value increases with the asset price. Adding or subtracting the premium shifts the profit graph vertically. The graph is a compact way to display the exercise threshold, break-even point, and direction of change for each position.
This tutorial uses a per-share graph for a call held to expiration. It is aligned with the SOA Exam FM (Financial Mathematics) syllabus. The horizontal axis is the underlying asset’s price at expiration, and the vertical axis is dollars per share. Transaction costs, taxes, and interest earned or paid on the premium are ignored. The profit convention here is expiration payoff minus the premium for a purchased call, or premium received minus the call payoff for a written call.
A payoff graph records the option’s expiration cash flow, without including the premium paid or received when the position was opened. A profit graph includes that premium. For a long call, profit is therefore the payoff graph shifted down by \(c\). For a written call, profit is the payoff graph shifted up by \(c\). Do not confuse the strike, where the payoff changes direction, with the break-even point, where profit equals zero.
Build the Graph in Labeled Steps
For either position, first draw and label the axes: \(S_T\) on the horizontal axis and dollars per share on the vertical axis. Mark \(K\) on the horizontal axis. The payoff is zero at and below \(K\), and its magnitude changes at a constant rate above \(K\). To graph profit, apply the premium shift and mark the horizontal zero-profit line. For a purchased call, the profit crosses that line at \(K+c\); the written call has the same break-even price under this convention.
Place \(K\) on the horizontal \(S_T\)-axis. It is the point at which the call’s payoff changes from zero to a positive amount for the buyer.
For a long call, use zero payoff through \(K\), then a line rising one dollar for each one-dollar increase in \(S_T\). For a written call, reflect that payoff across the horizontal axis.
Subtract \(c\) from the long-call payoff at every price, or add \(c\) to the written-call payoff at every price.
Draw the zero-profit line, mark \(K+c\), and calculate a value on each side of the strike and break-even point to confirm the segments.
The slopes are useful checks on the drawing. As \(S_T\) rises below \(K\), neither the long nor written call’s payoff changes, so both payoff slopes are zero there. Above \(K\), the long-call payoff slope is \(+1\), and the written-call payoff slope is \(-1\). Shifting a line vertically to include the premium changes its height, not its slope. Thus the profit slopes are also zero below \(K\), then \(+1\) for the long call and \(-1\) for the written call above \(K\).
Why the Graph Has These Segments
At expiration, exercising a call means paying \(K\) to receive an asset worth \(S_T\). If \(S_T\leq K\), exercise does not produce a positive exercise value, so the call payoff is zero. If \(S_T>K\), the value received in excess of the exercise price is \(S_T-K\). These two cases give the long-call payoff formula. The writer has the opposite expiration cash flow, so the written-call payoff is the negative of the long-call payoff.
The profit formulas follow by including the initial premium under the stated convention. For a buyer, the premium is a cost; for a writer, it is a receipt. Above the strike, setting either position’s profit to zero gives the same threshold:
This solution is in the region \(S_T>K\), as required for using the above-strike profit expressions, because \(c>0\) implies \(K+c>K\). Below the strike, the long call’s profit is \(-c\), while the writer’s profit is \(c\); neither is zero under the assumption of a positive premium. The break-even point is consequently to the right of the strike on the horizontal axis.
Worked Examples
Worked Example: Plotting a Purchased Call’s Payoff and Profit
A hypothetical purchased call has strike \(K=\$50\) and premium \(c=\$6.50\) per share. Construct its payoff and profit graph, labeling the strike, break-even point, zero-profit line, and slopes. Also calculate the payoff and profit at \(S_T=\$65\).
Method. First construct the payoff using the exercise value, then subtract the premium at every price to obtain the profit. This avoids treating the strike itself as the profit break-even point.
The strike is \(\$50\), and the break-even price is \(K+c=\$50+\$6.50=\$56.50\). For \(S_T\leq \$50\), payoff is \(\$0\) and profit is \(\$0-\$6.50=-\$6.50\). Above \(\$50\), payoff is \(S_T-\$50\), and profit is \(S_T-\$50-\$6.50=S_T-\$56.50\). The payoff and profit slopes are both zero below the strike and \(+1\) above it.
| \(S_T\) | Payoff: \(\max(S_T-50,0)\) | Profit: payoff \(-6.50\) |
|---|---|---|
| \(\$0\) | \(\$0\) | \(-\$6.50\) |
| \(\$50\) | \(\$0\) | \(-\$6.50\) |
| \(\$56.50\) | \(\$6.50\) | \(\$0\) |
| \(\$65\) | \(\$15\) | \(\$8.50\) |
To check the requested value at \(\$65\), the payoff is \(\$65-\$50=\$15\), and profit is \(\$15-\$6.50=\$8.50\). Plot the payoff through \((50,0)\), then rising; plot the profit through \((50,-6.50)\) and \((56.50,0)\), then rising. The calculation confirms that the profit is zero at the labeled break-even point.
Worked Example: Plotting a Written Call
A hypothetical investor writes a call with \(K=\$72\) and receives \(c=\$4.80\) per share. Construct the payoff and profit segments and find the writer’s profit if \(S_T=\$84\).
Method. Negate the buyer’s payoff to obtain the written-call payoff, then add the premium received. This prevents the common sign error of drawing a written call with the buyer’s upward-sloping payoff.
