CFA Level I Exam · Option Replication Using Put-Call Parity

Fiduciary Call vs Protective Put: Payoffs and Parity

Updated 7 October 2026 · Fact-checked

A fiduciary call is a European call plus a risk-free bond that pays the exercise price X at expiry. A protective put is the underlying asset plus a put with the same X. Both pay max(S_T, X) at expiry, so both cost the same today: c + X/(1+r)^T = S₀ + p.

Understand Fiduciary Call and Protective Put

Start with two simple portfolios. Both are built from options and basic assets. Both end up with the same payoff at expiry, whatever the asset price does.

The fiduciary call is a European call option plus a zero-coupon risk-free bond. The bond has a face value equal to the exercise price X and matures on the option's expiry date. Its price today is X ÷ (1 + r)^T. The name comes from the idea of a prudent, safe holding: the bond guarantees you at least X at expiry, and the call adds upside.

The protective put is one unit of the underlying asset plus a European put on it with the same exercise price X and the same expiry. The put acts as insurance. If the asset falls below X, the put pays the shortfall, so your floor is X.

Check the payoffs at expiry. If S_T > X: the fiduciary call pays the call (S_T − X) plus the bond X, which is S_T. The protective put pays S_T and the put is worthless, so S_T. If S_T ≤ X: the fiduciary call has a worthless call and the bond pays X. The protective put pays S_T plus the put (X − S_T), which is X. In both cases the payoff is max(S_T, X).

Identical payoffs in every state mean identical cost today, or there would be arbitrage. That equality is put-call parity: c + X/(1+r)^T = S₀ + p. On a payoff diagram, both portfolios are flat at X until S_T reaches X, then rise one-for-one with the asset. The difference is only in the build: one uses a call and a bond, the other uses the asset and a put.

Key formulas to remember

Fiduciary call payoff at expiry: max(S_T, X) = max(S_T − X, 0) + X; Call payoff plus the bond's face value X.
Protective put payoff at expiry: max(S_T, X) = S_T + max(X − S_T, 0); Asset value plus put payoff. Floor is X.
Put-call parity (European options, no income on asset): c + X ÷ (1 + r)^T = S₀ + p; Fiduciary call cost on the left, protective put cost on the right. Same X and T for both options.
Present value of the bond: PV(X) = X ÷ (1 + r)^T; Use the risk-free rate r with the same compounding as T. T is in years.
Rearranged for call or put: c = S₀ + p − X ÷ (1 + r)^T; p = c + X ÷ (1 + r)^T − S₀; Use to find a missing option price.

How to solve Fiduciary Call and Protective Put questions

Use this method for any question on fiduciary calls, protective puts or the parity link between them.

1Identify the two portfolios. Fiduciary call = call + bond paying X. Protective put = asset + put.
2Confirm the conditions: European options, same X, same expiry, and the same underlying.
3Compute the bond's present value: X ÷ (1 + r)^T. Match T to the rate's units, in years.
4Write the parity equation: c + PV(X) = S₀ + p.
5Substitute the known values and solve for the missing one.
6For payoff questions, split into S_T > X and S_T ≤ X and compute each portfolio's value.
7Check that the answer is sensible: option prices cannot be negative, and the two sides must match.
8Eliminate options that reverse the sign of PV(X) or use X without discounting.

Quickest way: Parity in one line

When to use it: Any numerical question giving three of c, p, S₀ and PV(X) and asking for the fourth.

Remember: call + bond = put + stock.
Compute PV(X) first on the calculator: X ÷ (1 + r)^T. On the BA II Plus, enter 1.04 (that is, 1 + r as a decimal, using your own r), press yˣ, enter T, press =, then press 1/x, then × X =.
Move terms to isolate the unknown. For a call: c = S₀ + p − PV(X).
Eliminate any option that is a negative price or ignores discounting.
For payoff questions, answer is max(S_T, X) for both portfolios.

Common mistakes in Fiduciary Call and Protective Put

Using X instead of its present value X ÷ (1 + r)^T in the parity equation.
The payoff uses X, so students carry it into the cost equation.
Fix: Payoffs use X at expiry. Costs today use PV(X). Always discount.
Thinking the fiduciary call and protective put have different payoffs.
The portfolios look different in construction.
Fix: Both equal max(S_T, X). Check both states, above and below X.
Mixing up which side holds the stock.
Both sides have two instruments and the names are easy to confuse.
Fix: Fiduciary call: call + bond. Protective put: stock + put. The put pairs with the stock that it protects.
Applying parity with different exercise prices or expiries for the call and put.
Questions list several options and students pick the wrong pair.
Fix: Parity needs the same X and T, and the same underlying. Match them first.
Using the wrong T, such as months instead of years.
Rushing under time pressure.
Fix: Convert to years before discounting. Six months is T = 0.5.

Worked examples

Example 1

A European call on a stock trades at $6.00. A European put with the same strike of $50 and expiry in six months trades at $4.20. The stock is at $52 and the risk-free rate is 4% a year (annual compounding). Is the fiduciary call's cost equal to the protective put's cost? Compute both.

Show the solution

PV(X) = 50 ÷ (1.04)^0.5. Square root of 1.04 is 1.019804, so PV(X) = 50 ÷ 1.019804 = 49.0290.
Fiduciary call cost = c + PV(X) = 6.00 + 49.0290 = 55.0290.
Protective put cost = S₀ + p = 52 + 4.20 = 56.20.
The costs differ: 55.03 versus 56.20.

Answer: No. The fiduciary call costs about $55.03 and the protective put costs $56.20, so the protective put is more expensive and parity is violated.

Example 2

A stock is at €40. A European put with strike €44 and one year to expiry costs €3.00. The risk-free rate is 5% a year. Using parity, what should the European call with the same strike and expiry cost? Options: A) €1.10 B) €2.10 C) €3.00

Show the solution

PV(X) = 44 ÷ 1.05 = 41.9048.
Parity: c = S₀ + p − PV(X) = 40 + 3.00 − 41.9048.
c = 1.0952, which rounds to €1.10.
Match to the options: €1.10 is option A. Option C (€3.00) is just the put price, and using X undiscounted would give a negative call price.

Answer: €1.10, option A.

Exam tips

Know the payoff max(S_T, X) for both portfolios. Questions often ask which portfolio has a floor at X.
Always discount X with the risk-free rate. Options that use X undiscounted are traps.
Parity holds for European options with the same strike and expiry only. Watch for mismatches.
Use the parity equation to find a missing price, then eliminate options that are negative or break the identity.
A payoff diagram question: both portfolios are flat at X until S_T = X, then slope up at one.

Practice questions from Option Replication Using Put-Call Parity

Fiduciary Call and Protective Put in other exams

The same ground in other exams, if you are preparing for more than one or want another angle on it.

Fiduciary Call and Protective Put: frequently asked questions

What is the difference between a fiduciary call and a protective put?

A fiduciary call is a call plus a risk-free bond paying X. A protective put is the underlying plus a put. They are built differently but give the same payoff at expiry, max(S_T, X), so they have the same cost today.

Why must a fiduciary call and protective put cost the same?

They produce identical payoffs in every state at expiry. If their prices differed, you could buy the cheaper one, sell the dearer one and lock in a risk-free profit. Arbitrage forces the prices equal.

What does the payoff diagram of a fiduciary call look like?

It is flat at X while the asset price at expiry is below X. Above X it rises one-for-one with the asset price. The protective put gives the same shape.

Does put-call parity work for American options?

Not as an exact equality. The standard parity relationship is for European options, because early exercise can change values.