CFA Level I Exam · Option Replication Using Put-Call Parity

Replicating Options, Stock and Bonds Using Put-Call Parity

Updated 7 October 2026 · Fact-checked

Put-call parity for European options says c + X/(1+r)^T = S + p. Each instrument can be rebuilt from the other three by rearranging it. A long call equals long put, long stock and a borrowed amount of PV(X). Signs flip for short positions.

Understand Replicating Options, Underlying and Bond Positions

Put-call parity links four things: a European call, a European put, the underlying asset and a risk-free bond. The call and put must have the same underlying, strike X and expiry T. The bond is a zero-coupon bond that pays X at expiry, so its price today is X ÷ (1 + r)^T.

The idea comes from two portfolios with the same payoff at expiry. A fiduciary call is a long call plus the bond. A protective put is a long put plus the underlying. Whatever the stock price at expiry, both are worth the larger of S_T and X. If two portfolios pay the same at expiry, they must cost the same today, or there is arbitrage.

So you have one equation: c + PV(X) = S + p. You can solve it for any one term. Isolate the instrument you want. Each term on the other side is a position you must hold. A plus sign means buy or go long. A minus sign means sell, short or borrow.

The bond term is special. A long bond means you lend, which you do by buying a zero-coupon bond. A short bond means you borrow PV(X) today and repay X at expiry.

To replicate a short position, flip every sign of the long replication. A synthetic short call is short put, short stock, long bond. This works because the payoffs are exactly opposite.

Key formulas to remember

Put-call parity (European): c + X ÷ (1 + r)^T = S + p; Same underlying, strike and expiry. With continuous compounding, PV(X) = X × e^(−rT). Assumes no dividends or carry on the underlying.
Synthetic long call: c = p + S − X ÷ (1 + r)^T; Buy put, buy stock, borrow PV(X).
Synthetic long put: p = c − S + X ÷ (1 + r)^T; Buy call, short stock, lend PV(X) by buying the bond.
Synthetic long underlying: S = c − p + X ÷ (1 + r)^T; Buy call, sell put, buy the bond.
Synthetic risk-free bond: X ÷ (1 + r)^T = S + p − c; Long stock, long put and short call at the same strike is a riskless (conversion-type) position that pays X at expiry.
Short positions: Reverse every sign of the long replication; For example, short call = −p − S + PV(X).

How to solve Replicating Options, Underlying and Bond Positions questions

Use this method for any question asking what position replicates an instrument, or what the synthetic position costs.

1Write put-call parity: c + PV(X) = S + p. Check that the options share strike, expiry and underlying.
2Compute PV(X) = X ÷ (1 + r)^T if numbers are given. Use the correct time in years.
3Isolate the target instrument on one side. Move the other terms across and change their signs.
4Read each term: plus means buy or lend, minus means sell, short or borrow.
5If the target is a short position, reverse every sign of the long version.
6Substitute numbers to get the synthetic value or price. Compare it with the market price if asked about mispricing.
7Check the answer by testing one payoff at expiry, such as S_T above X.

Quickest way: Move-and-flip shortcut

When to use it: Use it for conceptual MCQs that ask which combination replicates an instrument.

Memorise the base: Call + Bond = Stock + Put.
Isolate the target. Terms that were on the opposite side from the target keep their sign. Terms that were on the same side as the target flip sign.
Example: to isolate c, p and S were on the opposite side and stay positive. PV(X) was on the same side as c, so it becomes negative: c = p + S − PV(X).
Match the option with the three choices. Two will have a wrong sign, so eliminate them.

Common mistakes in Replicating Options, Underlying and Bond Positions

Treating the bond term as the stock price instead of PV(X).
Students see 'bond' and forget it is a zero-coupon bond paying the strike.
Fix: Always write the bond as X ÷ (1 + r)^T, not as S.
Getting the sign wrong on the bond when replicating a call.
Moving a term across the equals sign changes its sign, and it is easy to skip.
Fix: Isolate the call: c = p + S − PV(X). The minus means borrow, not lend.
Using different strikes or expiries for the call and put.
Parity is memorised as a formula without its conditions.
Fix: Confirm both options are European with the same X, T and underlying before applying parity.
Using the wrong time period, such as months instead of years.
Rates are annual but the expiry is given in months.
Fix: Convert T to years first, for example 6 months = 0.5.
Forgetting to reverse all signs for a short position.
Students flip only the option and keep the stock and bond as long.
Fix: A short synthetic is the exact opposite of the long one, so flip every term.

Worked examples

Example 1

A European put on a stock trades at $4.00. The stock price is $50, the strike is $50, expiry is 1 year and the annual risk-free rate is 5%. What is the price of the synthetic call made from the put, the stock and the bond? A) $5.38 B) $6.38 C) $7.38

Show the solution

Rearrange parity: c = p + S − X ÷ (1 + r)^T.
PV(X) = 50 ÷ 1.05 = 47.619.
c = 4.00 + 50 − 47.619 = 6.381, which rounds to $6.38.
This matches option B.

Answer: B) $6.38. Buy the put, buy the stock, and borrow $47.62 today.

Example 2

An investor is long a call and short a put, both with strike €100 and the same expiry, and holds a zero-coupon bond paying €100 at expiry. Which instrument does this combination replicate? A) Long stock B) Long put C) Short stock

Show the solution

From parity, S = c − p + PV(X).
Long call, short put and long bond match c − p + PV(X).
Check at expiry with S_T = €120: call pays 20, short put pays 0, bond pays 100, total 120 = S_T.
Check with S_T = €80: call 0, short put −20, bond 100, total 80 = S_T.

Answer: A) Long stock. The combination pays S_T in every state.

Exam tips

Expect three-option MCQs asking which position replicates an instrument. Use Call + Bond = Stock + Put and eliminate answers with wrong signs.
Check the bond direction carefully. Borrowing and lending are the usual traps in the options.
Numerical items need PV(X) first. Compute it before touching any other term.
Verify with a quick payoff test at one stock price when you are unsure of a sign.
Remember parity is for European options and assumes no dividends unless the question adjusts the stock price.

Practice questions from Option Replication Using Put-Call Parity

Replicating Options, Underlying and Bond Positions: frequently asked questions

How do I create a synthetic call with put-call parity?

Rearrange c + PV(X) = S + p to get c = p + S − PV(X). Buy the put, buy the stock and borrow PV(X). The payoff matches a long call at expiry.

How do I create a synthetic risk-free bond using options?

Buy the stock, buy a put and sell a call, all with the same strike and expiry. Parity gives PV(X) = S + p − c. The package pays X at expiry whatever the stock price does.

What is a synthetic long stock?

It is a long call, a short put and a long zero-coupon bond paying X at expiry, so S = c − p + PV(X). It pays the stock price at expiry.

Does put-call parity work for American options?

The exact equality is for European options. American options can be exercised early, so parity holds only as an inequality or approximation.