Let $(X,Y)$ be a martingale on $\mathbb R$ and $\psi:\mathbb R\to\mathbb R$ be a convex function. Then it follows by Jensen's inequality that
$$\mathbb E[\psi(X)]~~\le~~ \mathbb E[\psi(Y)]$$
and if $\psi$ is strictly convex and $\mathbb E[\psi(X)]=\mathbb E[\psi(Y)]$, then $X=Y$ almost surely. Now let us consider a special convex function $\psi(x)=(x-K)^+$ (that is not strictly convex), my question is the following:
If $\mathbb P[X\neq Y]>0$ and $\mathbb E[(X-K)^+]=\mathbb E[(Y-K)^+]$, could we show that $X, Y\le K$ or $X, Y\ge K$?