Past exam of the mathematics course of the University of Cambridge / 2024 / iii / Paper 211 / 2 / d / Solution

Define the piecewise-linear functionOn $[K_j,K_{j+1})$, its slope is $g^{\prime }(K_j)$. Since $g$ is convex, $g^{\prime }$ is nondecreasing and hence $h^{\prime }(s)\le g^{\prime }(s)$ wherever the derivatives exist. As $h(0)=g(0)$, integration gives $h(s)\le g(s)$ for every $s\ge 0$.

Positive no-arbitrage pricing, the forward identity ${\mathbb{E}}_{Q^T}[S_T\mid {\mathcal{F}}_t]=F_t^T$, and the call-price formula now give