Put-Call Parity As A Forward-Price Identity

From Put-Call Parity to a Forward Price

Put-call parity is more than a relationship between two option premiums. Its right-hand side, the stock price minus the present value of the strike, is also the current value of a forward position with that strike as its delivery price. This interpretation connects option prices to the price at which an asset can be bought or sold for delivery at a future date.

The distinction between a forward price and a forward contract’s current value matters. A forward price is the delivery price that makes a new contract worth zero at initiation. A contract with a different delivery price can have a nonzero value before expiration. Put-call parity lets us express both ideas using the prices of European calls and puts.

Definition: Consider a non-dividend-paying asset with current price $S_0$, and a forward contract requiring its buyer to pay $K$ at time $T$ in exchange for the asset. The contract’s value to the long side at time 0 is $S_0-Kv^T$, where $v=1/(1+i)$ and $i$ is a constant effective annual risk-free rate. The fair forward price $F_{0,T}$ is the delivery price that makes this value zero.

Throughout this tutorial, assume European calls and puts on the same non-dividend-paying asset, with the same strike $K$ and expiration $T$. Assume frictionless trading and that borrowing and investing are possible at the same constant risk-free rate $i$. Time $T$ is measured in years, so the discount factor is $v^T$. The results do not apply unchanged if the asset pays dividends or if the options have different exercise terms.

The Forward-Price Identity

Let $C_0$ and $P_0$ be the current call and put prices. European put-call parity under these assumptions is

$$ C_0-P_0=S_0-Kv^T. $$

The right-hand side is exactly the value of a long forward contract with delivery price $K$. The identity can therefore be written in forward form. The fair forward price $F_{0,T}$ is found by setting the value of a newly initiated contract to zero:

0=S_0-F_{0,T}v^T \qquad\Longrightarrow\qquad F_{0,T}=\frac{S_0}{v^T}=S_0(1+i)^T.

Substituting $S_0=F_{0,T}v^T$ into parity gives the forward-price identity:

C_0-P_0=v^T(F_{0,T}-K).

Thus, the call-minus-put difference is the discounted difference between the fair forward price and the options’ strike. If the strike is below the fair forward price, the difference is positive; if it is above, the difference is negative. At a strike equal to the fair forward price, call and put prices are equal under these assumptions.

Why the Identity Holds

At expiration, a long call and a short put with the same strike together have payoff $S_T-K$. A long forward with delivery price $K$ has that same payoff: the long side receives an asset worth $S_T$ and pays $K$. The stock-and-bond position consisting of one share and borrowing the present value of $K$ also has expiration value $S_T-K$. The positions therefore have matching expiration cash flows.

Under the stated assumptions, the current value of the stock-and-bond position is $S_0-Kv^T$. Matching the option position to that position gives parity, and recognizing the stock-and-bond value as the value of a long forward gives its forward interpretation.

\underbrace{C_0-P_0}_{\text{call minus put}} = \underbrace{S_0-Kv^T}_{\text{long forward value}} = \underbrace{v^T(F_{0,T}-K)}_{\text{discounted forward-price difference}}.

This is a valuation identity, not a statement that a forward contract with delivery price $K$ must be worth zero. Only a contract whose delivery price equals $F_{0,T}$ has zero value at initiation. A contract with another delivery price has value $S_0-Kv^T$ to its long side.

Worked Example: Relating a Forward Value to an Option-Price Difference

A non-dividend-paying asset is worth $120. The constant effective annual risk-free rate is 25%, and expiration is one year away. A European call with strike $100 costs $52, and the matching European put costs $12. Find the fair forward price, the value of a long forward with delivery price $100, and check parity.

Method. First calculate the one-year discount factor and fair forward price. Then compare the option-price difference with the value of the forward having delivery price $100. This avoids treating the delivery price as though it were the forward contract’s current value.

The discount factor is

v=\frac{1}{1+0.25}=0.8.

The fair forward price is

F_{0,1}=\frac{120}{0.8}=150.

A long forward with delivery price $100 has current value

$$ S_0-Kv=120-(100)(0.8)=120-80=40. $$

The option-price difference is

$$ C_0-P_0=52-12=40. $$

The forward form gives the same result:

v(F_{0,1}-K)=0.8(150-100)=0.8(50)=40.

The calculations agree: the option-price difference is $40, the current value of the long forward with delivery price $100. The fair forward price is $150; it is not the $40 value of that existing contract.

Inferring the Forward Price from Options

Rearranging the forward-price identity gives a direct way to infer a forward price from observed option prices:

F_{0,T}=K+\frac{C_0-P_0}{v^T}.

