Solution

Past exam of the mathematics course of the University of Cambridge / 2008 / iii / Paper 39 / 5 / ii / Solution

{title2=$h^S=V_S,\quad h^B=(V-SV_S)/B$}

If $V$ satisfies the <pricing equation for a local volatility model>, hold $h^S_t=V_S(t,S_t)$ stocks and $h^B_t=[V(t,S_t)-S_tV_S(t,S_t)]/B_t$ units of the <bank account>. Their value is $V(t,S_t)$. The <pricing equation for a local volatility model> and <Itô formula> give
$$
dV(t,S_t)=rV(t,S_t)dt+\sigma(S_t)S_tV_S(t,S_t)dW_t^Q=h^B_t\,dB_t+h^S_t\,dS_t.
$$
Thus this is a <self-financing strategy>, with terminal value $g(S_T)$. It is a <delta hedge>, and nonnegative $V$ makes its wealth an <admissible trading strategy>. The stock and bank wealth amounts are $S_tV_S$ and $V-S_tV_S$, rather than the asset-unit holdings.

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 Incoming links: Delta replication from a local-volatility pricing equation