What is put-call parity?
For European options on a stock with no dividends, same strike and expiry, a call minus a put equals the stock price minus the strike's present value. If the put is overpriced, sell it, buy the call, short the stock and lend that present value.
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Example answer
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Put-call parity is the relationship between the price of a call and a put on the same stock with the same strike and expiry. The call minus the put equals the stock price minus the strike.
It works because you can build the same payoff out of calls, puts and the stock. If it didn't hold there'd be an arbitrage, and traders would do it until the prices came back in line. So if I know the call price, I can work out the put price.
What holds it back
It names the right instruments but says "minus the strike" where it should be the present value of the strike, gives no conditions, and never says which payoffs are the same.
Put-call parity is the relationship between the price of a call and a put on the same stock with the same strike and expiry. For European options on a stock that pays no dividends, the call minus the put equals the stock price minus the present value of the strike. The reason is that two portfolios have the same payoff. One is a call plus enough cash to pay the strike at expiry. The other is a put plus the stock. At expiry both are worth the stock price or the strike, whichever is higher. Two things with the same payoff must cost the same today. Say the stock is at $100, the strike is $100, expiry is in one year and the risk-free rate is 5%. The present value of the strike is about $95.24. So the call should cost $4.76 more than the put. If it didn't hold there'd be an arbitrage, and traders would do it until the prices came back in line.
Put-call parity is the relationship between the price of a call and a put on the same stock with the same strike and expiry. For European options on a stock that pays no dividends, the call minus the put equals the stock price minus the present value of the strike. The reason is that two portfolios have the same payoff. One is a call plus enough cash to pay the strike at expiry. The other is a put plus the stock. At expiry both are worth the stock price or the strike, whichever is higher. Two things with the same payoff must cost the same today. Say the stock is at $100, the strike is $100, expiry is in one year and the risk-free rate is 5%. The present value of the strike is 100 divided by 1.05, about $95.24. So the call should cost $4.76 more than the put. With the call at $10, the put has to be $5.24. If it didn't hold there'd be an arbitrage. Say the put traded at $6. I'd sell the put, buy the call, short the stock and lend $95.24. That brings in $106 and costs $105.24, so I keep 76 cents today. At expiry the loan pays me $100. The long call and short put mean I buy the stock at $100 wherever it ends up, and that share closes the short. I owe nothing and took no risk, and traders would do it until the prices came back in line. Two caveats. With dividends I'd subtract their present value from the stock price. And American options can be exercised early, so for them parity holds only as a range.
The numbers, step by step
Stock at $100, strike $100, one year to expiry, a 5% risk-free rate compounded once a year, European options and no dividends.
- Present value of the strike$100 ÷ 1.05
- $95.24
- Call minus put$100 − $95.24
- $4.76
- Put price if the call is $10$10 − $4.76
- $5.24
- Cash in: sell put at $6, short stock$6 + $100
- $106
- Cash out: buy call, lend PV of strike$10 + $95.24
- $105.24
- Riskless profit today$106 − $105.24
- $0.76
What the interviewer listens for
- Can you state the formula with the present value of the strike?
- Do you know it's for European options with the same strike and expiry?
- Can you explain why it holds using two portfolios with the same payoff?
- Could you describe the trade if prices were out of line?
Common mistakes
- Leaving out discounting and saying the call minus the put equals the stock price minus the strike.The strike is paid at expiry, so it enters at its present value.
- Applying the equality to American options or to a dividend-paying stock without saying that early exercise and dividends change it.
- Stating the formula from memory with no reason behind it, so a follow-up about which two portfolios have the same payoff has no answer.
- Getting the arbitrage backwards by buying the expensive side.If the put is too dear, the trade sells the put and buys the call.
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