Option Replication Using Put-Call Parity

Derivatives. Worth 5 to 8 percent of the exam. One session: the lesson, the rules, the method, then the questions.

DerivativesOption Replication Using Put-Call Parity
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The lesson

The video lesson for this unit is recorded and waiting to be published. Until it is, the rules and the method below carry everything this session needs; watching is a way of hearing it, not the only way of getting it.

The reading

Five to ten minutes on this one unit: what the exam wants, the idea in plain words, then straight into the trap and the practice.

The exam wants you to calculate any one of C, P, S or PV(X) from put-call parity given the other three, construct the six named synthetic positions, and determine the correct four-leg arbitrage trade when parity is violated.

Put-call parity links a European call, a European put, the underlying and a risk-free bond into one equation, because two specific portfolios are built to have identical payoffs at expiration: C + PV(X) = P + S. The left side, a fiduciary call, is a long call plus a risk-free bond investment of PV(X) that grows to exactly X by expiration. The right side, a protective put, is a long put plus a long position in the underlying itself. If the stock finishes above the strike, both portfolios pay the stock price. If it finishes below the strike, both portfolios pay exactly X. Because the payoffs match in every scenario, the two portfolios must carry the same value today, and that equality is the whole formula.

The strike price must always enter the equation as its present value, never at face value. PV(X) = X / (1 + r)^T under discrete compounding, or X x e^(-rT) under continuous compounding, whichever convention a question specifies. The raw strike price X only ever appears in the description of the expiration payoff, never inside the present-value form of the parity equation itself, and substituting X directly for PV(X) is the exam's single most frequently planted wrong answer.

Rearranging the same equation solves for whichever of the four variables a question leaves out: P = C + PV(X) - S to solve for the put, C = P + S - PV(X) to solve for the call, S = C - P + PV(X) to solve for the stock. Subtracting P and PV(X) from both sides of the original equation gives the put-call forward parity relationship, C - P = S - PV(X), which says a synthetic forward, long call, short put, same strike and expiration, replicates a position economically equivalent to holding the underlying financed at the risk-free rate. Every other synthetic position, synthetic call, synthetic put, synthetic stock, synthetic bond, comes from rearranging the same formula to isolate a different term, and each one almost always carries a bond leg; a synthetic position with no bond component in the answer choices is very likely missing a piece.

When the two sides of the equation are not equal, the correct arbitrage sells the entire overpriced portfolio and buys the entire underpriced one, never a single instrument that looks mispriced on its own. If the fiduciary call side exceeds the protective put side, the fiduciary call is overpriced: sell the call and borrow PV(X), while buying the put and buying the stock. If the protective put side is larger instead, the trade reverses: sell the put and short the stock, while buying the call and investing PV(X). The trade is always four legs, executed together, never one leg in isolation.

This equality holds exactly only for European-style options, because its derivation depends on holding both portfolios all the way to expiration and confirming the payoffs match at that one point in time. An American option's holder can exercise early, which breaks the guaranteed payoff match the derivation depends on, and the exact equation becomes an inequality instead. A question that specifies European options is telling you the equation applies exactly as written; American options are the exception this exam expects you to catch.

Put-call parity as a balanced scale Protective put stock + put Fiduciary call call + bond
A protective put and a fiduciary call must be worth the same today, because they pay off the same at expiration. If the scale tips, an arbitrage trade brings it back level.

Worked in full

A European put option with a strike price of $50 and one year to expiration trades at $6. The underlying stock trades at $48. The risk-free rate is 5 percent annually, discretely compounded. What is the value of the corresponding European call, by put-call parity? PV(X) = $50 / (1.05)^1 = $47.62. Solving parity for the call: C = P + S - PV(X) = $6 + $48 - $47.62 = $6.38. The call should be worth about $6.38.

The same problem, one step removed

Same inputs: put price $6, strike $50, stock $48, risk-free rate 5 percent, one year to expiration. Compute PV(X) first, then rearrange parity to solve for the call yourself.

The trap

Using the strike price X directly instead of its present value PV(X) is the exam's most frequently placed wrong answer; the formula always requires the strike discounted back to today at the risk-free rate, never its face value.

What this unit turns on

Read these before the questions, not after them. Everything here traces to this module's own lesson and to the 2026 outline.

Put-call parity links a call, a put, the underlying, and a risk-free bond into one equation because two specific portfolios must have identical payoffs at expiration

The formula is C + PV(X) = P + S for European options on a non-dividend-paying underlying, where PV(X) is the present value of the strike price. The left side, the fiduciary call, is a long call plus a risk-free bond investment of PV(X) that grows to X by expiration. The right side, the protective put, is a long put plus a long position in the underlying. At expiration, tracking both portfolios across every possible stock price shows they always pay identically, S when S exceeds X, X when S is below X, so their present values, and therefore their costs today, must be equal or a riskless arbitrage would exist.

