Calculating present value using Euler's number
$e^{-rt}$ is the continuous discounting factor while $(1 + r)^{-t}$ is its discrete counterpart.

Identically/equivalently, there is the continuous manner of computing factors of variation, e.g. $1 + \ln \frac{x_{0+t}}{x_0}$, and its discrete counterpart $\frac{x_{0+t}}{x_0}$.

It all depends on whether you think you are dealing with a continuous quantity or not.

In asset pricing (or finance in general) the underlying notion is that of continuous (interest) compounding.