Why is $(S_t-K)^{+}$ a supermartingale when $S_t$ is supermartinagale

Question

To prove that $(S_t-K)^{+}$ a supermartingale with $S_t$ strictly positive supermartinagale and $K$ a constant, I started like this :

$S_t$ is supermartinagale, thus : $$E[S_t|S_0] < S_0$$ $$E[S_t - K|S_0] < S_0 - K$$

at this point I'm trying to apply Jensen Inequality but seems like it doesn't lead me to the right direction. Any help?

Answer 1

(2017.09.16) Is it me or is the plague of silent revenge downvotes spreading? Yet one more...

The result cannot hold.

If it was true, for any $X$ and $Y$ integrable, the property $E(X\mid Y)\leqslant Y$ almost surely would imply that $E(X^+\mid Y^+)\leqslant Y^+$ almost surely, which in turn implies that $E(X^+\mid Y^+)=0$ almost surely on $[Y^+=0]=[Y\leqslant0]$, which is equivalent to the fact that $X\leqslant0$ almost surely on $[Y\leqslant0]$.

But this is absurd, as the case when $P(Y=0)=1$, $P(X>0)\ne0$, $E(X)\leqslant0$, shows.

Answer 2

A counterexample is Brownian motion and $K=0$. Brownian motion is a martingale, hence also a supermartingale. But $E B_t^+ > 0$ for $t>0$ so $B_t^+$ is not a supermartingale.