Practice: Option Replication Using Put-Call Parity
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DerivativesOption Replication Using Put-Call Parity
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Question 1Exam level
A European call option on a non-dividend-paying stock has a premium of $8. The stock trades at $52, the exercise price is $50, the risk-free rate is 4% per year, and the option expires in one year. Using put-call parity, the value of a European put option with the same exercise price and expiration is closest to:
How sure are you?
Correct: A. Put-call parity: C + PV(X) = P + S. Rearrange to P = C + PV(X) - S. PV(X) = 50 / 1.04 = $48.08. P = 8 + 48.08 - 52 = $4.08. The correct answer is A.
B. You might use X instead of PV(X): P = 8 + 50 - 52 = $6.00, then rounds to $6.08. The strike price must be discounted. Using X instead of PV(X) is the single most common put-call parity error on the CFA exam.
C. You might subtract the call value from the stock price: P = S - C - X = 52 - 8 - 50 = -6 (takes absolute value), or uses wrong formula arrangement. Formula arrangement is C + PV(X) = P + S. Every term must be on the correct side before solving.
Unit: option-replication-using-put-call-parity
Question 2Harder
Using put-call parity, which of the following positions is most likely a synthetic long call?
How sure are you?
Correct: C. From C + PV(X) = P + S, rearrange: C = P + S - PV(X). A synthetic long call = long put + long stock + SHORT risk-free bond (borrow PV(X)). The minus PV(X) term means borrowing. You receive cash today equal to PV(X). Option B (long put + long stock) is a protective put, not a synthetic call.
A. Long put + short stock + long bond = synthetic short call or synthetic put. You might confuse directions. This combination replicates a short call or long put in a different way, not a synthetic long call.
B. Long put + long stock looks similar to C = P + S. Students forget the minus PV(X) (the borrow) term. Without the bond, you hold a protective put (P + S), not a synthetic call (P + S - PV(X)).
Unit: option-replication-using-put-call-parity
Question 3Harder
A trader observes: C = $5, P = $3, S = $48, X = $50, r = 5%, T = 1 year. Which statement best describes the situation?
How sure are you?
Correct: B. Check parity: PV(X) = 50/1.05 = $47.62. Left side: C + PV(X) = 5 + 47.62 = $52.62. Right side: P + S = 3 + 48 = $51.00. Left side ($52.62) > right side ($51.00). The fiduciary call (left side) is overpriced. Arbitrage: sell the overpriced side (sell call, sell bond = borrow PV(X)), buy the cheap side (buy put, buy stock). Profit = 52.62 - 51.00 = $1.62 per share, risk-free at expiration.
A. You might eyeball the numbers and assume the difference is a bid-ask spread or rounding. Parity is violated: $52.62 ≠ $51.00. A $1.62 gap on a $48 stock is material and creates riskless profit.
C. Student reverses the direction. Sees the call is related to the overpriced side but labels the wrong instrument as the problem. The entire left-side PORTFOLIO (call + bond) is overpriced. The individual call price is part of the overpriced side, but you must sell the entire fiduciary call (call + bond = sell call + borrow), not just the call alone.
Unit: option-replication-using-put-call-parity
Question 4Exam level
Put-call parity is formally expressed as C + PV(X) = P + S. What does the LEFT side of this equation represent, most likely?
How sure are you?
Correct: B. The fiduciary call = long call (C) + investment of PV(X) in a risk-free bond that matures to X at expiration. The RIGHT side (P + S) is the protective put: long put + long stock. Both portfolios produce identical payoffs at expiration in all scenarios. Hence they must have the same present value. That is put-call parity.
A. You might reverse the labels. They associate 'put' with the left side because of the word 'parity' suggesting symmetry. The protective put is on the RIGHT side: P + S. The left side has no put. It has a call plus a bond.
C. A synthetic forward = long call + short put. Students confuse 'bond' and 'short put' because both involve cash-like components. The fiduciary call has a LONG bond (lending PV(X)), not a short put. A synthetic forward has no bond.
Unit: option-replication-using-put-call-parity
Question 5Exam level
A European put option with X = $55 and T = 6 months trades at $6. The stock trades at $50. The risk-free rate is 6% per year (continuous compounding). The European call price from put-call parity is closest to:
How sure are you?
Correct: B. C = P + S - PV(X). Using continuous compounding: PV(X) = 55 × e^(-0.06 × 0.5) = 55 × e^(-0.03) = 55 × 0.97045 = $53.37. C = 6 + 50 - 53.37 = $2.63. 03 = $53.40, giving C = $2.60. Close but different. The question specifies 'continuous compounding,' which requires e^(-rT) notation.
A. You might use T = 6 (months as a whole number) instead of T = 0.5 (years): e^(-0.06×6) = 0.698, PV(X) = 38.38, C = 6 + 50 - 38.38 = 17.62. Clearly wrong. Or uses discrete compounding with T = 6 months interpreted as 6 periods. T must be in years when the risk-free rate is quoted annually. 6 months = 0.5 years.
