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Thread: Colculation Geometric Brownian Motion with deposits.

  1. #1
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    Colculation Geometric Brownian Motion with deposits.

    Hi all.
    There is a way of colculation a future standart diviation of a portfolio if you know the daily standart diviation and the mean.


    page 2 last formula.

    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....

    thank you, please help.

  2. #2
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    Re: Colculation Geometric Brownian Motion with deposits.

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

    dS_t = \mu S_tdt + \sigma S_t dW_t

    Suppose you invest X_0 amount of money at time 0, then you will long \frac {X_0} {S_0} unit of asset. Note S_0 is non-random if we are at time 0.

    At the end of each time period with length \Delta t, you further invest X amount of money, which essentially will long \frac {X} {S_{i\Delta t}} units of asset at the end of the i-th period, in addition to the previous holding, for i = 1, 2, \ldots, n.

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

    \frac {X_0} {S_0} + \sum_{i=1}^n \frac {X} {S_{i\Delta t}}

    units of asset, and your wealth at time T \geq n \Delta t will be just

    \left[\frac {X_0} {S_0} + \sum_{i=1}^n \frac {X} {S_{i\Delta t}}\right]S_T

    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.

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