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Let $Q$ be the unique equivalent martingale measure. The risk-neutral pricing formula isFor the usual nonnegative strike $K$ and maturities $u\ge t$, the monotonic bank account gives $K/B_u\le K/B_t$. ThereforeThe second inequality follows directly since the conditional expectation of a positive part is at least both zero and the conditional expectation of its argument; it is also conditional Jensen's inequality. Taking expectations proves $C(u,K)\ge C(t,K)$, the maturity monotonicity of calls with nonnegative strikes.

If $K_1\le K_2$, then $(S_T-K_1{)}^{+}\ge (S_T-K_2{)}^{+}$ pointwise; positive discounting and expectation prove monotonicity of a European call price in strike. For $0\le a\le 1$, the pointwise inequalitygives convexity of a European call price in strike after the same operations. HenceThe maturity assertion uses $K\ge 0$; it need not hold for a negative strike. For example, in the deterministic complete market $S_t=B_t=e^{rt}$ with $r>0$, a negative-strike call has price $1-K{e}^{-rT}$, which decreases with $T$. The monotonicity assertions are non-strict, as equality can occur.

For $K_1<K_2$, the lower-strike payoff dominates:If $C_t^{T,K_1}<C_t^{T,K_2}$, buy the cheaper lower-strike call and sell the higher-strike call. This gives positive initial consumption and a nonnegative terminal payoff, an arbitrage. Hence the monotonicity of a European call price in strike gives $C_t^{T,K_1}\ge C_t^{T,K_2}$.