Apply <Itô formula> to $V(t,S_t)$. Its Brownian coefficient is
$$
\sigma S_t\frac{\partial V}{\partial s}(t,S_t).
$$
The self-financing portfolio's Brownian coefficient is $\pi_t\sigma S_t$. Since $S_t\sigma>0$, equality of the two value processes forces
$$
\boxed{\pi_t=\widetilde V(t,S_t),
\qquad
\widetilde V(t,s)=\frac{\partial V}{\partial s}(t,s).}
$$
Thus the stock holding is the claim's <option delta>.

Solution

Past exam of the mathematics course of the University of Cambridge / 2019 / iii / Paper 211 / 3 / d / Solution

{title2=$\Delta$}

The option delta is the derivative of an option value with respect to the underlying asset price. In a smooth complete diffusion model, it is also the number of underlying units in the replicating <delta hedge>.

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