Correlated Bond Default
Formula Reference
Probability of double default
\begin{eqnarray}\label{eqA100}
p_{11} & = & p_xp_y + r\sqrt{p_x(1-p_x)p_y(1-p_y)}\\
p_x & = & \mbox{probability of 1st bond defaulting}\nonumber\\
p_y & = & \mbox{probability of 2nd bond defaulting}\nonumber\\
r & = & \mbox{correlation coefficient of the two bonds}\nonumber
\end{eqnarray}
Probability of 1st bond defaulting and 2nd not defaulting
\begin{equation}\label{eqA110}
p_{10} = p_x - p_{11} = p_x(1-p_y) - r\sqrt{p_x(1-p_x)p_y(1-p_y)}
\end{equation}
Probability of 2nd bond defaulting and 1st not defaulting
\begin{equation}\label{eqA120}
p_{01} = p_y - p_{11} = p_y(1-p_x) - r\sqrt{p_x(1-p_x)p_y(1-p_y)}
\end{equation}
Probability of neither bond defaulting
\begin{eqnarray}\label{eqA130}
p_{00} & = & 1 - p_{11} - p_{10} - p_{01}\\
& = & (1-p_x)(1-p_y) + r\sqrt{p_x(1-p_x)p_y(1-p_y)}\nonumber
\end{eqnarray}