Solving Option Prices with Put–Call Parity

Using Put–Call Parity to Find an Unknown

Put–call parity is useful not only as a relationship among option prices, but also as a direct method for finding a missing price. If the stock price, exercise price, maturity, interest rate, and one option price are known, parity can determine the other option price. It can also be rearranged to solve for the stock price or the present value of the exercise price.

The key is to keep the terms in their proper places. The stock price and option prices are values today; the exercise price is paid at expiration, so it must be discounted before it is compared with those current values.

Definition: For a non-dividend-paying stock, a European call and European put on the same stock, with the same exercise price $K$ and expiration time $T$, satisfy $C_0+PV(K)=P_0+S_0$. Here $C_0$, $P_0$, and $S_0$ are the call price, put price, and stock price today, and $PV(K)$ is the present value of the exercise price. With an effective annual risk-free rate $i$, $PV(K)=K/(1+i)^T$. This form assumes frictionless markets, no arbitrage, and the same risk-free borrowing and lending rate. It does not apply in this form when the stock pays dividends before expiration or when the options have different exercise prices or expiration dates.

Rearrange Before Substituting

Write the identity first, identify the unknown, and then isolate it. This prevents a common sign error: the call and put do not enter the equation with the same sign when one is isolated.

$$ C_0+PV(K)=P_0+S_0 $$

\begin{aligned} C_0 &= P_0+S_0-PV(K),\\ P_0 &= C_0+PV(K)-S_0,\\ S_0 &= C_0+PV(K)-P_0,\\ PV(K) &= P_0+S_0-C_0. \end{aligned}

When the interest rate and time to expiration are given, calculate $PV(K)$ before using these rearrangements. Under the effective annual rate convention, $T$ is measured in years. For instance, if expiration is in six months, use $T=0.5$; do not treat six months as one year.

Worked Example: Solve for a Call Price

A non-dividend-paying stock is at $S_0=\$102.00$. A European put with exercise price $K=\$100$ and expiration in one year is quoted at $P_0=\$8.40$. The matching European call has an unknown price. The effective annual risk-free rate is $i=4\%$, with annual compounding. Find the call price under put–call parity.

Method and assumptions. The options have the same stock, exercise price, and expiration, and the stock pays no dividends. Apply $C_0=P_0+S_0-PV(K)$. All option and stock prices are dollars per share.

Step 1: Discount the exercise price. With $T=1$,

PV(K)=\frac{100}{(1.04)^1} =\frac{100}{1.04} =\$96.153846\ldots

Step 2: Substitute in the rearranged identity.

C_0=8.40+102.00-96.153846\ldots =\$14.246153\ldots

Step 3: Check. Using the unrounded value, $C_0+PV(K)=14.246153\ldots+96.153846\ldots=\$110.40$. The other side is $P_0+S_0=8.40+102.00=\$110.40$. The identity balances. Rounded to the nearest cent, the call price is \$14.25.

Common mistake avoided. The exercise price is paid at expiration, not today. Subtracting $K=\$100$ in place of its present value would give $8.40+102.00-100=\$10.40$, which fails the parity check.

Why the Rearrangements Work

Each rearrangement is ordinary algebra applied to an equality. The equality itself has a no-arbitrage justification: at expiration, the difference between the call payoff and put payoff is exactly the stock price minus the exercise price.

\max(S_T-K,0)-\max(K-S_T,0)=S_T-K

To see this, consider the two possible regions. If $S_T\ge K$, the call pays $S_T-K$ and the put pays zero. If $S_T<K$, the call pays zero and the put pays $K-S_T$, so call payoff minus put payoff is $-(K-S_T)=S_T-K$ again. Thus a long call and short put have the same expiration payoff as a long stock position combined with borrowing an amount whose repayment is $K$ at expiration. The amount borrowed today is $PV(K)$.

If two portfolios have identical payoffs at expiration, they must have the same value today in a frictionless, no-arbitrage market. Otherwise, an investor could buy the cheaper portfolio and sell the more expensive one, leaving the same future payoff on both sides and a price difference today. Equating the values gives $C_0-P_0=S_0-PV(K)$, which rearranges to put–call parity. This payoff argument also explains why discounting the exercise price is essential.

Confirm the contract match.
Check that the call and put are European options on the same non-dividend-paying stock, with the same exercise price and expiration.

Find the present value needed.
If $K$, $i$, and $T$ are supplied, calculate $K/(1+i)^T$. Keep the rate and time units consistent.

Isolate the requested quantity.
Rearrange $C_0+PV(K)=P_0+S_0$ before inserting the data.

Check and round.
Substitute unrounded values into both sides of parity. Round the requested price only after the check.

Solving for a Put or the Stock Price

The same procedure applies when a put or stock price is unknown. A useful check is to return the calculated value to the original identity, rather than relying only on whether the result seems plausible. The following examples use effective annual rates, with $T$ stated in years.

Worked Example: Solve for a Put Price at a Two-Year Expiration

A non-dividend-paying stock is priced at $S_0=\$74.00$. A European call with $K=\$80$ and expiration in two years costs $C_0=\$9.25$. The matching European put price is unknown. The effective annual risk-free rate is $i=5\%$, compounded annually. Find the put price.

Method. Rearrange parity as $P_0=C_0+PV(K)-S_0$, and discount the exercise price for the full two-year term.

Step 1: Calculate the present value.

