I was reading the old, but still interesting paper "The volatility smile and its implied tree" by Derman and Kani. I have a two questions about the derivation of the $2n+1$ equations, both of them regarding Arrow-Debreu price. On page 6, is shown in figure 4 how the model is set up. On the next page the authors writes down:
$$C(K,t_{n+1})=\exp{(-r\Delta t)}\sum_{j=1}^n\{\lambda_jp_j+\lambda_{j+1}(1-p_{j+1})\}\max{\{S_{j+1}-K,0\}}$$
where $C(K,t_{n+1}$ denotes the price of a call option with strike $K$ and expiry and expiry $t_{n+1}$. The first term in the sum are the probabilities. However, since $p_j$ is already the risk neutral probability why do the authors multiply them by $\lambda_j$? $\lambda_j$ is the known Arrow-Debreu price at node $(n,i)$. I've never heard of an Arrow-Debreu price. After checking the web it is still unclear to me, what the reason is for this equation.
Moreover using the above equation, there should be a $\lambda_{n+1}$, which is not the case! So is it just set to $0$?