CFA Level I Exam · Pricing and Valuation of Options

Put-Call Parity Formula for CFA Level I

Updated 7 October 2026 · Fact-checked

Put-call parity links the prices of a European call, a European put, the underlying and a risk-free bond with the same strike and expiry. A fiduciary call (call + bond) must equal a protective put (put + underlying): c + X/(1+r)^T = p + S0. Solve for the missing piece; if prices break it, arbitrage exists.

Understand Put-Call Parity

Start with two portfolios. The first is a fiduciary call: one European call plus a zero-coupon risk-free bond that pays the strike price X at expiry. The second is a protective put: one European put plus one unit of the underlying asset. Both use the same strike and expiry.

Now compare payoffs at expiry. If the underlying ends above X, the fiduciary call pays ST (the call is worth ST − X, plus X from the bond). The protective put pays ST (the put expires worthless and you hold the asset). If the underlying ends below X, the fiduciary call pays X (call worthless, bond pays X). The protective put pays X (put pays X − ST, plus the asset worth ST). So both portfolios pay max(ST, X) in every state.

Two portfolios with identical payoffs in all states must cost the same today. Otherwise you could buy the cheap one, sell the expensive one and lock in a risk-free profit. This is the law of one price. That gives parity: c + X/(1+r)^T = p + S0.

The relationship is only exact for European options on the same underlying with the same strike and expiry. It also assumes no dividends or other cash flows on the underlying, unless you adjust. You can rearrange parity to build synthetic positions, for example a synthetic call is p + S0 − X/(1+r)^T.

Key formulas to remember

Put-call parity: c + X ÷ (1 + r)^T = p + S0; European options, same strike X and expiry T, no cash flows on the underlying. r is the annual risk-free rate, T in years.
With continuous compounding: c + X × e^(−rT) = p + S0; Use when the question gives a continuously compounded rate.
Fiduciary call and protective put: Fiduciary call = c + PV(X); Protective put = p + S0; Both pay max(ST, X) at expiry.
Synthetic call: c = p + S0 − X ÷ (1 + r)^T; Long put, long underlying, borrow PV of X.
Synthetic put: p = c − S0 + X ÷ (1 + r)^T; Long call, short underlying, lend PV of X.
Synthetic underlying: S0 = c − p + X ÷ (1 + r)^T; Long call, short put, long bond.
Synthetic bond: X ÷ (1 + r)^T = p + S0 − c; Long put, long underlying, short call.
Parity with known dividends: c + PV(X) = p + S0 − PV(dividends); Subtract the present value of dividends paid before expiry from S0. Equivalent form uses the forward price: c + PV(X) = p + PV(F0(T)).

How to solve Put-Call Parity questions

Use this method for any put-call parity question, whether it asks for a missing price, a synthetic position or an arbitrage.

1Check the options are European with the same strike and expiry on the same underlying. If not, parity does not apply exactly.
2Write the identity: c + PV(X) = p + S0, adjusting S0 for the PV of any dividends before expiry.
3Compute PV(X) = X ÷ (1 + r)^T. Make sure T is in years and r matches that period.
4Plug in the known values and solve for the unknown. Keep the equation balanced; move terms across with signs flipped.
5For a synthetic position, rearrange so the target instrument stands alone. Positive terms are long, negative terms are short or borrowing.
6For arbitrage, compute both sides. Buy the cheaper side and sell the more expensive side. The profit today equals the difference.
7Sanity check: the answer should be positive, and a call should not exceed S0.

Quickest way: Left side versus right side

When to use it: Use when you have all four prices and must pick the arbitrage or find the missing price quickly.

Compute PV(X) once. On a BA II Plus: enter N, I/Y, FV = X, PMT = 0, then CPT PV (ignore the sign).
Left = c + PV(X). Right = p + S0 (less PV of dividends if given).
Equal: no arbitrage. Left higher: sell the fiduciary call side, buy the protective put side. Right higher: do the reverse.
For a missing price, set Left = Right and subtract. Then check which option has the plausible magnitude.