The strike price must always be discounted to its present value before entering the formula, never used at face value

PV(X) = X / (1+r)^T under discrete compounding, or X multiplied by e^(-rT) under continuous compounding, whichever convention the question specifies. The raw strike price X appears only in the expiration payoff description, never in the present-value form of the parity equation itself; substituting X directly for PV(X) is the exam's most frequently placed wrong-answer trap on this topic, and the word 'continuously compounded' in a question stem is the explicit signal to use the exponential form rather than the discrete one.

When parity is violated, the correct arbitrage sells the entire overpriced portfolio and buys the entire underpriced portfolio, not a single mispriced instrument

Compute both sides of the equation numerically: if the fiduciary call side (C + PV(X)) exceeds the protective put side (P + S), the fiduciary call is overpriced, sell the call and borrow PV(X) (a short bond), while buying the put and buying the stock; if the protective put side exceeds the fiduciary call side, the reverse trade applies, sell the put and short the stock, while buying the call and investing PV(X) (a long bond). The profit locked in today equals the exact difference between the two sides, and it is realized regardless of where the stock price ends up at expiration, because the four legs' payoffs cancel exactly.

Rearranging the parity equation produces a forward-parity relationship and a set of named synthetic positions, each with a specific sign for the bond leg

Subtracting P + PV(X) from both sides of C + PV(X) = P + S gives C - P = S - PV(X), the put-call forward parity relationship, stating that a synthetic forward (long call, short put, same strike and expiration) replicates a position economically equivalent to holding the underlying financed at the risk-free rate. Rearranging further for other combinations produces the full set of synthetic positions, for example a synthetic long call equals long put plus long stock plus a short bond (borrowing PV(X)); the sign on the bond term always indicates the correct direction, a positive PV(X) term means invest (lend), a negative PV(X) term means borrow, and treating this sign carelessly is a frequent source of error when constructing a synthetic position.

The trick

Always discount the strike to PV(X); using X directly is the exam's most common trap

Write PV(X) = X/(1+r)^T (or X e^(-rT) for continuous compounding) as a first, separate step before substituting into the parity formula, rather than plugging the strike price straight in.

Fiduciary call = call plus cash (bond); protective put = put plus price (stock); the labels are easy to reverse

The left side of C + PV(X) = P + S contains no put at all, it is a call and a bond; the right side contains no call, it is a put and the stock.

In an arbitrage question, sell the entire overpriced portfolio and buy the entire underpriced portfolio, never just the single option that looks mispriced

The trade is always four legs; identifying which whole side is more expensive, then executing all of that side's legs together, is what locks in the riskless profit.

The method

The order to work a question of this type in, every time, before you touch the numbers.

  1. For a missing-price question, write C + PV(X) = P + S first, compute PV(X) as a separate step using the compounding convention specified in the question, then rearrange and solve for the unknown.
  2. For a fiduciary-call-versus-protective-put question, remember the left side is call plus bond and the right side is put plus stock, and never swap these labels.
  3. For an arbitrage-direction question, compute both sides numerically, identify the larger (overpriced) side, and sell every leg of that side while buying every leg of the smaller (underpriced) side.
  4. For a synthetic-position question, rearrange the formula algebraically for the desired instrument, then read the sign on the PV(X) term literally: positive means invest (long bond), negative means borrow (short bond).

Two worked examples, then you are on your own

The first is worked in full. The second gives you the setup and stops. After that the questions give you nothing, which is the point: the help fades on purpose, so the last thing you practise is the thing the exam actually asks of you.

Worked in full

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:

Answer 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.

Your turn, setup given

Put-call parity is formally expressed as C + PV(X) = P + S. What does the LEFT side of this equation represent, most likely?

For a missing-price question, write C + PV(X) = P + S first, compute PV(X) as a separate step using the compounding convention specified in the question, then rearrange and solve for the unknown.

The practice run

Pick an answer, say how sure you are, then reveal. Being sure and wrong is the most useful thing that can happen in a session, so answer honestly: it sends the unit back to learning and puts it at the front of your revision queue.

Question 1Harder

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 2Harder

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 3Exam 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 4Exam 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 5Harder

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 6Exam 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 7Exam 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 8Harder

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 9Exam 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 10Exam 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 11Above 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 12Above 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.

Unit: option-replication-using-put-call-parity