Past exam of the mathematics course of the University of Cambridge / 2014 / iii / Paper 38 / 4 / b / Solution

Buy half a call at each neighboring strike and sell one call at the middle strike. Its cost isFor fixed terminal stock price $s$, the function $K\mapsto (s-K{)}^{+}$ is convex. Since $K_i$ is the midpoint, the payoffis nonnegative for every $s$. More explicitly, it is zero outside $[K_{i-1},K_{i+1}]$, equals $(s-K_{i-1})/2$ on $[K_{i-1},K_i]$, and equals $(K_{i+1}-s)/2$ on $[K_i,K_{i+1}]$. The negative cost and nonnegative payoff produce an arbitrage. This is the butterfly-spread arbitrage for nonconvex call prices.