An investor invests 900 dollars in a stock. Each trading day, the investment has probability 0.5 of increasing by 10 dollars and probability 0.5 of decreasing by 8 dollars.
The changes in price in different trading days are mutually independent. Calculate the probability that the investment has a value greater than 969 dollars at then end of 252 trading days (about 1 years).
Using the laws of large numbers, I understand E(X) = 10 * 0.5 + (-8) * 0.5 = 1 and $$\bar{X}_n = \frac{X_1+....+X_n}{n} = E(X)$$ where n = 252. How would I use this information to find the expected investment value at the end of 252 trading days?
I would greatly appreciate any help that can point me in the right direction for this question.Thank you!
You are right with the expected value of the change after one day. The expected value of the change after two day days is $0.25\cdot 20+0.25\cdot 2+0.25\cdot 2+0.25\cdot (-16)=5+0.5+0.5-4=2$. So afer $n$ days the expected value of the change is $n$. Thus the expected value of the stock after 252 trading days is $900+252=1152$
And the variance of the change after one day is
$Var(X_{0,1})=0.5\cdot (10-1)^2+0.5\cdot (-8-1)^2=0.5\cdot 81+0.5\cdot 81=81$
$X_{i,i+1}$ is the random variable for the change from day $i$ to day $i+1$.
The random variables $X_i$ are mutually independent. That means that $Var\left(\sum\limits_{i=1}^n X_i \right)=\sum\limits_{i=1}^n Var\left(X_i \right)$
Therefore the variance of the stock after 252 day is $81\cdot 252$. Now apply the central limit theorem.
Last hint: $P(X>969)=1-P(X\leq 968)$