I read in paper Zhang(2010) Zhang(2010).pdf (305.2 KB) that the author set default probability of bank using a log-normal distribution written as {{\phi }_{t}}=cdf\left( {{\Delta }_{t}},\sigma \right) on page 11, in which {\Delta }_{t} is an endogenous variable and represents for aggregate capital ratio. I feel very confused that why the author set a variable as the mean of this cdf instead of a parameter, and how should I write this equation in Matlab?

I am not sure what the problem is. There is a variable on the right-hand side, so it’s a variable on the left as well. In Dynare, you have a normcdf-function. See 4. The model file — Dynare 5.1 documentation

Sorry if I did not make myself clear. Generally the log-normal distribution function is written as {{\phi }_{t}}=cdf\left( {\mu},\sigma \right). While according to the content in that paper, it seems that the author used a endogenous variable {\Delta }_{t} as the mean of the distribution instead of a fixed parameter. That’s why I feel confused. It seems that the original independent variable is treated as given and the original mean is treated as independent variable.

As the author says “The health of the banking sector as a whole will depend largely on the variation of aggregate capital ratio. With a higher aggregate ratio, the distribution moves to the right, and fewer banks will fall short of the 8 percent threshold and thus default, and vice versa.” If {\Delta }_{t} is the independent variable of this function, then bank default rate {\phi }_{t} will increase along with {\Delta }_{t}, which seems conflict with the statement in that paper.

Another paper Funke et al.(2015) Funke et al.(2015).pdf (874.6 KB) displays the same setting on page 23, and they also said the capital adequacy ratio {\Delta }_{t} is the mean of distribution.

The CDF of the log normal distribution is characterized by

the mean of the distribution \Delta_t

the standard deviation \sigma

the cutoff of 8% for the CDF (\int_{-\infty}^{8\%})
The aggregate distribution shifts its mean, causing a higher or lower share to fall below the cutoff and go bust.