What is the relationship between a protective put and a fiduciary call, and how does this connect to put-call parity?

Put-call parity is one of the most elegant results in derivatives, and understanding it through the protective put / fiduciary call equivalence is the CFA curriculum's preferred approach.

The Two Portfolios

Protective Put = Long Stock + Long Put

• You own the stock and buy insurance (the put) against decline
• Payoff at expiration: max(S_T, K)

Fiduciary Call = Long Call + Long Risk-Free Bond (face value K)

• You hold a call option and enough cash (invested at the risk-free rate) to exercise it
• Payoff at expiration: max(S_T, K)

Since both portfolios have identical payoffs in every state of the world, they must have the same price today (or else arbitrage).

Payoff Table — Ellsworth Industries Options (K = $50)

Scenario	S_T	Protective Put (Stock + Put)	Fiduciary Call (Call + Bond)
S_T = $30	$30	$30 + ($50 - $30) = **$50**	$0 +$50 = $50
S_T = $40	$40	$40 + ($50 - $40) = **$50**	$0 +$50 = $50
S_T = $50	$50	$50 +$0 = $50	$0 +$50 = $50
S_T = $60	$60	$60 +$0 = $60	$10 +$50 = $60
S_T = $80	$80	$80 +$0 = $80	$30 +$50 = $80

Every row matches. This proves: S + p = c + PV(K)

Put-Call Parity Formula

c + PV(K) = p + S

Rearranging:

• c = p + S - PV(K)
• p = c - S + PV(K)

Arbitrage Example

Suppose for Ellsworth options (K = $50, T = 1 year, Rf = 4%):

• Stock: $48
• Call: $5.00
• Put: $4.50
• PV(K) = 50/1.04 = $48.08

Protective put cost = $48 +$4.50 = $52.50** Fiduciary call cost =$5.00 + $48.08 = **$53.08

The fiduciary call is overpriced by $0.58. Arbitrage: sell the fiduciary call (write call, short the bond) and buy the protective put (buy stock, buy put). Lock in$0.58 risk-free profit.

Exam tip: Put-call parity questions come in two forms: (1) compute the missing option price, or (2) identify an arbitrage when parity is violated. Practice both.

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