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

Use a telescoping replication of a stock-price sum. Hold $T-t+1$ shares during interval $(t-1,t]$; at time $t$, sell one share and keep its proceeds in the bond. Start with $T$ shares and no cash, costing $T{S}_0$.

After the time-$t$ rebalance, the stock holdings are $T-t$ and the cash holdings are ${\sum }_{u=1}^t{S}_u$, so wealth isThe sale of one share exactly funds the cash increase, making the strategy self-financing. Equivalently,At time $T$ there are no remaining shares, and the cash equals the claim. All stock positions over trading intervals are predictable.

For every integer $m\in \{0,\dots ,N\}$,Thus the static replication on a finite terminal support consists of one share and two calls of every listed strike, held to maturity:The strike-$N$ call contributes zero at maturity but is harmless in this identity. The portfolio costs $S_0+2{\sum }_{K=1}^N{C}_0(K)$. No distributional assumption on the stock is needed beyond its stated terminal support.

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.