 Solution

The delta replication from a local-volatility pricing equation uses the asset-unit holdingsTheir wealth is exactly $V(t,S_t)$. Moreover,where the final equality is the calculation in part (i). Thus the holdings are a self-financing strategy with terminal wealth $g(S_T)$, proving replication. Since $V\ge 0$, this replicating strategy has nonnegative wealth and is an admissible trading strategy. The equality of trading gains is unchanged by replacing the drift with its physical-measure value: the stock drift in Itô formula and in the stock holding changes by the same amount. If ${\pi }_t$ denotes wealth fractions, then, wherever $V>0$, its stock fraction is $S_t{V}_S/V$ and its bank fraction is $1-S_t{V}_S/V$.