PV(K)=\frac{80}{(1.05)^2} =\frac{80}{1.1025} =\$72.562358\ldots

Step 2: Solve for the put.

P_0=9.25+72.562358\ldots-74.00 =\$7.812358\ldots

Step 3: Check. The left side of parity is $9.25+72.562358\ldots=\$81.812358\ldots$. The right side is $7.812358\ldots+74.00=\$81.812358\ldots$. Rounded to the nearest cent, $P_0=\mathbf{\$7.81}$.

Common mistake avoided. Discounting for only one year would use $80/1.05=\$76.190476\ldots$ and produce a different put price. The stated expiration is two years away, so the exponent must be $T=2$.

Worked Example: Solve for the Stock Price

A European call and put on the same non-dividend-paying stock both expire in one year and have $K=\$110$. Their prices are $C_0=\$12.60$ and $P_0=\$7.10$. The effective annual risk-free rate is $i=5\%$. Find the stock price today.

Method. Isolate the stock price: $S_0=C_0+PV(K)-P_0$. The stock is not the same quantity as the exercise price, so do not substitute $K$ for $S_0$.

Step 1: Discount the exercise price for one year.

PV(K)=\frac{110}{1.05} =\$104.761904\ldots

Step 2: Calculate the stock price.

S_0=12.60+104.761904\ldots-7.10 =\$110.261904\ldots

Step 3: Check. $C_0+PV(K)=12.60+104.761904\ldots=\$117.361904\ldots$, and $P_0+S_0=7.10+110.261904\ldots=\$117.361904\ldots$. Rounded to the nearest cent, $S_0=\mathbf{\$110.26}$.

Common mistake avoided. Using $S_0=P_0+PV(K)-C_0$ reverses both option signs. The correct rearrangement comes from subtracting $P_0$ from both sides and then subtracting $C_0$.

Inferring the Present Value of the Exercise Price

Sometimes the option and stock prices are known, and the requested quantity is $PV(K)$. In that case, parity gives the present value directly. If the exercise price, interest rate, and expiration are also provided, use them to verify that the inferred present value is consistent with the discounting assumptions.

Worked Example: Infer and Verify the Present Value

A non-dividend-paying stock is priced at $S_0=\$114.30$. Matching European options have exercise price $K=\$108$ and expire in one year. The call costs $C_0=\$26.80$, and the put costs $P_0=\$12.50$. The effective annual risk-free rate is $i=8\%$. Infer $PV(K)$ from the prices and verify it using the rate.

Method. Rearrange parity as $PV(K)=P_0+S_0-C_0$. Then independently calculate the discounted exercise price as a check.

Step 1: Infer the present value from the quoted prices.

PV(K)=12.50+114.30-26.80=\$100.00

Step 2: Verify using the stated rate and maturity.

\frac{K}{(1+i)^T} =\frac{108}{(1.08)^1} =\$100.00

Step 3: Check parity. The call side is $26.80+100.00=\$126.80$; the put-and-stock side is $12.50+114.30=\$126.80$. The inferred value matches the stated discounting assumptions exactly.

Common mistake avoided. Using $PV(K)=K$ would give $\$108$, even though the exercise price is paid one year from now. The correct present value is $\$100$.

Common Exam Traps

Parity calculations are short, but they can be derailed by a sign, timing, or contract mismatch. Before accepting a result, check the structure as well as the arithmetic.

Wrong sign when isolating a price. For a call, $C_0=P_0+S_0-PV(K)$; for a put, $P_0=C_0+PV(K)-S_0$. Write the original equality before moving terms.
Using the exercise price instead of its present value. The exercise price is paid at expiration. Discount it to today using the specified rate and time.
Misreading the time unit. Under an effective annual rate, express $T$ in years. A two-year expiration requires $(1+i)^2$, not $(1+i)$.
Rounding too early. Carry extra digits through the present-value and parity calculations, and round the final requested price to cents.
Applying parity to mismatched contracts. The call and put must have the same underlying, exercise price, and expiration, and this version assumes no dividends.

A final equality check is often the fastest way to detect a mistake. Substitute the unrounded answer into $C_0+PV(K)=P_0+S_0$. If the two sides differ beyond rounding, revisit the rearrangement, discounting period, or inputs.

Check Your Understanding

For each problem, use the stated effective annual rate and assume the options satisfy the conditions for European put–call parity. Round final dollar amounts to the nearest cent.

A non-dividend-paying stock has $S_0=\$95.00$. A one-year European put with $K=\$100$ costs $P_0=\$6.80$. If $i=5\%$, find the matching call price. (Answer: $\$6.56$)
A stock has $S_0=\$142.00$, and a European call with $K=\$150$ costs $C_0=\$17.50$. The matching put expires in two years, and $i=4\%$. Find its price. (Answer: $\$14.18$)
Matching one-year European options have $K=\$75$, $C_0=\$8.90$, and $P_0=\$13.40$. If $i=6\%$, find the stock price. (Answer: $\$66.25$)
A non-dividend-paying stock is at $S_0=\$114.30$; matching one-year options have $C_0=\$26.80$ and $P_0=\$12.50$. If $K=\$108$ and $i=8\%$, infer $PV(K)$ from the option prices and verify it by discounting. (Answer: $\$100.00$)
A non-dividend-paying stock is at $S_0=\$131.50$. A European put with $K=\$120$ and expiration in three years costs $P_0=\$14.20$. If $i=4\%$, find the matching call price. (Answer: $\$39.02$)

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