Lesson 2 of 4 6–9% of the exam

By Daniil Tarakanov Updated Oct 21, 20262027 curriculum

Building synthetic positions with put–call parity

Rearrange S + p = c + PV(X), where PV is present value, to isolate the position you want; the other side is its synthetic copy. A long put = long call + long bond + short share. A covered call (share minus call) = long bond minus put.

6 min cram, 26 min full read. Based on the 2027 CFA® Program Level I curriculum, Derivatives, Learning Module 9. Also part of the 2026 curriculum, so it applies to the November 2026 exam too.

Tailor this to you: take the self-assessment

Before this: Put–call parity: fiduciary call = protective put

On the exam

Explain put–call parity for European options

Explain put–call parity as replication. Expect 'which combination is equivalent to…' for a long put, long share, short call or covered call, and one numeric question that solves for a synthetic position's cost.

How deeply to learn each part

Know the idea: Understand it in general. It frames other topics, but you will not calculate with it.
Use it: Understand it well enough to solve a question with it. Recognize when it applies.
Memorize: Learn it by heart. The exam will not give it to you.

You know this one. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still do.

You tested strong here. The full lesson is folded away. The takeaways, formulas and checkpoint below are your quick refresher, and the checkpoint is a fair test of whether you still know it.

How can you build any one of the four positions from the other three?

Rearrange put–call parity so the position you want stands alone. The other side is its synthetic position : a package of the other instruments that has the same payoff and, by parity, the same cost.

Start from S + p = c + PV(X), where PV(X) is the present value of the exercise price. Write B for the bond paying X, so S + p = c + B. Move a term to the other side and its sign flips.

Target	Equivalent to	Equation
Long share	long call + short put + long bond	S = c − p + PV(X)
Long call	long share + long put + borrow PV(X)	c = S + p − PV(X)
Long put	long call + long bond + short share	p = c + PV(X) − S
Long bond	long share + long put + short call	PV(X) = S + p − c

The first row is a synthetic share , the second a synthetic call , the third a synthetic put and the last a synthetic bond . The tree puts every row in one place.

Every position built from the other three

S + p = c + BB is the bond paying X; move a term across and flip its sign

Long share Memorize S = c − p + B
- Long call +c
- Short put −p
- Long bond paying X +B
Long call Memorize c = S + p − B
- Long share +S
- Long put +p
- Short bond (borrow PV(X)) −B
Long put Memorize p = c + B − S
- Long call +c
- Long bond paying X +B
- Short share −S
Long bond Memorize B = S + p − c
- Long share +S
- Long put +p
- Short call −c

Loading the interactive

For a short position, flip every sign. A short call is a short share, a short put and a long bond: from c = S + p − B you get −c = −S − p + B. A short put is a long share, a short call and a short bond (borrowing PV(X)).

Synthetic share Use it

S_0 = c_0 - p_0 + X(1 + r)^{-T}

Synthetic bond Use it

X(1 + r)^{-T} = S_0 + p_0 - c_0

Rearrange itPut–call parity: solve for any input

The question asks for

A share trades at 48.00. A European call on it, with an exercise price of 50.00 and 0.5 years to expiry, costs 4.10. The risk-free rate is 3.00% and the share pays no income. The price of the European put with the same exercise price and expiry is closest to:

Share price, S₀Exercise price, XRisk-free rate, r (annual)%Time to expiry, T (years)Call price, c₀

p_{0} = c_{0} + X (1 + r)^{- T} - S_{0}

4.10 + 50.00 × (1 + 0.03)^−0.5 − 48.00 = 4.10 + 49.27 − 48.00 = 5.37

Put price, p₀5.37

On your calculator

1.03y^x0.5+|-=×50=49.27
+4.1−48=5.37

y^x power · +|- change sign

1.03ENTER0.5CHSy^x50×49.27
4.1+48−5.37

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Exam tip To read a synthetic position, write S + p = c + B and cover the term you want. Whatever is left, with signs flipped for terms that crossed the equals sign, is the recipe.

Why is a covered call a bond minus a put?

A covered call is a long share plus a short call on it. Parity says S − c = PV(X) − p, which is a long bond and a short put.

Check the payoffs at expiry. The covered call pays S_T − max(S_T − X, 0) = min(S_T, X). The bond minus put pays X − max(X − S_T, 0) = min(S_T, X). Same payoff: it follows the share down but is capped at X.

Covered call equals bond minus put Memorize

S_0 - c_0 = X(1 + r)^{-T} - p_0

Go deeper Why dealers and arbitrageurs care about synthetics

A dealer who sells a put can hedge by buying a call, buying the bond and shorting the share. Because every instrument can be copied by the others, any price gap gets traded away, which keeps prices aligned. A long call plus a short put with the same strike is also a synthetic forward, the topic of the next lesson.

Memorize this

MemorizeThe synthetic table

Target	Equivalent to	Equation
Long share	long call + short put + long bond	$S = c - p + PV(X)$
Long call	long share + long put + borrow $PV(X)$	$c = S + p - PV(X)$
Long put	long call + long bond − share	$p = c + PV(X) - S$
Long bond	long share + long put + short call	$PV(X) = S + p - c$
Covered call	long bond + short put	$S - c = PV(X) - p$

Memory hook Write S + p = c + B. Cover the thing you want, move the rest across and flip the signs. Shorts: flip every sign.

Know it cold Short call = short share + short put + long bond. The bond always pays $X$ , never $S$ .

