Put-call parity · RiskAtoms Academy

Know these first

Math Atoms

∑ Exponents

By the end you will be able to

1Use put-call parity to link call, put, share and bond prices.
2Explain why a call plus a bond has the same payoff as a put plus the share.
3Spot when quoted option prices break parity and describe the arbitrage.
4Build a synthetic call, put or share from the other three instruments.

Why it matters

If you know the price of a call, you already know the price of the matching put. No model, no forecast, no view on volatility: just the share price and an interest rate. Traders use this one line every day to check option prices against each other.

Core explanation

Two portfolios, one payoff

Build two portfolios, using European options with the same strike K and the same expiry T:

Portfolio A: call + bond

One call, plus a zero-coupon bond that pays K at expiry. If the share ends above K, use the bond's K to exercise the call: you hold the share. If it ends below K, the call lapses and you hold K in cash.

Portfolio B: put + share

One put, plus the share. If the share ends above K, the put lapses and you hold the share. If it ends below K, use the put to sell the share for K: you hold K in cash.

Either way, both portfolios end up with whichever is larger, the share or K. Two things that pay exactly the same in every outcome must cost the same today, or someone could buy the cheaper one, sell the dearer one and keep the difference with no risk. That is put-call parity. The diagram below plots both payoffs. Switch between them: the lines lie exactly on top of each other.

Payoff diagramPortfolio A against portfolio B

Legs

Long put, strike 100

Long share

Portfolio A: call + bondTap or drag the chart to move the line

Price of the underlying at expiry110

At an expiry price of 110

Payoff110.00

Leg by leg

Long put, strike 100payoff 0.00
Long sharepayoff 110.00

Synthetics. Because the four prices are tied together, any one instrument can be copied with the other three. A copy like this is a synthetic position, and copying a payoff is called replication. Click each tile to see the recipe.

Make one from the other three

Click a tile

Buy a put and the share, and borrow the present value of K. Same payoff as a call: C = P + S₀ − PV(K).

The fine print. Parity in this form needs European options (exercised only at expiry) with the same strike and expiry, on a share that pays no dividends before expiry. With dividends you subtract their present value from the share price; for American options parity gives a range rather than an exact equation.

The formal version

Write $C$ and $P$ for the call and put prices today, $S_0$ for the share price, $K$ for the strike, $r$ for the annual interest rate (compounded annually) and $T$ for the years to expiry.

The formula, for reference

C + \frac{K}{(1+r)^T} = P + S_0 \qquad\Longleftrightarrow\qquad C - P = S_0 - \frac{K}{(1+r)^T}

With a continuously compounded rate (L1.3.05), as in Hull, the bond term is $Ke^{-rT}$ . With a known dividend, replace $S_0$ by $S_0 - D$ , where $D$ is the present value of the dividend.

Try it

Set a share, a strike and a call price, and parity gives the put price. Then move the dealer's put quote away from it and read the trade that profits. Pushing the call price very low shows a case parity rules out completely.

Try itParity lab

Price both options with Black-Scholes →

Share price, S₀100.00

Strike, K100.00

Interest rate, r (annual)5.00%

Time to expiry, T (years)1.00

Call price, C10.45

Dealer's put quote7.00

PV of strike95.24

Parity put price5.69

P = C − S₀ + K ÷ (1 + r)ᵀ

The put is too dear. Sell it for 7.00. Build the same payoff for 5.69: buy the call, short-sell the share and lend 95.24. You pocket 1.31 today, and at expiry the two sides cancel.

Worked example

Pricing a put from a call

A share trades at $100 and pays no dividends. A 1-year European call with strike $100 costs $10.45. The interest rate is 5% a year, compounded annually.

Parity in four steps

Step 1 of 5

1
Present value of the strike
$100 \div 1.05 = 95.2381$

Regulatory lens · looking ahead

Banks must check that the prices they use to value their positions are sound, through independent price verification and, in the EU, prudent valuation rules. Parity is one of the simplest checks: call and put prices fed into a bank's systems should agree with it. Later levels cover valuation control and model risk.

Sparks

SparkParity came first

Hans Stoll set out the link between put and call prices in the Journal of Finance in 1969, before the Black-Scholes formula of 1973. Parity needs no model of how share prices move, which is why it holds so well in practice.

SparkConversions and reversals

Option traders have names for the parity trades: buying the share, buying a put and selling a call is a “conversion”; the opposite is a “reversal”. Because they are almost riskless, they earn only tiny margins, and that keeps listed option prices close to parity.

Key takeaways

A call plus a bond paying K has the same payoff as a put plus the share: both are worth max(S_T, K).
So C + K ÷ (1 + r)ᵀ = P + S₀, for European options with the same strike and expiry, no dividends.
Know three of the four prices and parity gives the fourth.
If quotes break parity, sell the dear side, buy the cheap side, and lock in the gap.
Any of call, put, share or bond can be built synthetically from the other three.

Reaction

Check what stuck

Question 1 · objective 1

A share trades at 50. A 1-year European put with strike 50 costs 3. The interest rate is 4% a year, compounded annually, and there are no dividends. What should the matching call cost, to two decimal places?

Question 2 · objective 1

A 1-year European call and put with strike 100 cost 8 and 5. The rate is 5% a year, compounded annually. What share price does parity imply, to two decimal places?

Question 3 · objective 2

Put-call parity in the form C + K ÷ (1 + r)ᵀ = P + S₀ holds exactly for:

AAny call and put on the same shareBEuropean call and put with the same strike and expiry, on a share paying no dividends before expiryCAmerican options onlyDOptions whose strike equals the share price

Question 4 · objective 3

A put is quoted above its parity price. Which trade locks in a riskless profit?

ABuy the put, sell the call, buy the share, borrowBSell the put, buy the call, short-sell the share, lend the present value of the strikeCBuy both the put and the callDThere is no trade; parity is only approximate

Question 5 · objective 2

Portfolio A holds a call with strike 100 and a zero-coupon bond paying 100 at the call's expiry. At expiry the share is at 80. What is portfolio A worth?

Glossary

Put-call parity: The rule that a European call plus a bond paying the strike costs the same as a European put plus the share.
Zero-coupon bond: A bond that pays one amount on one date and nothing before. Its price is that amount discounted to today.
Fiduciary call: A call plus a zero-coupon bond paying the strike. At expiry it is worth the larger of the share price and the strike.
Protective put: A share plus a put on it. At expiry it is worth the larger of the share price and the strike.
Synthetic position: A position built from other instruments that has the same payoff as the one it copies.
Replication: Copying the payoff of one instrument with a combination of others.

Deep Dives & Origins

Deep Dives

John C. Hull, Options, Futures, and Other Derivatives, 11th edition (Pearson, 2021), the chapter “Properties of Stock Options”. Intermediate.

Origins

John C. Hull, Options, Futures, and Other Derivatives, 11th edition, Pearson, 2021.
Hans R. Stoll, “The Relationship Between Put and Call Option Prices”, Journal of Finance 24(5), 1969.

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