 Solution

Differentiate the backward equation for $V$ with respect to the stock state, and put $w=V_S$. The product derivative of $a^2/2$ is $a{a}^{\prime }$, givingThe assumed smoothness and bounded derivatives put $w$ in the uniqueness class for the equation defining $U$. Therefore $U=V_S$, and ${\pi }_t=V_S(t,S_t)$ is the option delta. Substituting in the Itô integral from part (a) and using $d{S}_t=a(S_t)d{W}_t$ gives the self-financing representationThe boundedness hypotheses justify both stochastic integrals. This derivation does not divide by $a$ and remains valid even at states with zero diffusion amplitude.