Put-Call Parity

The Most Important Relationship in Options

Put-call parity is a no-arbitrage constraint that ties together the prices of European calls, European puts, the underlying stock, and a risk-free bond. It requires no assumptions about how stock prices move — it follows purely from the absence of free money. Understanding it deeply will teach you more about options pricing than memorizing any formula.

The relationship is:

C − P = S − K · e^−rT

where C is the call price, P is the put price, S is the current stock price, K is the shared strike, r is the continuously compounded risk-free rate, and T is time to expiration in years. The term K · e^−rT is the present value of K — what you'd need to invest today at the risk-free rate to have exactly K at expiration.

The Proof: Two Portfolios, One Payoff

Consider two portfolios constructed today and held to expiration at time T:

Portfolio A: Buy one European call at strike K. Invest K · e^−rT in a risk-free bond.

Portfolio B: Buy one European put at strike K. Buy one share of the stock.

What does each portfolio pay at expiration? Let S_T denote the stock price at time T.

If S_T > K (call expires in-the-money):

• Portfolio A: Call pays S_T − K. Bond matures to K. Total: S_T.
• Portfolio B: Put expires worthless (worth 0). Stock worth S_T. Total: S_T.

If S_T < K (put expires in-the-money):

• Portfolio A: Call expires worthless. Bond matures to K. Total: K.
• Portfolio B: Put pays K − S_T. Stock worth S_T. Total: K − S_T + S_T = K.

In both cases, Portfolio A and Portfolio B pay identical amounts at expiration: max(S_T, K). Since they have identical future cash flows with certainty, they must cost the same today — otherwise you could buy the cheap portfolio, short the expensive one, and lock in a risk-free profit. This argument is called the law of one price.

Therefore: C + K · e^−rT = P + S, which rearranges to C − P = S − K · e^−rT.

What Happens When Parity Breaks?

In 2010, researchers documented persistent put-call parity violations in equity options during the 2008 financial crisis. When Lehman Brothers was failing, put prices on financial stocks were dramatically elevated relative to calls — a violation of parity that couldn't be easily arbitraged away because shorting the required stocks was impossible (regulators had temporarily banned short selling in financial stocks). This is a perfect example of how no-arbitrage relationships depend on the ability to execute all legs of the trade freely.

In normal markets, any violation is exploited and closed within milliseconds by algorithmic traders. The parity relationship is one of the most tightly enforced in all of finance.

Synthetic Positions

Rearranging put-call parity gives you four synthetic equivalences that every options trader knows:

• Synthetic long stock: C − P + K · e^−rT = S. Buy call, sell put, invest PV(K) → same payoff as owning the stock.
• Synthetic call: C = P + S − K · e^−rT. If calls are expensive, you can replicate one using a put plus stock.
• Synthetic put: P = C − S + K · e^−rT. If you know the call price, you can derive the put price exactly.
• Box spread: Buy a call spread and sell a put spread at the same strikes. The payoff at expiration is always K₂ − K₁ (a constant), so the box must trade at its present value. Box spreads are used by market makers to borrow and lend at the implied options rate.

Dividends and Early Exercise

The parity formula above assumes no dividends. When the underlying pays a dividend with present value PV(D), the relationship becomes:

C − P = S − PV(D) − K · e^−rT

This matters enormously in practice. In the days before a large dividend, deep ITM call holders sometimes exercise early to capture the dividend — a situation where American and European prices diverge. The dividend-adjusted parity helps predict when this will happen.

A Numerical Example

AAPL is trading at $190. A 90-day call with K = $190 trades at $8.50. The risk-free rate is 5%. No dividends. What should the put trade at?

Using P = C − S + K · e^−rT:

K · e^−rT = 190 · e^{−0.05 × 0.25} = 190 · 0.9876 = $187.64

P = $8.50 − $190 + $187.64 = $6.14

If the market is quoting the put at $7.50, that's a violation. You'd short the put at $7.50, buy the call at $8.50, short the stock at $190, and invest $187.64 at the risk-free rate. Your net cash inflow today: $7.50 − $8.50 + $190 − $187.64 = $1.36. At expiration, all legs net to zero regardless of where AAPL trades. Free money.