What is the relationship between a protective put and a fiduciary call, and how does this connect to put-call parity?
I'm studying put-call parity for CFA Level I and I understand the formula c + PV(K) = p + S, but I'm struggling to see why a protective put and a fiduciary call always have the same payoff. Can someone show me both sides with a payoff table?
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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