How does put-call parity work for European options and why can't it be violated?

Question

AcadiFi · Accepted Answer

Put-call parity for European options on a non-dividend stock states: C + PV(K) = P + S, where C is call price, P is put price, K is strike, S is spot, and PV(K) = K*e^(-rT). The relationship holds because both sides replicate the same terminal payoff of max(S_T, K).

Worked example: Stock Nimbus Analytics trades at $58, a 6-month European call with K=$60 costs $3.40, risk-free rate is 4%. Parity implies P = C + K*e^(-rT) - S = 3.40 + 60*e^(-0.04*0.5) - 58 = 3.40 + 58.812 - 58 = $4.212.

If the market put traded at $5.00, arbitrage: short the put ($5.00 in), short stock ($58 in), buy call ($3.40 out), buy T-bill maturing at $60 ($58.812 out). Net cash today = $5.00 + $58 - $3.40 - $58.812 = $0.788 risk-free profit. At maturity both synthetic portfolios pay off identically, so the $0.788 is locked in.

```mermaid
flowchart LR
  A[Call + PV K] --> B[Same payoff]
  C[Put + Stock] --> B
  B --> D[Max S or K]
```

The parity relationship requires European exercise (no early exercise optionality) and no dividends. Violations create instant arbitrage, which is why market prices converge.

How does put-call parity work for European options and why can't it be violated?

Master Part I with our FRM Course

Related Questions

Practice Questions