Option Replication Using Put-Call Parity

Derivatives, LOS weight share 0.5 percent of the 365 Level I learning outcomes.

DerivativesOption Replication Using Put-Call Parity

The formula uses the present value of the strike price, never the strike price itself, and the exam plants the answer a candidate gets by forgetting to discount as one of the wrong choices on almost every put-call parity question.

Before you watch

Answer these three first. Getting them wrong now is normal, and it helps the lesson stick. Reveal the answers when you're done, then read on.

1. A European call on a non-dividend-paying stock trades at $8. The stock is $52, the strike is $50, the risk-free rate is 4%, and the option expires in one year. Using put-call parity, the value of the corresponding European put is closest to:

Answer: B. Put-call parity: C + PV(X) = P + S, so P = C + PV(X) - S. PV(X) = 50 / 1.04 = $48.08. P = 8 + 48.08 - 52 = $4.08. Using the raw strike price ($50) instead of its present value is the single most common error on this formula.

2. The left side of the put-call parity equation, C + PV(X), is best described as:

Answer: B. The fiduciary call is a long call plus an investment of PV(X) in a risk-free bond that matures to exactly X at expiration; it sits on the left side of the equation. The right side, P + S, is the protective put (long put plus long stock). Reversing these two labels is a frequently tested error.

3. Put-call parity holds as an exact equality only for:

Answer: B. Put-call parity is derived from the fact that two specific portfolios have identical payoffs only if held to expiration with no early exercise possible; European options cannot be exercised early, preserving the exact equality. American options may be exercised early, which can break the equivalence, leaving only inequality bounds rather than exact parity. Whether an option trades on an exchange or OTC is irrelevant to the relationship.

The lesson

The video lesson for this unit is recorded and waiting to be published. Everything it teaches is written out below.

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.

Learning outcomes covered by this module

Verbatim from the 2026 CFA Level I topic outline. Every practice question and key rule below is tagged to one of these where the stem and explanation make the match clear.

  1. explain put-call parity for European options
  2. explain put-call forward parity for European options

Key rules

Written from this module's own lesson and the 2026 CFA Level I topic outline, in teaching order, each tagged to the learning outcome it belongs to where that is clear.

LOS 01

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.

LOS 01

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.

LOS 01

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.

LOS 02

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

Authored only where a key rule has an arbitrary number, list, or formula shape worth a memory device; a module with none of those has no tricks here, on purpose.

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

Authored, ordered steps for answering this module's question types; a calculation module's calculator-dependent step ends with a bracketed BA II Plus keystroke sequence.

  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).

One card

Condensed from the key rules and tricks above, nothing new. What you'd want on one index card the night before.

Practice questions

Pick an answer, say how sure you are, then reveal. Every wrong choice gets its own explanation. 10 question(s) available for this unit.

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 9Above 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 10Above 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

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