Valuation of Cox-Ross-Rubinstein Model

Question

We have a Cox-Ross-Rubinstein model with parameters $u$ ("up"), $d$ ("down") , $r$ (interest rate) and $q$ (equivalent martingale probability) $(q=(1+r-d)(u-d)^{-1})$ . We have a contingent claim with payoff $$ X=S_1(N)^c $$ where $S_1(N)$ is the final price, and $c$ is a positive integer. I need to show that the initial valuation of a claim is: $$ \pi_X(0) = S_1(0)^c(1+r)^{-N}\left(u^cq+d^c(1-q)\right)^N $$

I know that \begin{align} \pi_X(0) & = E_Q[\frac{X}{S_0(N)}] \\ & = (1+ r)^{-N}E_Q[S_1(N)^c] \\ & = (1+ r)^{-N} \sum_{j=0}^N(S_1(0)u^jd^{N-j})^c{N \choose k}q^j(1-q)^{N-j} \end{align} and then the $S_1(0)$ can be taken out which gives me the first part, but then I'm not really sure how to proceed. I don't see how the "T choose k" bit is going to disappear, or how we can get rid of the summation sign.

Answer 1

There was an error in your expected value, which I have corrected - the probabilities and the binomial coefficient (the "N choose k") should not be raised to the power $c$. With that correction, it is a simple application of the Binomial theorem: \begin{eqnarray} \left(u^cq+d^c(1−q)\right)^N&=&\sum_{j=0}^N {N \choose j}(u^cq)^{j}(d^c(1−q))^{N-j}\\ &=&\sum_{j=0}^N {N \choose j} \left(u^j d^{N-j}\right)^c q^j (1-q)^{N-j} \end{eqnarray}