Every position built from the other three

S + p = c + BB is the bond paying X; move a term across and flip its sign

Long share Memorize S = c − p + B
- Long call +c
- Short put −p
- Long bond paying X +B
Long call Memorize c = S + p − B
- Long share +S
- Long put +p
- Short bond (borrow PV(X)) −B
Long put Memorize p = c + B − S
- Long call +c
- Long bond paying X +B
- Short share −S
Long bond Memorize B = S + p − c
- Long share +S
- Long put +p
- Short call −c

Synthetic share

S_0 = c_0 - p_0 + X(1 + r)^{-T}

Quick example

Call 6.80, put 4.10, X = 70, r = 2%, T = 0.5. PV(X) = 69.31. Synthetic share = 6.80 − 4.10 + 69.31 = 72.01.

On your calculator

1.02y^x0.5+|-=×70= 69.31
+6.8−4.1= 72.01

y^x power · +|- change sign

1.02ENTER0.5CHSy^x70× 69.31
6.8+4.1− 72.01

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Synthetic bond

X(1 + r)^{-T} = S_0 + p_0 - c_0

Quick example

Share 36, put 2.90, call 3.40. The bond paying X is worth 36 + 2.90 − 3.40 = 35.50 today.

On your calculator

36+2.9−3.4= 35.5

36ENTER2.9+3.4− 35.5

ENTER enter (on the HP, puts the number on the stack)

Covered call equals bond minus put

S_0 - c_0 = X(1 + r)^{-T} - p_0

Quick example

Share 70, X = 75, one year, r = 4%, put 6.50. Cost of the covered call = 75 ÷ 1.04 − 6.50 = 72.12 − 6.50 = 65.62.

On your calculator

75÷1.04= 72.12
−6.5= 65.62

75ENTER1.04÷ 72.12
6.5− 65.62

ENTER enter (on the HP, puts the number on the stack)

How it is asked

Which combination is equivalent to a long put?

A long put is equivalent to…?

p = c + PV(X) − S: long call + long bond + short share. Trap: share + call (that is not a put).
Covered call equivalent

Long share + short call has the same payoff as…?

S − c = PV(X) − p: long bond + short put. Trap: bond + put.
Cost of a synthetic position

S = 55, put 6.20, X = 60, r = 3.5%, T = 0.5. Call?

55 + 6.20 − 58.98 = 2.22. Trap: 1.20 (X not discounted).
Short position

A short call is equivalent to…?

Short share + short put + long bond: every sign flipped.

Exam traps

Forgetting to flip signs when moving a term across the equals sign.
Calling the covered call 'share + call'. It is share minus call (you sell the call).
Using a bond that pays S instead of X: the bond always pays the exercise price.
Leaving out the discounting: the synthetic bond costs PV(X) today, not X.

Checkpoint

5 exam-style questions. About 90 seconds each.

1A long put option is equivalent to:
2A covered call position (long share, short call) has the same payoff as:
3A share is USD 72. A three-month European put with an exercise price of USD 80 costs USD 9.40, and the risk-free rate is 4%. The share pays no income. The no-arbitrage price of the European call with the same strike and expiry is closest to:
4A short call position is equivalent to:
5A share trades at USD 90, a one-year European put with an exercise price of USD 100 costs USD 12.00 and the risk-free rate is 5%. The share pays no income. The cost of a covered call on the share, written with the same strike and expiry, is closest to:

Key takeaways

Each of S, c, p and the bond can be built from the other three.
Move a term across the equation and flip its sign. For a short position, flip every sign.
Covered call = long bond − put: capped upside at X, losses as the share falls below X.
Synthetic and actual positions have the same payoff, so with parity they have the same cost.

Synthetic share

S_0 = c_0 - p_0 + X(1 + r)^{-T}

Quick example

Call 6.80, put 4.10, X = 70, r = 2%, T = 0.5. PV(X) = 69.31. Synthetic share = 6.80 − 4.10 + 69.31 = 72.01.

On your calculator

1.02y^x0.5+|-=×70= 69.31
+6.8−4.1= 72.01

y^x power · +|- change sign

1.02ENTER0.5CHSy^x70× 69.31
6.8+4.1− 72.01

ENTER enter (on the HP, puts the number on the stack) · CHS change sign · y^x power

Synthetic bond

X(1 + r)^{-T} = S_0 + p_0 - c_0

Quick example

Share 36, put 2.90, call 3.40. The bond paying X is worth 36 + 2.90 − 3.40 = 35.50 today.

On your calculator

36+2.9−3.4= 35.5

36ENTER2.9+3.4− 35.5

ENTER enter (on the HP, puts the number on the stack)

Covered call equals bond minus put

S_0 - c_0 = X(1 + r)^{-T} - p_0

Quick example

Share 70, X = 75, one year, r = 4%, put 6.50. Cost of the covered call = 75 ÷ 1.04 − 6.50 = 72.12 − 6.50 = 65.62.

On your calculator

75÷1.04= 72.12
−6.5= 65.62

75ENTER1.04÷ 72.12
6.5− 65.62

ENTER enter (on the HP, puts the number on the stack)

All Derivatives formulas, every lesson

Practice exam-style questions on Put–Call Parity and Option Replication

Next lesson Put–call forward parity

Questions about this lesson?

Where this comes back

Put–call forward parity Put–Call Parity and Option Replication
Introduction to Risk Management Portfolio Construction

Questions candidates ask

How do you create a synthetic put option?

Rearrange parity: p = c + PV(X) − S. Buy the call, buy a risk-free bond that pays X, and sell the share short. It has the same payoff and the same cost as the put.

What is a synthetic long stock position?

A long call, a short put with the same strike and expiry, and a long risk-free bond paying X. By parity, S = c − p + PV(X), so the package costs the same as the share and pays the same at expiry.

Why does a covered call behave like a short put?

Parity gives S − c = PV(X) − p. A covered call is a long bond plus a short put, so it has the same capped-upside, falling-share risk as selling a put and lending PV(X).

Spotted an error? Tell us and we will fix it.