= American quadratic-payoff option in a binomial market
{title2=$V_t=S_t^2\lambda^{T-t}$}
In a <discrete-time binomial market> with $B_t=(1+r)^t$ and one-step <stock> factors $1\pm\varepsilon$, use the <risk-neutral probability> $p=(1+r/\varepsilon)/2$, where $0\le r<\varepsilon<1$. For the exercise payoff $S_t^2$, the discounted one-step reward multiplier has mean
$$
\lambda=\frac{1+2r+\varepsilon^2}{1+r}>1.
$$
The <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 $S_0^2\lambda^T$.
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