I was just wondering if someone could verify whether these are the two boundary conditions for a Call Spread Black-Scholes PDE.
The first one I have is:
$max(S_{T} - K_{1}, 0) - max(S_{T}-K_{2},0)$
While the second boundary condition I have is:
$S_{t} - K_{1}e^{-r(T-t)} + K_{2}e^{-r(T-t)} - S_{t} = (K_{2}-K_{1})e^{-r(T-t)}$
Is this correct? Thanks in advance
EDIT: Rather than create a new question, I thought I should ask it here: Should a call spread at time $t_{0}$ always take on a symmetric shape? I have the graph of a call spread PDE at time $t_{0}$ for spot values between 0 to 20, with zero interest rate, and with strikes $K_{1} = 9$, $K_{2} = 11$. Does the shape of this call spread look okay, or does it have to be perfectly symmetrical? Thanks!