$[4]$ Consider the $N-$period binomial lattice. Denote by $S_n^j$ for $0nN$ and $0jn$ be the price of the underlying at time $t_n$ and state $j($i$.$e$.,j$ ups$).$ A European Straddle Option has payoff at time $T$

max$\{K-S_N,0\}+$max$\{S_N-K,0\}.$

Denote by $V(S_n^j,K,t_n)$ be the fair value of the straddle option at time $t_n$ in state $j.$ Use induction $($over $n)$ to show that for any constant $>0,$

$V(S_n^j,K,t_n)=V(S_n^j,K,t_n),n=0,1$dots,$N,j=0,1,...,n$