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Math Help - GBM and wiener process

  1. #1
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    GBM and wiener process

    Wiener Process and GBM process

    I have these two problems on hand and I don't think I have the basic idea

    Consider the generalized Wiener Process

    dx=0.1dt+0.3dz
    suppose the initial value f x is x0=1
    a)calculate the expected value of x after 2 years
    b)calculate the standard deviation of x in 2 years
    c)determine the 95%confidence interval for x after 2 months.
    d. consider a European call option on this asset with an exercise price of $2 and expiry date in 3 years. What is the probability that this option will be exercised at expiry? show your work.
    e) what is the probability that a European put option put option on this asset with the same maturity and exercise price will be exercised?

    My work
    E(dx)=0.1*2=0.2 Ex=1+0.2=1.2
    variance E(dx^2)=0.3^2*2=0.18
    SD=sqrt0.18

    and I am not sure about the rest

    second question
    consider the GBM process:
    dS1 = 0.1S1dt+0.3S1dz
    suppose the initial value of S1 after 2 years.
    b.Calculate the stadard deviation of S1 after 2 years
    c. what can you say about the distribution of X1=log(S1) after 2 years?
    what are its expected value and standard deviation?
    d. Find a 95% confidence interval fr X1 after 2 years. Use your result to calculate a 95% confidence interval for S1

    I don't get it, what is the difference btw wiener and GBM?I cannot find any source that tell me the exact difference

    I don't know how to solve these questions, and I am at my wits end.

    Help!
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  2. #2
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    Re: GBM and wiener process

    Hey noblewhale.

    Geometric Brownian Motion is a specific kind of process that is a function of a Wiener Process. A Wiener process has specific properties and is basically a normal distribution with the properties of additive variance, independence for independent intervals, the first value is 0 and there's another one I can't recall currently.

    For the first one (and the second but you have a different variable), you have a random variable that is a function of t where for every value of t you get a random variable in itself and this is a stochastic process.

    You need to solve for X(t) (i.e. the distribution or definition of the random variable) and then find the expectation given particular values of t (think about each t being an individual random variable).

    There are two main ways in probability to do this: the first is through stochastic calculus and the 2nd one is through the transformation process to get the distribution of a random variable if it's some kind of function of another (you should get some kind of log-normal distribution if you do it this way).

    What have you covered with respect to the above concepts?
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  3. #3
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    Re: GBM and wiener process

    Hi Chiro:

    First of all thank you for your reply.

    I am scratching my head when I read your response. Basically I understand what a wiener process is. It is some process with stochastic additive for change in t. The change is normally distributed.

    Geometric Brownian Motion is a case that somehow involves the logarithm (I am not sure how...)

    We have only covered the basic definition of the two processes. I tried to read through wikipidia to understand them & I am still clueless about the relationship btw GBM and Wiener. I guess my confusion comes from not able to find an example.

    I can only say that I can technically name the properties of both processes, but unable to apply them.
    :S

    NW
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  4. #4
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    Re: GBM and wiener process

    The idea of these particular processes is that you have a Stochastic process that has a time component (deterministic) and a random component (typically a Weiner process) and that the process evolves over time taking into account the deterministic and non-deterministic components.

    For example if the dWt was multiplied by 0, then you would have a deterministic first order differential equation with dX = adt which gives dX/dt = a so X(t) = at + C but if you have a random component, then this will affect the evolution of the process by adding in "random changes" at each time-step which will make it "jagged" as a result.

    So what happened was that mathematicians like Ito developed a way to calculate derivatives of stochastic processes in the above form (dX = a(X,t)dt + b(X,t)dt) where one could find the derivatives of a particular variable with respect to both the time and the Weiner process variables of which one is a random variable and the other is deterministic.

    With this lemma, you then use this to help get to a solution for X(t).

    In a geometric brownian motion, and in this kind of situation, your distribution will be a log-normal one where X(t) = S_0*e((mu - [1/2]*sigma^2)*t + sigma*Wt).

    The proof for this involves using the various forms of Ito's lemma and knowledge of integrating deterministic and non-deterministic parts (on top of differentiating).

    If you are studying this in class and you haven't gone through this, then that is a problem you need to address.
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  5. #5
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    Re: GBM and wiener process

    after days of confusion, thank you
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