Probability of geometric Brownian motionI want to solve the following question: Let X(t) be the price of JetCo stock at time t years from the present. Assume that X(t) is a geometric Brownian motion with zero drift and volatility σ = 0.4 / y r. If the current price of JetCo stock is 8.00 USD, what is the probability that the price will be at least 8.40 USD six months from now.Here is my attempt: Since X(t) is a geometric Brownian motion, log(X(t)) is a regular Brownian motion with zero drift and σ = 0.4 / y r. We want to find the probability that log ⁡ ( X ( 1 / 2 ) ) ≥ log ⁡ ( 8.40 ) given that log ⁡ ( X ( 0 ) ) = log ⁡ ( 8.00 ).What can I do from here?

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Probability of geometric Brownian motionI want to solve the following question: Let X(t) be the price of JetCo stock at time t years from the present. Assume that X(t) is a geometric Brownian motion with zero drift and volatility   σ  =  0.4      /    y  r. If the current price of JetCo stock is 8.00 USD, what is the probability that the price will be at least 8.40 USD six months from now.Here is my attempt: Since X(t) is a geometric Brownian motion, log(X(t)) is a regular Brownian motion with zero drift and   σ  =  0.4      /    y  r. We want to find the probability that   log  ⁡  (  X  (  1      /    2  )  )  ≥  log  ⁡  (  8.40  ) given that   log  ⁡  (  X  (  0  )  )  =  log  ⁡  (  8.00  ).What can I do from here?

Jeroronryca · Accepted Answer

Step 1You are on the right track so far, but you have the drift wrong. The drift of log(X(t)) is, from Ito&#039;s lemma,   −      1    2        σ    2    ..After that you use the fact that since it&#039;s a brownian motion starting from log(8.0) the distribution is   log  ⁡  (  X  (  t  )  )  ∼  N      (    log    ⁡    (    8    )    −          1      2              σ      2        t    ,          σ      2        t    )    ..Step 2So let Z be normal with mean   log  ⁡  (  8  )  −      1    2        σ    2        (          1      2        )   and variance       σ    2        (          1      2        )    .. To finish the problem, you need to compute   P  (  Z  &amp;gt;  log  ⁡  (  8.40  )  )  ..

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