For \(S_T\leq \$72\), the call expires without exercise: written payoff is \(\$0\), and writer profit is \(\$4.80\). For \(S_T>\$72\), written payoff is \(-\left(S_T-\$72\right)\), and profit is \(\$4.80-\left(S_T-\$72\right)\). The break-even price is \(K+c=\$72+\$4.80=\$76.80\). The payoff and profit are flat below the strike; above it, both have slope \(-1\).
| \(S_T\) | Written payoff: \(-\max(S_T-72,0)\) | Writer profit: \(4.80-\max(S_T-72,0)\) |
|---|---|---|
| \(\$60\) | \(\$0\) | \(\$4.80\) |
| \(\$72\) | \(\$0\) | \(\$4.80\) |
| \(\$76.80\) | \(-\$4.80\) | \(\$0\) |
| \(\$84\) | \(-\$12\) | \(-\$7.20\) |
At \(S_T=\$84\), the call payoff to its buyer is \(\$84-\$72=\$12\), so the writer’s payoff is \(-\$12\). The writer keeps the \(\$4.80\) premium, giving profit \(\$4.80-\$12=-\$7.20\). The plotted profit crosses zero at \(\$76.80\), as the equation confirms.
Worked Example: Comparing Payoff and Profit Graphs for Both Positions
A hypothetical call has \(K=\$30\) and premium \(c=\$2\) per share. Prepare a comparison that would allow a reader to sketch both long and written payoff and profit graphs. Identify their slopes and calculate the four values at \(S_T=\$45\).
Method. Calculate the buyer’s payoff first, use its negative for the writer’s payoff, and then apply the premium in the correct direction for each profit. This avoids confusing the opposite payoff signs with the premium adjustment.
The strike is \(\$30\); the break-even price for each profit graph is \(K+c=\$32\). For \(S_T\leq \$30\), long payoff is \(\$0\), long profit is \(-\$2\), written payoff is \(\$0\), and writer profit is \(\$2\). Above \(\$30\), the long payoff and profit rise with slope \(+1\); the written payoff and profit fall with slope \(-1\).
| \(S_T\) | Long payoff | Long profit | Written payoff | Writer profit |
|---|---|---|---|---|
| \(\$20\) | \(\$0\) | \(-\$2\) | \(\$0\) | \(\$2\) |
| \(\$30\) | \(\$0\) | \(-\$2\) | \(\$0\) | \(\$2\) |
| \(\$32\) | \(\$2\) | \(\$0\) | \(-\$2\) | \(\$0\) |
| \(\$45\) | \(\$15\) | \(\$13\) | \(-\$15\) | \(-\$13\) |
At \(\$45\), long payoff is \(\$45-\$30=\$15\), and long profit is \(\$15-\$2=\$13\). The writer’s payoff is \(-\$15\), and writer profit is \(\$2-\$15=-\$13\). The table also checks both profit graphs at \(\$32\): each is zero there, even though the two payoff values are equal in magnitude and opposite in sign.
Common Graphing Mistakes
- Putting the break-even point at the strike. The strike is where the payoff segment changes. A positive premium means profit is still \(-c\) for the buyer and \(c\) for the writer at the strike. The profit zero occurs at \(K+c\).
- Giving a written call a positive slope above \(K\). The writer’s expiration payment is the negative of the buyer’s payoff. Above the strike, written payoff and written profit therefore have slope \(-1\), not \(+1\).
- Moving the payoff graph when including the premium. Payoff does not include premium. The premium shifts the profit graph vertically; it does not change the payoff graph or the slope.
- Using the wrong sign for the premium. The buyer pays \(c\), so subtract it from payoff. The writer receives \(c\), so add it to the writer’s payoff to find profit.
A reliable sketch should show the strike and break-even as distinct horizontal-axis labels, and the zero-profit level as a horizontal line. Use one point below the strike, the strike itself, and a point above break-even as a sign and slope check. These checks are particularly useful when a problem asks for a graph rather than only a formula.
Check Your Understanding
For each problem, identify the requested graph features and use the stated profit convention. Values are per share.
- A purchased call has \(K=\$25\) and \(c=\$3.50\). State its break-even price and profit at \(S_T=\$35\). (Answer: Break-even \(\$28.50\); profit \(\$6.50\).)
- An investor writes a call with \(K=\$110\) and \(c=\$7\). Give the profit slopes below and above the strike, the break-even price, and profit at \(S_T=\$120\). (Answer: Slopes \(0\) and \(-1\); break-even \(\$117\); profit \(-\$3\).)
- A purchased call has \(K=\$64\) and \(c=\$5.25\). Give its payoff and profit at \(S_T=\$50,\ \$64,\ \$69.25,\) and \(\$80\). (Answer: Payoffs \(\$0,\ \$0,\ \$5.25,\ \$16\); profits \(-\$5.25,\ -\$5.25,\ \$0,\ \$10.75\), respectively.)
- A call with \(K=\$90\) and \(c=\$6.50\) is considered as either a long or written position. At \(S_T=\$105\), give each position’s payoff and profit, and state the common profit break-even price. (Answer: Long payoff \(\$15\), long profit \(\$8.50\); written payoff \(-\$15\), writer profit \(-\$8.50\); break-even \(\$96.50\).)
- An investor writes a call with \(K=\$42\) and \(c=\$3\). State the writer’s profit at \(S_T=\$47.25\), the point where the profit graph crosses zero, and the slope of the profit segment above the strike. (Answer: Profit \(-\$2.25\); zero at \(S_T=\$45\); slope \(-1\).)