Maturity monotonicity of calls with nonnegative strikes ($C(u,K)\ge C(t,K)(u\ge t,K\ge 0)$)

If the bank account $B$ is positive and nondecreasing and $S/B$ is a martingale under an equivalent martingale measure $Q$, then for $K\ge 0$ the European call price $C(T,K)$ is nondecreasing in $T$. Put $M_t=S_t/B_t$. For $u\ge t$, $K/B_u\le K/B_t$ and hence $(M_u-K/B_u{)}^{+}\ge (M_u-K/B_t{)}^{+}$. Conditional convexity gives ${\mathbb{E}}_Q[(M_u-K/B_t{)}^{+}\mid {\mathcal{F}}_t]\ge (M_t-K/B_t{)}^{+}$. Taking expectations and multiplying by $B_0$ proves the claim. The result is not generally true for negative strikes: with $S_t=B_t=e^{rt}$ and $r>0$, a negative-strike call costs $1-K{e}^{-rT}$, decreasing in maturity.