C. You might use simple interest: PV(X) = 55/(1 + 0.06×0.5) = 55/1.03 = $53.40, C = 6 + 50 - 53.40 = $2.60. Rounds up incorrectly. The question specifies continuous compounding. Simple interest gives a slightly different number and is the wrong method.
Unit: option-replication-using-put-call-parity
Question 6Exam level
Which of the following most likely explains why put-call parity applies only to European options and not American options?
How sure are you?
Correct: A. Put-call parity derives from the fact that two portfolios with identical payoffs at expiration must have the same present value. With American options, early exercise is possible. The arbitrage that enforces parity relies on holding both portfolios to expiration. Early exercise disrupts this. An American put might be exercised early (especially deep in-the-money), giving its holder more than the European formula implies. This destroys the exact equality, leaving only inequality bounds.
B. It is true that American options are worth at least as much as European options (early exercise premium), so candidates reason that higher prices break the equation. Higher prices alone do not break parity. What breaks parity is that the early exercise RIGHT changes the payoff structure before expiration. The inequality relationship still holds, but exact equality does not.
C. You might confuse instrument type (American/European) with trading venue (exchange/OTC). Whether options are exchange-traded or OTC is irrelevant to put-call parity. European options exist on exchanges; American options exist OTC. The distinction is exercise style, not venue.
Unit: option-replication-using-put-call-parity
Question 7Harder
An analyst uses put-call parity to construct a synthetic long stock position. Which combination is most likely correct?
How sure are you?
Correct: A. From C + PV(X) = P + S, rearrange for S: S = C - P + PV(X). In position terms: long stock = long call + short put + LONG bond (invest PV(X), i.e., lend PV(X) at the risk-free rate). Option A states: 'long call + short put + invest PV(X)'. This matches exactly. Not borrow.
B. Long call + long put is a straddle. A completely different strategy for volatility bets. A straddle has convex payoff and does not replicate the linear payoff of stock ownership.
C. Option C says 'borrow PV(X)' instead of 'invest PV(X).' Borrowing vs investing is the critical sign distinction. From S = C - P + PV(X): the +PV(X) term means you OWN a bond (invest/lend). Borrowing would be -PV(X), which would create a synthetic forward, not synthetic stock.
Unit: option-replication-using-put-call-parity
Question 8Exam level
A fiduciary call portfolio and a protective put portfolio on the same stock have these current values: Fiduciary call = $14.20, Protective put = $12.80. An arbitrageur should most likely:
How sure are you?
Correct: B. Put-call parity requires fiduciary call = protective put. Here fiduciary call ($14.20) > protective put ($12.80). The fiduciary call is overpriced. Arbitrage: sell the overpriced portfolio (sell the call, sell the bond = borrow PV(X)), buy the underpriced portfolio (buy the put, buy the stock). Profit = $14.20 - $12.80 = $1.40 per unit, locked in today. The positions offset exactly at expiration regardless of the final stock price.
A. Buy the fiduciary call. It is larger in value so seems 'better.' Candidates confuse 'bigger' with 'underpriced.'. A larger current price means OVERPRICED. You sell overpriced assets in arbitrage, not buy them.
C. A $1.40 gap might seem like a bid-ask spread on a large-notional trade. The question states no transaction costs. On the CFA exam, assume frictionless markets unless told otherwise. A $1.40 gap creates a $1.40 riskless profit. Always exploit it.
Unit: option-replication-using-put-call-parity
Question 9Exam level
All else equal, if the risk-free interest rate increases, which effect on put-call parity is most accurate?
How sure are you?
Correct: B. PV(X) = X / (1+r)^T. As r increases, PV(X) falls (denominator grows). From C = P + S - PV(X): as PV(X) falls, for the equation to hold, C must rise or P must fall. Economically: a higher risk-free rate makes the deferred payment of the exercise price (the benefit of owning a call vs the stock) more valuable. Higher r, then higher call value, lower put value. This matches option B.
A. Intuition: 'higher rates mean everything is worth more, so PV(X) goes up.' Rates and prices move inversely for bonds. PV(X) is a present value. It moves INVERSELY with rates. Higher r, then lower PV(X). Bond pricing fundamentals: prices fall when rates rise.
C. Students who do not see the bond component in the formula assume interest rates are irrelevant to option pricing at Level 1. The bond component (PV(X)) is explicitly rate-sensitive. Put-call parity directly links option pricing to the risk-free rate through this term.
Unit: option-replication-using-put-call-parity
Question 10Harder
A stock is priced at $60, and a 1-year European call with X = $60 is priced at $7.50. The 1-year risk-free rate is 5%. A European put with X = $60 is priced at $6.00. Which statement is most likely correct?
How sure are you?
Correct: B. Check parity: PV(X) = 60/1.05 = $57.14. Left side (fiduciary call): C + PV(X) = 7.50 + 57.14 = $64.64. Right side (protective put): P + S = 6.00 + 60 = $66.00. Right side ($66.00) > left side ($64.64). The protective put is overpriced. Arbitrage: sell the right side (sell put, short stock), buy the left side (buy call, invest PV(X)). Profit = 66.00 - 64.64 = $1.36 risklessly. The overpriced instrument is the put (within the protective put portfolio).
