Put–call forward parity

In short

• A long share can be replicated by a long forward plus a risk-free bond paying $F_0(T)$ at T. The bond costs $F_0(T)(1+r)^{-T}$, which equals $S_0$ for an underlying without income or costs.
• A synthetic protective put is that synthetic share plus a long put. It pays $\max(S_T, X)$, like the protective put and the fiduciary call.
• Put–call forward parity: $F_0(T)(1+r)^{-T} + p_0 = c_0 + X(1+r)^{-T}$.
• Rearranged: $p_0 - c_0 = [X - F_0(T)](1+r)^{-T}$. A long put plus short call equals a long bond paying X plus a short forward.
• If $X = F_0(T)$, the put and the call have the same price.