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Lesson 1 of 5 · 13 min

Fiduciary call, protective put and put–call parity

A call plus a bond paying X and a share plus a put pay exactly the same at expiration, so they must cost the same today: $S_0 + p_0 = c_0 + X(1+r)^{-T}$ .

Video lesson · 6 min. Watch it first, then use the text below to go deeper.

In short

A fiduciary call is a long European call plus a risk-free zero-coupon bond that pays X at expiration. It costs $c_0 + X(1+r)^{-T}$ .
A protective put is a long share plus a long European put on it. It costs $S_0 + p_0$ .
At expiration both portfolios are worth $\max(S_T, X)$ : X if the share ends below the exercise price, $S_T$ if it ends above.
Identical payoffs in every state mean identical prices today. This is put–call parity: $S_0 + p_0 = c_0 + X(1+r)^{-T}$ .
Parity needs European options on the same underlying with the same exercise price and expiration, and (in this reading) an underlying with no income or costs.
Rearranged, parity gives the no-arbitrage put price from a traded call, or the call price from a traded put.

Two ways to get upside with a floor

Imagine an investor who wants to profit if a share rises but never wants to end up with less than the exercise price X. There are two simple ways to build that.

The first is the fiduciary call. She buys a European call with exercise price X and puts the present value of X into a risk-free zero-coupon bond. The bond matures at exactly X on the expiration date, so cash is there to pay for the share if the call is worth exercising. Cost today: $c_0 + X(1+r)^{-T}$ .

If the share ends above X, she exercises: the call pays $S_T - X$ and the bond pays X, so she has $S_T$ . If the share ends below X, the call expires worthless and she keeps the bond's X. Either way she holds the larger of $S_T$ and X.

Adding the bond lifts the call's payoff by X = 75 everywhere. The result is flat at 75 below the exercise price and rises one-for-one above it:

\max(S_T, X)

The protective put pays the same

The second route is the protective put: buy the share and buy a European put on it with the same X and expiration. Cost today: $S_0 + p_0$ .

If the share ends below X, she exercises the put and sells the share for X. If it ends above X, the put expires worthless and she simply holds a share worth $S_T$ . Again she ends up with the larger of $S_T$ and X.

The table compares the two portfolios in every possible state at expiration. Line by line the pieces differ, but the totals are always the same.

Value at expiration, same X and T
Position	$S_T < X$ (put exercised)	$S_T = X$	$S_T > X$ (call exercised)
Share	$S_T$	$S_T$	$S_T$
Long put	$X - S_T$	0	0
Protective put total	X	X (= $S_T$ )	$S_T$
Long call	0	0	$S_T - X$
Bond paying X	X	X	X
Fiduciary call total	X	X (= $S_T$ )	$S_T$

The put tops the share up to 75 whenever it falls below the exercise price (at 60 the put pays 15). The violet line is exactly the fiduciary call's line from the previous figure.

Same payoffs, same price: put–call parity

No-arbitrage pricing says that two portfolios with identical future cash flows in every state must trade at the same price today (ignoring transaction costs). The fiduciary call and the protective put pass that test, so their costs must be equal.

That equality is put–call parity: $S_0 + p_0 = c_0 + X(1+r)^{-T}$ . Read it as: share plus put equals call plus bond. It links four prices, so if you know any three you can solve for the fourth.

The conditions matter. Both options must be European, written on the same underlying, with the same exercise price and the same expiration date, and the bond must mature on that date paying X. The reading also assumes the underlying pays no dividends and has no carry costs.

Solving for the put gives $p_0 = c_0 + X(1+r)^{-T} - S_0$ ; solving for the call gives $c_0 = S_0 + p_0 - X(1+r)^{-T}$ . These are the two most common exam calculations.

Example 1's numbers: both portfolios cost EUR 83.10. The put's price is whatever makes the right bar as tall as the left one.

Memory hook: the left side of $S_0 + p_0 = c_0 + PV(X)$ is the protective put, the right side is the fiduciary call. Each side has one option and one 'safe' asset (share or bond).

Formulas

Put–call parity: $S_0 + p_0 = c_0 + X(1+r)^{-T}$
S₀ = share price today, p₀ and c₀ = European put and call prices with the same X and T, X(1 + r)^(−T) = price of a risk-free zero-coupon bond paying X at T. Underlying has no income or costs.
Solving for the put or the call: $p_0 = c_0 + X(1+r)^{-T} - S_0 \qquad c_0 = S_0 + p_0 - X(1+r)^{-T}$
Use when one option's price is observed and you need the no-arbitrage price of the other.
Value of both portfolios at expiration: $S_T + \max(0,\, X - S_T) = \max(0,\, S_T - X) + X = \max(S_T,\, X)$
Left: protective put. Right: fiduciary call.

Worked examples

Example 1 Pricing a put from a call

A non-dividend-paying share trades at EUR 80. A six-month European call with an exercise price of EUR 75 costs EUR 9.20. The risk-free rate is 3%. What is the no-arbitrage price of the six-month EUR 75 put?

Example 2 Pricing a call from a put

A share trades at SEK 250. A three-month European put with X = SEK 260 costs SEK 14.10, and the risk-free rate is 2%. What should the three-month SEK 260 call cost?

Example 3 Why the conditions matter

An analyst tries to apply parity to a call with X = 100 expiring in June and a put with X = 110 expiring in September on the same share. Can she?

On the exam

Typical items give three of S₀, c₀, p₀ and the rate and ask for the fourth, or ask which portfolio is a fiduciary call or protective put. Distractors usually forget to discount X or flip a sign.

Using X instead of $X(1+r)^{-T}$ . The bond in the fiduciary call is bought today at the present value of X.
Putting the share on the call's side. Share goes with the put (protective put); bond goes with the call (fiduciary call).
Using T in months instead of years when discounting.
Applying parity to options with different exercise prices or expirations, or to American options.

Check yourself

Q1A share trades at USD 52. A one-year European call with X = USD 50 costs USD 6.00 and the risk-free rate is 5%. The no-arbitrage price of the matching put is closest to:
Q2A fiduciary call consists of:
Q3A share trades at JPY 3,000. A six-month European put with X = JPY 3,100 costs JPY 190 and the risk-free rate is 1%. The no-arbitrage call price is closest to:
Q4At expiration the share ends below the exercise price. The protective put and the fiduciary call are each worth:
Q5Put–call parity as presented in this reading is least likely to require that the call and put:

0/5Check yourself

Answer the 5 questions: +10 XP for each one you get right first time.

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