Answer:
[tex]P(X\geq 470)=0.9991[/tex]
Step-by-step explanation:
Present price of stock is s, if x is the times that the stock increases (by u) among the 1000 periods we can calculate the price after 1000 periods as follows:
[tex]s*u^n*d^{(1000-x)}[/tex] note that if the price increase n periods, so it should decreases in the other 1000-x periods.
We need that the price be 30% higher or 1.3 times the initial price. That requirement is expressed as follows:
[tex]s*u^n*d^{(1000-x)}>1.3s[/tex]
applying properties of exponents and reorganizing.
[tex]d^{1000}(\frac{u}{d})^x>1.3[/tex]
solving for x
[tex]1000 log(d)+ x log (\frac{u}{d})>log(1.3)[/tex]
[tex] x >\frac{log(1.3)-1000 log(d)}{log (\frac{u}{d})}[/tex]
Using the values given d= 0.99 and u=1.012
[tex] x > 469.208967[/tex]
So, we need at least 470 increase periods
Now we can calculate the probability of have 470 increase periods or more, taking into account that the distribution of x is binomial, the period can be and increase period with 0.52 of probability and we have 1000 total trials.
The formula for the binomial cumulative probability function is
[tex]F(x;p,n) = \Sigma_0^x \left(\begin{array}{c}n&i\end{array}\right)p^i(1-p)^{n-i}[/tex]
With this formula we can calculate the probability of obtain x successful trials or less ([tex]P(X\leq x)[/tex]). There, p is the probability of success (probability of increase period in this case), x is the maximum number of successful trials, n the number of total trials. We should calculate F with x=470, p=0.52 and n= 1000. The result is [tex]P(X\leq x)[/tex], but we need [tex]P(X\geq x)[/tex] so we can calculate that probability as
[tex]P(X\geq x)=1-P(X\leq x)[/tex]
In order to calculate the cumulative function we can use excel function BINOM.DIST
BINOM.DIST(x,trials,probability,cumulative)
BINOM.DIST(470,1000,0.52,TRUE)
We obtain that [tex]P(X\leq 470)=0.00087[/tex], so [tex]P(X\geq 470)=1-0.00087=0.9991[/tex]