This calculation uses the strike, the call and put prices, and the risk-free discount factor. Under the assumptions of parity, it should give the same forward price as $S_0/v^T$. For a fixed maturity, matching European call-put pairs at different strikes should also imply a common forward price. If the calculations disagree, check that the options match in underlying, strike, and expiration, and that the discount factor and rate period have been used consistently.

The formula uses the difference between the call and put prices, not their sum. It also divides that difference by the discount factor: the option-price difference is a present value, whereas $F_{0,T}-K$ is a difference between prices at time $T$.

Worked Example: Calculating an Implied Forward Price

A non-dividend-paying asset is worth $120. The risk-free rate is a constant effective annual rate of 25%, and European options expire in one year. At strike $140, the call costs $20 and the put costs $12. Calculate the forward price implied by the options and check it against the spot-based forward price.

Method. Calculate the option-price difference, divide by the one-year discount factor, and add the strike. This avoids adding a present-value difference directly to the strike without first converting it to a time-$T$ amount.

The discount factor is $v=1/(1.25)=0.8$. The call-put difference is

$$ C_0-P_0=20-12=8. $$

The implied forward price is

F_{0,1}=140+\frac{8}{0.8}=140+10=150.

Independently, the spot-based forward price is

\frac{S_0}{v}=\frac{120}{0.8}=150.

The two calculations agree. As a further parity check, the call-minus-put difference predicted from spot and strike is $120-(140)(0.8)=120-112=8$, matching the quoted option prices.

Worked Example: Checking Implied Prices at Two Strikes

A non-dividend-paying asset is worth $120. The constant effective annual risk-free rate is 25%, and both European option pairs expire in one year. At strike $140, the call and put prices are $20 and $12. At strike $160, they are $12 and $20. Calculate the forward price implied by each pair and assess whether they are consistent.

Method. Apply $F_{0,T}=K+(C_0-P_0)/v^T$ separately to each strike. The common mistake is to expect the option-price difference to be identical across strikes; the identity instead requires the resulting forward price to be identical.

The discount factor is $v=0.8$. At strike $140,

C_0-P_0=20-12=8, \qquad F_{0,1}=140+\frac{8}{0.8}=150.

At strike $160,

C_0-P_0=12-20=-8, \qquad F_{0,1}=160+\frac{-8}{0.8}=160-10=150.

Both pairs imply a forward price of $150, which also agrees with the spot-based value $120/0.8=150$. The differences in the option premiums are not the same across strikes, but each difference is consistent with the same forward price.

Common Exam Trap: Mixing Present and Future Values

A frequent error is to use $C_0-P_0=F_{0,T}-K$, omitting the discount factor. The right-hand side of that incorrect expression is a time-$T$ price difference; the option premiums are time-0 values. The correct relationship is $C_0-P_0=v^T(F_{0,T}-K)$.

For example, in the previous example at strike $140, the incorrect calculation would assert that the option difference is $150-140=10$. The actual difference is

v(F_{0,1}-K)=0.8(150-140)=8,

which agrees with $20-12=8$. To recover the forward price from the option difference, divide by $v^T$, rather than adding the undiscounted difference $C_0-P_0$ directly to the strike.

A second error is to call $S_0-Kv^T$ the forward price. It is the current value of a forward contract with delivery price $K$. The fair forward price is $S_0/v^T$, the delivery price that makes the contract’s current value zero.

Check Your Understanding

Use the forward-price identity and state whether each requested value is a forward price or a current contract value.

A non-dividend-paying asset is worth $96. The effective annual rate is 4%, and European options expire in one year. At strike $100, the call costs $8.50. Find the put price using parity, rounding to the nearest cent. (Answer: $8.65)
For a one-year option pair, the effective annual rate is 25%, the strike is $125, the call costs $16, and the put costs $11. Find the forward price implied by the options. (Answer: $131.25)
A non-dividend-paying asset is worth $150. The effective annual rate is 5%, and expiration is two years away. For strike $140, calculate $C_0-P_0$ required by parity and the fair forward price. Give the option difference to four decimal places and the forward price to the nearest cent. (Answer: $C_0-P_0=\$23.0159$; $F_{0,2}=\$165.38$)
The discount factor to expiration is $0.9$. A call-put pair at strike $90 has $C_0-P_0=9$; a second pair with the same expiration at strike $110 has $C_0-P_0=-9$. Find the forward price implied by each pair and state whether they agree. (Answer: Both imply a forward price of $100.)
An asset is worth $110 and the discount factor to expiration is $0.95$. Find the current value to the long side of a forward contract with delivery price $120, and state the matching call-put price difference under parity. (Answer: The long forward value and $C_0-P_0$ are both $-\$4$.)

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