I'm reading Chapter 13 of Hull's book and am stuck on how he got from stock returns to continuously compounded stock returns. As a recap, he built the generalized Wiener Process, which describes a change in some variable x is a function of a drift term and stochastic term.
dx = a * dt + b * ∈ * sqrt(dt)
where a * dt is the drift term, b * ∈ * sqrt(dt) is the stochastic term, and ∈ is a standard normal distribution N(0,1)
I get that part. However, that only describes the change in x not a return of x. So we need to make a modification. Under 13.3, he says we just multiply the drift and stochastic terms by x itself so that we can re-arrange the equation to be:
dx = a * x * dt + b * x * ∈ * sqrt(dt)
dx/x = a * dt + b * ∈ * sqrt(dt)
which allows us to describe the returns of x. I get all of that. What I don't get is the case where b is 0, which is when there is no stochastic process. He says the above equation will just become:
dx/x = a * dt
Which when you take the integral with bounds 0 and T, you get:
xt = x0 * exp (a * t)
How do you get to the last equation, which is the continuously compounded return? Isn't the derivative of exp(x) equal to itself? And thus if the anti-derivative has exp(x) in it, that means the equation itself has exp(x) in it? I don't get how exp(x) appears out of thin air.