Common mistakes in Put-Call Parity

Using X instead of its present value
The strike looks like a price, so it gets added directly.
Fix: Always discount X at the risk-free rate for time T. Only the bond term is discounted.
Ignoring dividends on the underlying
The basic formula is memorised without its no-cash-flow assumption.
Fix: If the stem gives dividends before expiry, subtract their present value from S0 before using parity.
Applying parity to American options
Candidates forget early exercise can break the identity.
Fix: Parity is exact only for European options. Check the stem.
Getting the arbitrage direction wrong
Candidates sell the cheaper portfolio by mistake.
Fix: Buy the cheaper side, sell the more expensive side. Then list each leg: option, underlying and borrow or lend.
Mixing up the time units
Expiry is given in months but the rate is annual.
Fix: Convert months to years first, for example 6 months is T = 0.5.

Worked examples

Example 1

A European call on a non-dividend-paying stock with strike $50 and six months to expiry trades at $6.20. The stock is $52 and the risk-free rate is 4% per year, compounded annually. What is the price of the European put with the same strike and expiry, according to put-call parity?

Show the solution

Parity: p = c + X ÷ (1 + r)^T − S0.
T = 0.5, so (1.04)^0.5 = 1.019804.
PV(X) = 50 ÷ 1.019804 = 49.0290.
p = 6.20 + 49.0290 − 52 = 3.2290.

Answer: The put is worth about $3.23. Options listed smallest to largest might be $2.18, $3.23, $4.29; choose $3.23.

Example 2

A European call (strike €40, one year) trades at €5.00 and a European put with the same terms trades at €3.00. The stock is €41 and pays no dividends. The risk-free rate is 5% per year. Is there an arbitrage, and what is the profit today?

Show the solution

PV(X) = 40 ÷ 1.05 = 38.0952.
Left side (fiduciary call) = 5.00 + 38.0952 = 43.0952.
Right side (protective put) = 3.00 + 41.00 = 44.00.
Right is higher, so the protective put is expensive and the fiduciary call is cheap.
Buy the call, lend €38.0952 (buy the bond), sell the put and short the stock, or equivalently sell the protective put.
Profit today = 44.00 − 43.0952 = 0.9048.

Answer: Yes. Buy the fiduciary call (call plus bond) and sell the protective put (sell the put, short the stock). The riskless profit is about €0.90 per share today, and the positions net to zero at expiry.

Exam tips

Memorise the identity as two named portfolios. Questions often ask which position is equivalent to a fiduciary call or protective put.
For synthetic-position questions, rearrange the equation and read off the signs. Negative terms mean short or borrow.
In arbitrage questions, find which side is cheaper first. Then eliminate options with the wrong direction. This usually leaves one answer.
Watch for dividends and for American-style wording. Either changes the answer, and these are classic trap details.
With three options and no penalty for wrong answers, always guess if a calculation runs long. Your first elimination step raises your odds.

Practice questions from Pricing and Valuation of Options

Put-Call Parity in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Put-Call Parity: frequently asked questions

What is the put-call parity formula for CFA Level I?

The formula is c + X ÷ (1 + r)^T = p + S0. It applies to European options with the same strike and expiry. If the underlying pays dividends, subtract their present value from S0.

Why must a fiduciary call equal a protective put?

Both portfolios pay max(ST, X) at expiry in every outcome. Identical payoffs must have identical prices, otherwise arbitrageurs would buy the cheaper one and sell the dearer one for a riskless profit.

How does put-call parity change with dividends?

Dividends reduce the underlying's price, which lowers the call and raises the put. Subtract the present value of dividends paid before expiry from S0, or use the forward price: c + PV(X) = p + PV(F0(T)).

Does put-call parity work for American options?

Not exactly. Early exercise can break the equality, so the identity is stated for European options. On the exam, check the option style before you apply it.