What is the value of $d_1$ when the probability of default is 50%
I know that: $$ \begin{aligned} d_2 &= 0 \\ \mathcal{N}(d_2) &= 50\%\\ 1- \mathcal{N}(d_2) &= \mathcal{N}(-d_2) = 50\% \end{aligned} $$ But I don´t know if $d_1 = 0$ or different.
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Since $d_1 = d_2 + \sigma\sqrt{\tau}$, you need to know the volatility of your asset value process. You typically estimate it from equity prices (see e.g. Hull's book).
