Let $S(0)$ be the initial price of the stock. Assume that in each period the stock can go either up by a factor of $u$ or down by a factor of $d$. Assume that the probability that the stock goes up is $p$ and the probability that it goes down is $1−p$ in each period, and that these events are independent. Determine the expected value of the stock after $n$ periods.
(Express your answer in closed form in terms of $u$, $d$, $p$, $n$).
I know how to find the expected value of the stock after one period, it would be: $$E[S(T)]=[S(0)(1+u) \times p] + [S(0)(1-d) \times (1-p)]$$
But I don't know how to find the expectation for $n$ periods.
Since they are independent is this correct?
$$E[S(T)]=\frac{([S(0)(1+u) \times p] + [S(0)(1-d) \times (1-p)])^n}{n}$$
Let $E[S_0,n]$ be the expected value after $n$ periods, starting from $S_0$. Since each price change is multiplicative we see that $$E[\lambda S_0,n]=\lambda E[S_0,n]$$
Now, looking at the first period we see the recursion: $$E[S_0,n]=p\times E[S_0(1+u),n-1]+(1-p)\times E[S_0(1-d),n-1]$$$$=(1+pu -(1-p)d)\times E[S_0,n-1]$$ Iterating shows that $$E[S_0,n]=(1+pu -(1-p)d)^n\times S_0$$
Sanity Check: if $u=d$ and $p=\frac 12$ then symmetry tells us that the expected value must always be $S_0$, no matter how many periods there are, and this is indeed implied by the formula.