= Strict local martingale failure of the law of one price
Fix $T>0$, let $X$ be <Brownian motion>, and let $\tau=\inf\{u:X_u=-1\}$. Set $q(t)=t/(T-t)$ for $t<T$ and
$$
B_t=1,\qquad S_t=2+X_{q(t)\wedge\tau}\quad(t<T),\qquad S_T=1.
$$
The hitting time is finite almost surely, so each price path reaches one before $T$ and stays there. Stopping additionally on hitting $n+2$ makes the <stock> price a bounded <martingale>; these stopping times increase to $T$. Hence $S$ is a positive <local martingale>, and the original probability measure is an <equivalent local martingale measure>, excluding admissible <arbitrage>. Nevertheless $S_T=B_T$ while $S_0=2\ne1=B_0$. The zero-cost short-stock/long-two-cash <portfolio> has terminal gain one but wealth $2-S_t$, which has no deterministic lower bound. It is not admissible, resolving the apparent contradiction.
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