the problem i have is how to calculate the standart diviation of the same portfolio with deposits.
lets say for example that i have a portfolio with a daily standart diviation of 1% and a daily mean return on 0.05%. i am starting with 1000$ and i add 1$ every day for 500 days. what will be the var or the standart diviation of the sum after 500 days.

i dont need the answer but the formula, if posible plese refer me to a paper on that matter. i need it for some calculation i am trying to do at work. meanwhile i am using a Monte-Carlo simulation and it is taking too muth time and it is not acurate....

Re: Colculation Geometric Brownian Motion with deposits.

First of all you have an asset with price following a geometric brownian motion, with SDE

Suppose you invest amount of money at time , then you will long unit of asset. Note is non-random if we are at time .

At the end of each time period with length , you further invest amount of money, which essentially will long units of asset at the end of the -th period, in addition to the previous holding, for .

At the end of the periods, under this strategy, you will totally long

units of asset, and your wealth at time will be just

The distribution of this one will not be nice; as we know that there will be no explicit / "closed-form" solution to Asian option which involve arithmetic means of log-normal random variables. I guess the variance can be calculated but you will need some careful calculation with the covariance. Sorry I am busy right now to work this out and see if I can try tomorrow.