Question

A model for the movement of a stock supposes that if the present price of the stock is s, then, after one period, it will be either us with probability p or ds with probability 1 − p. Assuming that suc- cessive movements are independent, approximate the probability that the stock’s price will be up at least 30 percent after the next 1000 periods if u = 1.012,d = 0.990, and p = .52.

Answer 1

Answer:

$P(X\geq 470)=0.9991$

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:

$s*u^n*d^{(1000-x)}$ 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:

$s*u^n*d^{(1000-x)}>1.3s$

applying properties of exponents and reorganizing.

$d^{1000}(\frac{u}{d})^x>1.3$

solving for x

$1000 log(d)+ x log (\frac{u}{d})>log(1.3)$

$x >\frac{log(1.3)-1000 log(d)}{log (\frac{u}{d})}$

Using the values given d= 0.99 and u=1.012

$x > 469.208967$

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

$F(x;p,n) = \Sigma_0^x \left(\begin{array}{c}n&i\end{array}\right)p^i(1-p)^{n-i}$

With this formula we can calculate the probability of obtain x successful trials or less ($P(X\leq x)$). 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 $P(X\leq x)$, but we need $P(X\geq x)$ so we can calculate that probability as

$P(X\geq x)=1-P(X\leq x)$

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  $P(X\leq 470)=0.00087$, so $P(X\geq 470)=1-0.00087=0.9991$