Questions: Put-Call Parity

Question 1

You observe that C − P > S − PV(K) for a European option pair with the same strike and expiration. What is the correct arbitrage response?

• A. Buy the call and sell the put — the call is underpriced relative to the put
• B. Sell the call, buy the put, buy the stock, and borrow PV(K) — sell the expensive portfolio, buy the cheap one
• C. Do nothing — small deviations from parity are always within the no-arbitrage band
• D. Buy the put and sell the call — the put is underpriced

Answer: (B)

If C − P > S − PV(K), then C + PV(K) > S + P: Portfolio A (call + bond) costs more than Portfolio B (stock + put), but both have identical payoffs. The arbitrage is to sell the expensive portfolio A (sell the call, borrow PV(K)) and buy the cheap portfolio B (buy the stock, buy the put). This locks in a riskless profit equal to the parity gap. Competition among arbitrageurs drives prices back toward equality.

Question 2

Why does put-call parity hold exactly for European options but only approximately for American options?

• A. American options have larger bid-ask spreads, introducing pricing errors
• B. European options can be exercised at any time; American options can only be exercised at expiration
• C. American put options may have value from early exercise that is not captured by the European parity formula
• D. Put-call parity applies equally to both — the distinction is only theoretical

Answer: (C)

American options can be exercised before expiration. An American put may be worth exercising early — particularly when the stock has fallen sharply and the time value of waiting is costly. This early-exercise premium adds value to the American put beyond what European parity predicts, breaking the equality. The no-arbitrage portfolios underlying parity assume the option is held to expiration; early exercise invalidates that assumption. Note also that option B has the direction of European vs. American backwards — European options can only be exercised at expiration.

Question 3

If two portfolios produce identical cash flows in every possible future state, they must have the same price today.

• T. True
• F. False

Answer: True

This is the law of one price — the foundation of no-arbitrage pricing. If the portfolios had different prices, you could buy the cheaper one and sell the more expensive one, earning a riskless profit today with zero net future obligation. Such an arbitrage opportunity cannot persist in a rational market. Put-call parity is derived directly from this principle: Portfolio A (call + bond) and Portfolio B (stock + put) both pay max(S_T, K) at expiration, so they must cost the same to construct today.

Question 4

Put-call parity implies that a call and a put with the same strike and expiration is expected to have the same price.

• T. True
• F. False

Answer: False

Put-call parity states C − P = S − PV(K), not C = P. Calls and puts have equal prices only in the special case where S = PV(K) — roughly when the stock price equals the discounted strike, which can occur for at-the-money options with very short time to expiration. In general, a call (which profits from price increases) and a put (which profits from price decreases) have different prices. Parity constrains their relationship, not their equality.

Question 5

Explain the no-arbitrage logic behind put-call parity: why must C − P equal S − PV(K)?

Model answer: Construct two portfolios: (A) buy a call + invest PV(K) in risk-free bonds; (B) buy the stock + buy a put. Portfolio A pays max(S_T, K) at expiration in all states: if S_T > K, exercise the call; if S_T ≤ K, the bond pays K. Portfolio B also pays max(S_T, K): if S_T > K, hold the stock; if S_T ≤ K, exercise the put. Identical payoffs in all states require identical prices: C + PV(K) = S + P, which rearranges to C − P = S − PV(K).

The derivation requires no assumptions about how stock prices move — only the no-arbitrage condition. This makes put-call parity a model-free result. Any deviation from parity reveals an explicit arbitrage: buy the cheap portfolio, sell the expensive one, and collect a riskless profit at construction. The parity relationship also enables synthetic replication: knowing any three of {C, P, S, PV(K)}, you can determine the fourth.