Hi all,
excuse me, I have a naif doubt. If I want to simulate a GBM (Geometric Brownian Motion), under for example the Euler scheme (but the same under the analytical solution, or whatever), i.e.
S(t+dt) = S(t) + mu(S_t,t)*dt + sigma(S_t,t)*sqrt(dt)*Z,
if for simplicity I set mu = 0, and sigma (the volatility) is an annualized term (if for example I compute it by a daily historical series, and then I multiply it by sqrt(260), assuming it is normal etc.), but I want to simulate the GBM with a monthly time step, I have to simply put dt=1/12 in that equation, to get a monthly time step simulation? I mean, my doubt is if instead I have to render "monthly" even sigma (i.e. multiply since the beginning sigma not by sqrt(260), but by sqrt(20)), and then using anyway dt=1/12... I mean, the dt = 1/12 acts as a monthly discretization on all the terms of that stochastic equation, assuming they have to be annualized?
Thanks a lot