For $K_1<K_2$, the lower-strike payoff dominates:
$$
(S_T-K_1)^+\geq(S_T-K_2)^+.
$$
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 $C_t^{T,K_1}\geq C_t^{T,K_2}$.

Solution

Past exam of the mathematics course of the University of Cambridge / 2023 / iii / Paper 211 / 3 / b / Solution

At a fixed maturity, no arbitrage makes a European call price nonincreasing in its strike because the lower-strike payoff dominates the higher-strike payoff state by state.


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