A. The numbers look close; candidates assume small differences are rounding. $66.00 vs $64.64 is a $1.36 discrepancy. Parity is violated and the arbitrage trade yields $1.36 risk-free.
C. You might identify that the right side is too high and tries to sell the right components, but labels it as 'call is overpriced'. Wrong label, wrong direction. The left side (fiduciary call) is CHEAPER. You BUY the left side. The call is not overpriced. The protective put portfolio is overpriced.
Unit: option-replication-using-put-call-parity
Question 11Exam level
At expiration, which of the following statements about a fiduciary call (C + PV(X)) and a protective put (P + S) is most likely correct?
How sure are you?
Correct: B. Track both portfolios at expiration. FIDUCIARY CALL: If S > X: call pays S-X, bond pays X. Total = S-X+X = S. If S < X: call expires worthless (0), bond pays X. Total = X. PROTECTIVE PUT: If S > X: put expires worthless (0), stock worth S. Total = S. If S < X: put pays X-S, stock worth S. Total = X-S+S = X. Both portfolios: pay S when S > X and pay X when S < X. Identical payoffs, then same present value, then put-call parity.
A. The call and put have opposite payoff shapes, so candidates assume the portfolios also differ at expiration. That is exactly the insight of put-call parity. The PORTFOLIOS are identical at expiration even though the individual instruments differ. The bond in the fiduciary call and the stock in the protective put work together to equalize payoffs.
C. The bond seems like an extra 'free' component that adds value. The bond is not free. It was purchased at cost PV(X). The bond's contribution is priced in, which is exactly why the two portfolios cost the same today.
Unit: option-replication-using-put-call-parity
Question 12Exam level
Which of the following is most likely NOT a required assumption for put-call parity to hold exactly?
How sure are you?
Correct: C. Put-call parity requires: (1) European options. Early exercise must be impossible so payoffs match only at expiration; (2) No dividends. Dividends paid during the option life reduce the stock price and distort the right side (P + S); (3) No transaction costs. Arbitrage enforces parity, but transaction costs create a no-arbitrage band instead of a single point. Whether options are exchange-traded or OTC is completely irrelevant to the mathematical relationship. Parity holds for any European options regardless of trading venue.
A. You might argue American options 'almost' satisfy parity. But 'almost' is not 'exactly.'. European-style is a hard requirement for exact parity. American options violate it by definition.
B. Candidates who know the basic formula C + PV(X) = P + S may not recall that dividends modify the relationship. When dividends are present, the formula becomes C + PV(X) = P + S - PV(Dividends). The no-dividend assumption is required for the standard formula to hold without adjustment.
Unit: option-replication-using-put-call-parity
Question 13Above the exam
An investor wants to replicate the payoff of a long call option using put-call parity, but only has access to the underlying stock, a risk-free bond, and a put option with the same strike and expiration, not the call itself. Combining the put-call parity relationship with basic algebra, the investor should most likely construct the synthetic long call by:
How sure are you?
Correct: B. Put-call parity states C + PV(X) = P + S, so C = S + P - PV(X). A synthetic long call is replicated by going LONG the stock, LONG the put, and effectively BORROWING the present value of the strike price (a negative bond position, i.e., -PV(X)), since subtracting PV(X) on the right side of the rearranged equation corresponds to borrowing that amount rather than investing it. This combination reproduces the call's payoff at every stock price using only the stock, a put, and financing.
A. Selling the put and buying a bond, with no stock position, does not correspond to any rearrangement of C = S + P - PV(X); a synthetic call specifically requires a LONG stock position combined with a long put and borrowing, not a short put position with no stock at all.
C. Buying the put and shorting the stock, with no financing, replicates a different payoff entirely (closer to a synthetic short call or another combination), not a long call; the correct replication requires a LONG stock position (not short) alongside the long put and the borrowing (financing) component.
Unit: option-replication-using-put-call-parity
Question 14Above the exam
Using put-call FORWARD parity, an investor wants to synthetically create a long forward contract position using options only, no direct forward or stock position. Combining put-call forward parity with the standard put-call parity relationship, the investor should most likely:
How sure are you?
Correct: A. Put-call forward parity shows that C - P = PV[F0(T) - X], meaning a long call combined with a short put at the same strike (long call minus short put, i.e., long call and short put simultaneously) replicates the payoff of a long forward contract at that strike price. This is the options-only synthetic forward construction: buy the call, sell (write) the put, same strike and expiration, no stock or direct forward position needed.
B. A long straddle (buying both a call AND a put at the same strike) replicates a bet on large price movement in EITHER direction, profiting from volatility, not a directional forward-like exposure; it does not replicate a forward position's payoff at all.
C. A short straddle (selling both a call and a put) profits from LOW volatility (the stock staying near the strike), which is a fundamentally different risk profile from a forward contract's linear, symmetric exposure to the underlying's price; it does not replicate a forward position either.