Put-Call Parity

beginner
parityarbitragefundamentals

The fundamental no-arbitrage relationship that links European call and put prices. If this equation breaks, risk-free profit exists. Drag the sliders and watch the curves prove it.

Put-Call Parity

The fundamental no-arbitrage relationship between calls, puts, and the underlying. If this equation breaks, free money exists.

At expiry (T = 0):
185$
120$250$
8.5$
0.5$40$
6.2$
0.5$40$
185$
120$250$
0Strike $185Spot $185-9.2k-6.1k-3.1k03.1k6.1k9.2k$111$136$160$185$210$234$259Stock Price at Expiry ($)
Long Call
Synthetic Call (Stock + Put)
Long Stock
Long Put
@ $185:Long Call: $-850Synthetic Call (Stock + Put): $-620Long Stock: +$0Long Put: $-620
185$
111$259$
Parity divergence of $230 — this is because . Adjust premiums to restore parity.
LONG CALL (REAL)

Buy a call option. Pay premium upfront. Profit when stock rises above strike + premium. Max loss = premium.

P&L = max(0, S - K) - C
SYNTHETIC CALL (STOCK + PUT)

Buy the stock AND buy a put at the same strike. The put insures your downside. Together they replicate a call. If parity holds, same cost, same payoff.

P&L = (S - S₀) + max(0, K - S) - P

Why does parity matter?

If , a risk-free arbitrage exists. Market makers continuously monitor this relationship. When it breaks (even briefly), they trade to capture the difference, which pushes prices back into equilibrium. This is why you can trust that option prices are internally consistent — parity is enforced by arbitrage, not by theory alone.

Understanding Put-Call Parity

The core idea

Put-call parity states that a European call and put with the same strike and expiry are linked by a simple equation. You can always replicate one using the other plus the underlying stock (and borrowing/lending). Because both portfolios produce identical payoffs at expiry, they must cost the same today — otherwise you could earn risk-free profit.

Two equivalent portfolios

PORTFOLIO A

Buy a call option + invest K · e-rT in a risk-free bond

Cost = C + K · e-rT
PORTFOLIO B

Buy a put option + buy the underlying stock

Cost = P + S₀

At expiry, both portfolios are worth exactly max(S, K). Since they have identical payoffs in every scenario, they must have the same cost today.

The arbitrage argument

IF C + K · e-rT < P + S₀

Portfolio A is cheap. Buy A, sell B (short stock, sell put, buy call). Pocket the difference. At expiry, the positions perfectly cancel out. Risk-free profit.

IF C + K · e-rT > P + S₀

Portfolio A is expensive. Sell A, buy B (buy stock, buy put, sell call). Pocket the difference. Same result — zero risk.

Practical implications

  • Market makers use parity to price one option from the other
  • Violations are short-lived — arbitrageurs close them instantly
  • American options don't obey exact parity (early exercise complicates things), but it's a useful bound
  • Dividends shift the relationship — replace S₀ with S₀ minus PV(dividends)
  • Understanding parity helps you see that a covered call and a short put are economically equivalent