I have a 3-period model; where $\beta_t$ is the value the savings account.
The stock prices $S_t$ are given by $S_1=\xi_1$ , $S_2=\xi_1\xi_2$ and $S_3=\xi_1\xi_2\xi_3$.Where $\xi_1,\xi_2,\xi_3$ are independent RV and take values {u,d} ie (up price movement, down price movement).
I have been given $H_3 = S_1 e^{S_2}$ is the price of a claim at time t = 3 and I need to prove the replicating portfolio is $(a_3,b_3)=(0,{H_3}/\beta^{3})$ and then find the price at time 2.
But I'm confused as to how to go about creating a replicating portfolio which would match this payout to discount it.
Any help would be appreciated.