Fiduciary call, protective put and put–call parity

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.