A European call costs $8 and the corresponding European put costs $3. The stock price is $100, the strike is $98, and PV(K) = $95. Does put-call parity hold?
ANo — C − P = $5 but S − PV(K) = $3, so there is an arbitrage opportunity
BNo — C − P = $5 but S − PV(K) = $7, so the call is overpriced
CYes — C − P = $5 and S − PV(K) = $5, so parity holds exactly
DYes — parity always holds by definition for European options regardless of prices
C − P = $8 − $3 = $5, and S − PV(K) = $100 − $95 = $5. Both sides equal $5, so put-call parity holds exactly and no arbitrage opportunity exists. If they differed — say, C − P = $6 but S − PV(K) = $5 — then a riskless profit would be available by selling the overpriced side (call+short put) and buying the underpriced side (long forward).
Question 2 Multiple Choice
An investor buys a straddle (buys a call and a put at the same strike K) for a total premium of $8. For what range of stock prices at expiration does the investor profit?
AOnly if the stock rises above the strike by more than $8
BIf the stock finishes exactly at the strike — both options are at-the-money and earn the premium back
CIf the stock finishes more than $8 above or more than $8 below the strike
DOnly if the stock falls below the strike by more than $8
A straddle has a V-shaped payoff: the combined payoff is |S_T − K| — the absolute distance from the strike. After paying $8 in total premium, the investor profits if |S_T − K| > $8, i.e., the stock finishes more than $8 away from the strike in either direction. This is why straddles are used before earnings: the direction of the move is unknown, but if the move is large enough to exceed the total premium paid, the straddle profits regardless of direction.
Question 3 True / False
Put-call parity holds for both European and American options traded on the same underlying stock.
TTrue
FFalse
Answer: False
Put-call parity holds exactly only for European options (exercisable only at expiration). American options carry an early exercise premium — the holder may exercise before expiration when advantageous — which breaks the exact replication argument. For American options, the relationship satisfies an inequality: S − K ≤ C − P ≤ S − PV(K), rather than an equality.
Question 4 True / False
A bull spread with strikes K₁ = $50 and K₂ = $60 and a net premium of $3 will always profit if the stock finishes above $60 at expiration.
TTrue
FFalse
Answer: True
A bull spread buys a call at K₁ = $50 and sells a call at K₂ = $60. Above K₂, both options are in-the-money: the long call pays S_T − 50 and the short call costs S_T − 60, for a net payoff of $10 regardless of how far above $60 the stock finishes. After paying the $3 net premium, the profit is a fixed $7. The payoff is capped and flat above K₂, and the break-even is at K₁ + premium = $53.
Question 5 Short Answer
Explain why put-call parity must hold for European options in liquid markets, using the concept of no-arbitrage.
Think about your answer, then reveal below.
Model answer: If C − P ≠ S − PV(K), you can construct a riskless profit with no capital at risk. A portfolio of long call plus short put has the same payoff at expiration as a long forward (obligation to buy at K): both pay S_T − K regardless of whether S_T is above or below K. A long forward today costs S − PV(K). If C − P > S − PV(K), sell the call+short-put portfolio and buy the forward, locking in an immediate profit with zero net risk. Competition among arbitrageurs eliminates any such gap, enforcing parity.
The replication argument is the backbone of no-arbitrage pricing: two portfolios with identical payoffs in every future state must have identical prices today. The long call plus short put replicates a forward, whose current value is S − PV(K). Any wedge between C − P and S − PV(K) is immediately exploitable by buying cheap and selling expensive. In liquid markets, arbitrageurs eliminate these gaps almost instantly — which is why put-call parity is an empirical near-equality rather than just a theoretical result, and why it holds without any model assumptions about return distributions.