Past exam of the mathematics course of the University of Cambridge / 2015 / iii / Paper 40 / 5 / c / Solution

The pathwise identity $(S_t-S_{t-1}{)}^2=S_t^2-S_{t-1}^2-2S_{t-1}(S_t-S_{t-1})$ telescopes toThis is the discrete realized-variance replication identity. Replicate $S_T^2$ using part (b). Add a self-financing portfolio with initial wealth $-S_0^2$ and stock holdings $-2S_{t-1}$ during $(t-1,t]$.

To give the cash positions explicitly, letThe added portfolio holds $G_{t-1}+2S_{t-1}^2$ units of the bond over that interval. Its starting value is $G_{t-1}$ and its change is exactly $-2S_{t-1}(S_t-S_{t-1})$. The combined holdings are thereforeThe terminal value equals the squared-increment sum by the telescoping identity. Its initial cost isThe dynamic stock hedge uses only prices known before each interval, while the calls remain static.