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So I was reading here where they calculate the expected value of an option at present given the expected value of the option in a year by calculating
$$C_0 = C_1 e^{(-r)}$$
where r is the interest rate. But where does e come from here? Shouldn't present value just be calculated using
$$C_0 = C_1 / (1 + r)$$
What am I missing?
$e^{-rt}$ 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.
