Below is a problem I did. I believe my answer is right but I would like somebody to check it for me.
Problem: Show that the Black-Scholes formula for call and put options satisfy put-call parity equation. The put-call parity equation is: $$ c + Ke^{-rT} = p + S_0 $$
We have: \begin{align*} c &= S_0 N(d_1) - Ke^{-rT}N(d_2) \\ p &= Ke^{-rT}N( -d_2) - S_0 N( -d_1 ) \\ c - p &= S_0 N(d_1) - Ke^{-rT}N(d_2) - \left( Ke^{-rT}N( -d_2) - S_0 N( -d_1 ) \right) \\ c - p &= S_0 N(d_1) - Ke^{-rT}N(d_2) - Ke^{-rT}N( -d_2) + S_0 N( -d_1 ) ) \end{align*} Note that: $$ N(d_1) + N(-d_1) = 1 $$ \begin{align*} c - p &= S_0 ( N(d_1) + N( -d_1 ) ) - Ke^{-rT}N(d_2) - Ke^{-rT}N( -d_2) \\ c - p &= S_0 ( N(d_1) + N( -d_1 ) ) - Ke^{-rT} ( N(d_2) + N( -d_2) ) \\ c - p &= S_0 - Ke^{-rT} \\ c + Ke^{-rT} &= p + S_0 \end{align*}
Is this reasoning correct?