3
$\begingroup$

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.

$\endgroup$
1
$\begingroup$

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}

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service, privacy policy and cookie policy

Not the answer you're looking for? Browse other questions tagged or ask your own question.