Past exam of the mathematics course of the University of Cambridge / 2010 / iii / Paper 39 / 5 / c

Let $L=\exp ((r-\frac{1}{2}{\sigma }^2)(T-t)+\sigma \sqrt{T-t}Z)$. Boundedness of $g^{\prime }$ and integrability of the lognormal distribution variable $L$ justify differentiation under the expectation, by dominated convergence:A differentiable increasing payoff has $g^{\prime }\ge 0$, and $L>0$. Thus the delta hedge hasMoreover $\mathbb{E}L=e^{r(T-t)}$, so $0\le {\pi }_t\le \Vert g^{\prime }{\Vert }_{\infty }$. At maturity the delta is $g^{\prime }(S_T)\ge 0$. The admissible trading strategy already constructed therefore uses nonnegative stock holdings throughout replication. After maturity, liquidate the stock position and retain the proceeds in the bank account, so ${\pi }_t=0$ for $t>T$ if holdings are to be defined for all times.