= Maturity monotonicity of calls with nonnegative strikes
{title2=$C(u,K)\geq C(t,K)\quad(u\geq t,\ K\geq0)$}
If the <bank account> $B$ is positive and nondecreasing and $S/B$ is a <martingale> under an <equivalent martingale measure> $Q$, then for $K\geq0$ the European call price $C(T,K)$ is nondecreasing in $T$. Put $M_t=S_t/B_t$. For $u\geq t$, $K/B_u\leq K/B_t$ and hence $(M_u-K/B_u)^+\geq(M_u-K/B_t)^+$. Conditional convexity gives $\mathbb E_Q[(M_u-K/B_t)^+\mid\mathcal F_t]\geq(M_t-K/B_t)^+$. 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 $1-Ke^{-rT}$, decreasing in maturity.
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