 Solution

Since the initial value and increments are integers, $m=S_T-K$ is an integer. If $m\le -1$, all three positive parts in the second difference are zero. If $m\ge 1$, the second difference is $(m-1)-2m+(m+1)=0$, including $m=1$. At $m=0$ it equals $0-0+1=1$. ThusThe original PDF has the first term $(S_T-K-1{)}^{+}$; the TeX's $(S_T-K+K{)}^{+}$ is an extraction error. This discrete call-price curvature identity isolates an individual integer state.