Why Compare These Two Positions?
An option’s expiration payoff depends on the terminal price of its underlying. A call gives its holder the right to buy at the strike, while a put gives its holder the right to sell at the strike. When each option is paired with a different set of assets, the separate payoff diagrams can combine into the same total cash flow.
The comparison in this tutorial is between a long call plus a risk-free bond that pays the strike at expiration and a long put plus one share of the underlying. The key is to compare their total expiration payoffs in every possible terminal-price region, including exactly at the strike. This is an equality of terminal cash flows, not a statement about the positions’ initial costs or profits.
The bond’s relevant amount in this comparison is its payment at expiration, not its market price before expiration. No interest-rate calculation is needed to compare these expiration payoffs. The calculation assumes that the options have the same underlying, strike, and expiration, and that the share and bond payment are measured in the same currency. It considers the terminal cash flows specified here, without adding any separate cash flows.
Compare the Payoffs by Terminal-Price Region
There are three cases to check: the share price finishes below the strike, exactly at the strike, or above it. In each case, calculate both components of each position, then add them. At the strike, neither option produces a positive payoff; that boundary still needs to be checked rather than inferred from one side.
| Terminal price | Call plus bond | Put plus share | Common payoff |
|---|---|---|---|
| \(S_T<K\) | \(0+K=K\) | \((K-S_T)+S_T=K\) | \(K\) |
| \(S_T=K\) | \(0+K=K\) | \(0+K=K\) | \(K\) |
| \(S_T>K\) | \((S_T-K)+K=S_T\) | \(0+S_T=S_T\) | \(S_T\) |
The comparison gives a common payoff of \(K\) when the share finishes at or below the strike, and \(S_T\) when it finishes above the strike. Thus, the common payoff is \(\max(K,S_T)\). That compact expression is a summary of the three cases, not a substitute for checking which option pays in each region.
Why the Payoffs Must Match
The reason is that the option’s positive payoff fills the amount needed to bring the other asset or payment up to the larger of \(S_T\) and \(K\). If the share is below \(K\), the bond already pays \(K\); the put adds exactly the shortfall \(K-S_T\) to the share’s terminal value. If the share is above \(K\), the share already has value \(S_T\); the call adds exactly the amount \(S_T-K\) to the bond payment.
The call pays \(\max(S_T-K,0)\), and the put pays \(\max(K-S_T,0)\).
Add \(K\) for the bond to the call payoff; add \(S_T\) for the share to the put payoff.
For \(S_T<K\), the call is zero and the put is \(K-S_T\). For \(S_T>K\), the put is zero and the call is \(S_T-K\). At \(S_T=K\), both option payoffs are zero.
Below or at \(K\), each position pays \(K\); above \(K\), each pays \(S_T\). The totals therefore agree in every case.
This case argument is a direct justification: it covers every possible \(S_T\), because a real terminal price must be below, equal to, or above \(K\). The equality is therefore not based on a likely price movement or on an average outcome. It holds state by state at expiration.
Worked Example: Checking All Three Regions
A European call and a European put are written on the same share, each with strike \(K=\$96\) and the same expiration. Compare one call plus a bond paying \(\$96\) at expiration with one put plus one share when the terminal share price is (a) \(\$71\), (b) \(\$96\), and (c) \(\$124\). Report each total expiration payoff.
Method. For each terminal price, compute the call payoff and add \(\$96\). Separately compute the put payoff and add the terminal share price. This checks the below-strike, boundary, and above-strike cases directly.
(a) \(S_T=\$71\). Since \(71<96\), the call payoff is \(\max(71-96,0)=\$0\), so call plus bond pays \(0+96=\$96\). The put payoff is \(\max(96-71,0)=\$25\), so put plus share pays \(25+71=\$96\). The two totals agree.
(b) \(S_T=\$96\). Both option payoffs are zero: \(\max(96-96,0)=0\) for the call and \(\max(96-96,0)=0\) for the put. Call plus bond pays \(0+96=\$96\), and put plus share pays \(0+96=\$96\). Equality holds at the boundary.
(c) \(S_T=\$124\). The call payoff is \(\max(124-96,0)=\$28\), so call plus bond pays \(28+96=\$124\). The put payoff is \(\max(96-124,0)=\$0\), so put plus share pays \(0+124=\$124\).
Check and trap avoided. The matched totals are \(\$96,\$96,\$124\), as the common payoff \(\max(K,S_T)\) predicts. Do not add the strike payment to the put-and-share position as well: in this comparison, that position contains the share, not the bond.
Scaling the Comparison to Several Units
The equality also holds when both positions are held in the same number of units. If there are \(n\) matched units, each containing one call and a bond paying \(K\), the first position’s payoff is \(n[\max(S_T-K,0)+K]\). The matching position has \(n\) puts and \(n\) shares, with payoff \(n[\max(K-S_T,0)+S_T]\). Since the per-unit totals agree for every terminal price, multiplying both by the same \(n\) preserves equality.
