# Merton model d1 and probability of default

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.

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).