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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 $S_0^2{\lambda }^T$.

Use the natural filtration ${\mathcal{F}}_t=\sigma (Z_1,\dots ,Z_t)$ and the usual integrable definition of the Snell envelope. Since the factors are positive, its terminal value is $U_T=Y_T$. If $U_{t+1}=Y_{t+1}{c}_{t+1}$, independence of random variables givesThe backward recursion therefore becomesThis proves the assertion by backward induction. All constants are finite and at least one. More explicitly,where the empty product is one. This is the Snell envelope of a multiplicative process. The standard Snell envelope hypothesis includes integrable rewards; independence of random variables and positivity then give finite first moments of the factors. If that hypothesis is dropped, an infinite-mean factor gives an infinite continuation value, so finite constants need not exist.

For the discrete-time binomial market, write $p=(1+r/\epsilon )/2$. The assumptions give $0<p<1$, positive bank account and stock prices, andThusThe finite discrete-time fundamental theorem of asset pricing states that a market with a strictly positive numéraire has no arbitrage if there is an equivalent probability measure under which all numéraire-denominated asset prices are martingales. The original probability measure is already such an equivalent martingale measure, proving no arbitrage.

The complete-market American-option pricing theorem identifies the discounted replication or superhedging value with the Snell envelope of the discounted exercise rewards. HereThe independent factors have common meanThe Snell envelope recursion consequently gives $c_t={\lambda }^{T-t}$. At every time before $T$, the continuation value strictly exceeds immediate exercise, so optimal exercise is at maturity. The undiscounted value process is $V_t=S_t^2{\lambda }^{T-t}$, and the time-zero replication cost isIn fact $V_t/B_t$ is a martingale, not just a supermartingale, because the Snell envelope has no exercise compensator before maturity.

To hedge the optimally exercised payoff, subtract the two next-state portfolio equations. The first-period delta hedge isThe corresponding bank account holding, since $B_0=1$, is $V_0-{\Delta }_0{S}_0=S_0^2{\lambda }^{T-1}(\lambda -2)$. Substitution into either next-state equation verifies both terminal values at time one. Repeating this two-state calculation at every node gives a self-financing strategy replicating the maturity payoff. It also covers earlier exercise: $V_t\ge S_t^2$ at all dates, so the seller can meet any earlier payment with the available wealth.