Worked Example: Comparing Four Matched Units
A portfolio holds four calls on a share with strike \(K=\$45\), together with four risk-free bonds each paying \(\$45\) at expiration. Compare it with four puts of the same strike and expiration together with four shares. Find each portfolio’s total expiration payoff when \(S_T=\$39\), \(S_T=\$45\), and \(S_T=\$58\).
Method. Calculate the payoff per matched unit in each region, then multiply each total by four. The unit count applies to the call, bond, put, and share together; changing the number of units on only one component would describe a different position.
At \(S_T=\$39\). Per unit, call plus bond pays \(\max(39-45,0)+45=0+45=\$45\). Put plus share pays \(\max(45-39,0)+39=6+39=\$45\). For four units, each portfolio pays \(4(45)=\$180\).
At \(S_T=\$45\). Per unit, the call payoff and put payoff are both zero. Each position pays \(\$45\) per unit, so each four-unit portfolio pays \(4(45)=\$180\).
At \(S_T=\$58\). Per unit, call plus bond pays \(\max(58-45,0)+45=13+45=\$58\). Put plus share pays \(\max(45-58,0)+58=0+58=\$58\). For four units, each portfolio pays \(4(58)=\$232\).
Check and trap avoided. Both portfolios produce \(\$180,\$180,\$232\) across the three cases. A common scaling error is to multiply the option payoff by four but count the bond or shares only once. Every component in this example is held in four matching units.
Worked Example: A Non-Integer Terminal Price
A pair of European options has strike \(K=\$82\). One position consists of three calls and three bonds paying \(\$82\) each at expiration. The other consists of three puts and three shares. Calculate and compare both total expiration payoffs if the share finishes at \(S_T=\$97.50\).
Method. Because \(\$97.50>\$82\), the call is in the money and the put has zero payoff. Calculate one matched unit first, then multiply by three.
For one call, the payoff is \(\max(97.50-82,0)=\$15.50\). Adding the bond payment gives \(15.50+82=\$97.50\). For one put, the payoff is \(\max(82-97.50,0)=\$0\). Adding one share gives \(0+97.50=\$97.50\).
For three matched units, the call-and-bond position pays \(3(97.50)=\$292.50\). The put-and-share position also pays \(3(97.50)=\$292.50\).
Check and trap avoided. Both totals equal three times the terminal share price because the share finishes above the strike. The bond contributes its \(\$82\) maturity payment, not its price at an earlier date.
Common Errors and What This Equality Does Not Say
A correct payoff comparison depends on matching the terms and counting each component consistently. These checks help keep terminal cash flows separate from other option calculations.
- Using the wrong region. Below the strike, the call is zero and the put pays \(K-S_T\). Above the strike, the put is zero and the call pays \(S_T-K\).
- Skipping the boundary. At \(S_T=K\), both options pay zero. Each whole position still pays \(K\), through the bond in one position and the share in the other.
- Confusing a bond’s face value with its earlier price. The bond contributes its specified maturity payment \(K\) at expiration. Its price at another date is not an expiration payoff.
- Comparing profit instead of payoff. Option premiums and other initial cash flows are not included here. A profit calculation must account for those separately.
- Failing to match the contracts. The options must have the same underlying, strike, expiration, and unit basis, and the paired bonds and shares must be counted in corresponding amounts.
The result established here concerns terminal cash flows only: the two matched positions pay the same amount for every possible \(S_T\). It does not, by itself, calculate either position’s initial cost or an investor’s profit. Those require valuation at the relevant date, a separate step from the expiration-payoff comparison.
Check Your Understanding
For each problem, compare the complete expiration payoffs of the specified matched positions.
- A call and put share strike \(K=\$74\). A bond pays \(\$74\) at expiration. If \(S_T=\$61\), calculate call plus bond and put plus one share. (Answer: \(\$74\) for each position.)
- For strike \(K=\$108\), compare five matched units of the two positions when \(S_T=\$137\). (Answer: \(\$685\) for each position.)
- For strike \(K=\$52\), calculate the total payoff of six matched units at each of \(S_T=\$44,\$52,\$69\). (Answer: \(\$312,\ \$312,\ \$414\) for each position, respectively.)
- A pair of options has strike \(K=\$160\). Two puts and two shares are held in one position; the matching position has two calls and two bonds each paying \(\$160\). Find both payoffs when \(S_T=\$145\), and state which region applies. (Answer: Below the strike; \(\$320\) for each position.)
- A call and put have strike \(K=\$83\). Three matched units are held, and \(S_T=\$101\). Calculate both portfolio payoffs, showing the per-unit amount first. (Answer: \(\$101\) per unit and \(\$303\) for each portfolio.)