If a stock goes up by a factor $u$ with probability $p$ and down by a factor $d$ with probability $1-p$, find the expected value of the stock after $n$ periods. Assume the periods are independent.
I drew a basic tree diagram for 3 periods just to get a sense of how it works.
I'm having trouble using math to describe the pattern.
Ie. for period $n=3$. letting $X(T)$ be the stock price random variable, I got $ E[X]= X(0)[(up)^3+3(up)^2d(l-p)+3upd^2(1-p)^2+d^3(1-p)^3]$
This looks like a binomial expansion, but I'm not sure how to generalize it for n periods. I know it will have n terms in the unsimplified polynomial and a unique 1st and last term.
Please let me know if this is even the right approach!
Let $A$ be the initial amount, and let $X_i=u$ with probability $p$, and $X_i=d$ with probability $1-p$. Then the amount after $n$ periods is $W$, where $$W=AX_1X_2\cdots X_n.$$ By independence, we have $$E(W)=AE(X_1)E(X_2)\cdots E(X_n).$$ For any $i$ we have $E(X_i)=pu+(1-p)d$, so $E(W)=A(pu+(1-p)d)^n$.