Updated for the 2026-2027 CFA® Level I curriculum.
Put-call parity links the price of a European call, a European put, the underlying asset, and a risk-free bond. When the call and put share the same strike price and expiration, their prices must satisfy one equation. If they do not, an arbitrage opportunity exists. This relationship shows up in Level I as both a conceptual question and a calculation question, so you need to know the formula and what it means.
Quick Answer
Put-call parity states that a fiduciary call (a call plus a risk-free bond paying the strike at expiration) has the same payoff as a protective put (the underlying plus a put). The formula is . If market prices do not match this equation, traders can build a synthetic position to profit from the mispricing without taking on price risk.
Key Takeaways About Put-Call Parity for European Options
Put-call parity applies only to European options with the same underlying, strike price, and expiration date.
The core equation is , where is the present value of the strike price.
A fiduciary call and a protective put produce identical payoffs at expiration, which is why their current prices must be equal.
Rearranging the formula creates synthetic calls, puts, underlying positions, and risk-free bonds.
Parity lets you solve for any one price (call, put, stock, or bond) if you know the other three.
If observed prices violate parity, an arbitrage profit is available until prices adjust.
Mixing up strike prices, maturities, or option style (American vs. European) breaks the relationship entirely.
What You Need to Know for CFA Level I
Recognize when a call and put qualify for put-call parity (same underlying, strike, and expiration).
Apply the formula to solve for any missing variable.
Explain why a protective put and a fiduciary call must have equal value.
Derive synthetic positions (synthetic call, put, underlying, and bond) from the parity equation.
Identify when parity is violated and describe the arbitrage trade that would correct it.
What Is Put-Call Parity?
Put-call parity is a pricing relationship, not a trading strategy. It says that once you fix the underlying, the strike price, and the expiration date, the prices of a European call and a European put are connected. You cannot set them independently.
The idea rests on two portfolios with identical payoffs at expiration:
Fiduciary call. Buy a call option and buy a risk-free, zero-coupon bond that pays the strike price at expiration.
Protective put. Buy the underlying asset and buy a put option with the same strike and expiration.
At expiration, both portfolios pay off exactly the same amount, no matter what the underlying price does. If two portfolios always produce the same payoff, they must have the same price today. that is the entire logic behind parity.
Put-Call Parity Formula
For a European call and put on a non-dividend-paying underlying, with the same strike price and expiration:
Where:
= price of the European call option
= price of the European put option
= current price of the underlying asset
= price of the underlying asset at expiration
= strike price (common to both options)
= present value of the strike price, discounted at the risk-free rate over the life of the options
The left side is the fiduciary call. The right side is the protective put. Because both sides produce the same payoff at expiration, they must cost the same today.
Why the Parity Relationship Holds
The relationship holds because of no-arbitrage pricing. If two positions guarantee identical future payoffs, they must trade at the same price now. If they didn't, you could sell the expensive position, buy the cheap one, and lock in a risk-free profit.
Check the payoffs at expiration under both possible outcomes:
Outcome | Fiduciary Call Payoff ( + bond) | Protective Put Payoff () |
|---|---|---|
Underlying price above strike | Call pays , bond pays → total = | Stock is worth , put pays 0 → total = |
Underlying price at or below strike | Call pays 0, bond pays → total = | Stock is worth , put pays → total = |
In both cases, the two portfolios pay exactly the same amount. That equality of payoffs is what forces equality of price today.
Synthetic Positions From Put-Call Parity
Once you accept the parity equation, you can rearrange it to isolate any single position. This creates four synthetic positions.
Rearranged Formula | Synthetic Position | Built From |
|---|---|---|
Synthetic call | Long put, long underlying, short bond | |
Synthetic put | Long call, short underlying, long bond | |
Synthetic underlying (stock) | Long call, short put, long bond | |
Synthetic bond | Long put, long underlying, short call |
Each synthetic position replicates the payoff of the actual instrument. This matters on the exam because questions often ask you to identify which combination of instruments recreates a specific payoff without naming it directly.
Using Put-Call Parity to Find a Missing Price
If you know three of the four variables in the parity equation, you can solve for the fourth. This is the most common Level I application.
Setup:
Stock price = $95
Strike price = $100
Call price = $8
Risk-free rate = 4%, options expire in 1 year
Step 1: Find the present value of the strike price.
Step 2: Rearrange the parity formula to solve for the put.
Given the call price, stock price, and risk-free rate, the put must be priced at $9.15 to prevent arbitrage. If the put trades at any other price in the market, one side of the parity equation is mispriced relative to the other.
Put-Call Parity and Arbitrage
Parity holds through market forces. If the equation does not balance, an arbitrage trade brings prices back in line.
Using the example above, suppose the put actually trades at $10.50 instead of the theoretical $9.15. The put is overpriced relative to parity.
Arbitrage response:
Sell the overpriced put for $10.50.
Buy the synthetic put instead: buy the call for $8, short the stock for $95, and invest PV = $96.15 in a risk-free bond.
Net cost of the synthetic put = 8 − 95 + 96.15 = $9.15.
Profit = $10.50 − $9.15 = $1.35 per option, locked in today with no exposure to where the stock price ends up.
This trade works because the payoffs offset exactly at expiration, regardless of the stock price. The $1.35 is captured immediately and does not depend on market movement. As more traders exploit this gap, the put price falls and the call, stock, or bond prices adjust until parity holds again.
Common Exam Traps
Using American options in the strict equation
Put-call parity as shown here applies only to European options. American options allow early exercise, which breaks the exact equality. Level I tests this distinction directly.
Forgetting to discount the strike price
A common error is plugging in instead of . This overstates or understates the missing variable and produces a wrong answer that still looks reasonable.
Mixing strikes or maturities
Parity only holds when the call and put share the identical strike price and expiration date. If a question uses different strikes or maturities, the equation does not apply.
Rearranging with the wrong sign
Solving for or requires careful sign management. Flipping a sign during rearrangement is the most common calculation mistake on this topic.
Practice Questions
A European call option on a non-dividend-paying stock is priced at $8. The stock currently trades at $95. Both the call and a European put with the same expiration have a strike price of $100. The risk-free rate is 4%, and both options expire in one year. Using put–call parity, the price of the put option is closest to:
$6.85
$9.15
$13.00
Correct Answer: B
Calculation:
Put-call parity requires . Solving for isolates the put price using the discounted strike, the stock price, and the call price. The correct present-value adjustment produces $9.15.
Option A: Adds the stock price instead of subtracting it, reversing the correct sign in the rearranged formula.
Option C: Uses the undiscounted strike price ($100) instead of its present value, overstating the put price.
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FAQs About Put-Call Parity for European Options
What is put-call parity in simple terms?
It is the pricing rule that connects a European call, a European put, the underlying asset, and a risk-free bond that share the same strike and expiration. Once three prices are known, the fourth is fixed.
Does put-call parity apply to American options?
No. The exact equation applies only to European options. Early exercise on American options breaks the equality shown here.
What happens if put-call parity is violated?
An arbitrage opportunity exists. Traders can build a synthetic position and an opposite actual position to lock in a riskless profit until prices adjust back to parity.