I have a question regarding the simulation of a GBM. I have found similar questions here but nothing which takes reference to my specific problem:

Given a GBM of the form

$dS(t) = \mu S(t) dt + \sigma S(t) dW(t)$

it is clear that this SDE has a closed form solution in

$S(t) = S(0) exp ([\mu - \frac{1}{2}\sigma^2]t + \sigma W(t))$

for a given $S(0)$.

Now, I have found sources claiming that in order to simulate the whole trajectory of the GBM, one needs to convert it to its discrete form (e.g., a similar question here or Iacus: "Simulation and Inference for Stochastic Differential Equations", 62f.). Yet, in Glasserman: "Monte Carlo Methods in Fin. Eng.", p. 94, I find that

$S(t_{i+1}) = S(t_i) exp ([\mu - \frac{1}{2}\sigma^2](t_{i+1}-t_i) + \sigma\sqrt{t_{i+1}-t_i} Z_{i+1})$

where $i=0,1,\cdots, n-1$ and $Z_1,Z_2,\cdots,Z_n$ are independent standard normals is an exact method (i.e., has no apprximation error from discretization).

I really don't understand what the difference between the two is, or put differently, if the exact method lets me simulate the whole trajectory, why would I bother converting it to the discrete form?

Maybe I'm just not seeing the point here but I'm really confused and grateful for any help!


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