Past exam of the mathematics course of the University of Cambridge / 2018 / iii / Paper 211 / 4 / b

For an integer $m$ and $d\in \{-1,0,1\}$, verify the single-step identityIf $m\ge 1$, the positive part is linear across the step and $f(m)=1$; if $m\le -1$, it remains zero and $f(m)=0$. At $m=0$, $d^{+}=(d+d^2)/2$, which is checked for the three permitted values of $d$.

Apply this with $m=S_{t-1}-K$ and $d=S_t-S_{t-1}$, and telescope over $t$. The result is the discrete Tanaka formulaThe first sum is a martingale transform; the second records the correction at the kink of the positive part. The algebraic identity itself needs only the integer values and permitted increments, not the martingale property.