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quidni
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Posts: 11
Joined: November 5th, 2008, 7:32 pm

Volatility annualization in a monthly step GBM simulation

April 1st, 2018, 9:46 am

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
 
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bearish
Posts: 5186
Joined: February 3rd, 2011, 2:19 pm

Re: Volatility annualization in a monthly step GBM simulation

April 1st, 2018, 6:16 pm

I think the simple answer to your question is "yes", although you add confusion when introducing sqrt(20). Also, you would almost never want to simulate a GBM directly with an Euler scheme, but rather do it in log space and exponentiate back as needed. 
 
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quidni
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Posts: 11
Joined: November 5th, 2008, 7:32 pm

Re: Volatility annualization in a monthly step GBM simulation

April 25th, 2018, 8:28 pm

Ok, thanks a lot, but assuming so I'm using the analytical/exact formula for the GBM (and not the Euler scheme), this means both the diffusion (sigma) and drift (mu) term have always to be put as annualized, and then the choice of dt (as 1/12 if I want a monthly time step simulation) makes me the desired time step, right?