American quadratic-payoff option in a binomial market ($V_t=S_t^2{\lambda }^{T-t}$)

In a discrete-time binomial market with $B_t=(1+r{)}^t$ and one-step stock factors $1\pm \epsilon$, use the risk-neutral probability $p=(1+r/\epsilon )/2$, where $0\le r<\epsilon <1$. For the exercise payoff $S_t^2$, the discounted one-step reward multiplier has meanThe Snell envelope of a multiplicative process gives the value $V_t=S_t^2{\lambda }^{T-t}$. Continuation strictly exceeds exercise before maturity, so the optimal exercise time is $T$. The replicating portfolio in a binomial market holds $2S_t{\lambda }^{T-t-1}$ shares on the next step, found by subtracting the two next-state values and dividing by the stock-price difference. At time zero the price is $S_0^2{\lambda }^T$.