I'm studying American Options. If I have $X=(X_n)$ an American option, it is not possible to determine a self-financing predictable strategy ($\alpha, \beta$) that replicates the option in sense that $V^{(\alpha,\beta)}_n=X_n$ for every $n=0,\ldots,N$.

This is simply due to the fact $\tilde{V}^{(\alpha,\beta)}$ is a $Q-$martingale while $X$ is a generic adapted process.

Let be: $$ \mathcal{A}^-_X=\{(\alpha,\beta) \in \mathcal{A} \mid V_N^{(\alpha,\beta)} \le X\}$$ the family of sub-replicating porfolios for the derivate $X$.

Why We can't define a strategy that sub-replicates $X$, for every $n=0,\ldots,N$?



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