= Solution
Assume a non-dividend-paying tradable asset follows <geometric Brownian motion> $dS=\mu S\,dt+\sigma S\,dW$, with constant volatility $\sigma$ and risk-free rate $r$. Trading is continuous, frictionless and divisible; borrowing and lending occur at the same rate, short selling is allowed, the strategy is self-financing, and there are no jumps or arbitrage opportunities. These are modelling assumptions, not assertions that market returns are exactly Gaussian or that hedging is costless in practice.
For a smooth option price $V(S,t)$, <Itô's formula> gives
$$
dV=\left(V_t+\mu S V_S+\tfrac12\sigma^2S^2V_{SS}\right)dt+\sigma S V_S\,dW.
$$
Replicate it by $\Delta$ shares and a cash account of value $b=V-\Delta S$. Self-financing requires $dV=\Delta\,dS+rb\,dt$. Matching the stochastic term fixes $\Delta=V_S$. Matching the remaining drift and cancelling the physical drift $\mu$ then gives
$$
\boxed{V_t+\frac12\sigma^2S^2V_{SS}+rSV_S-rV=0.}
$$
This is the <Black-Scholes equation>. The cancellation does not assert that the actual expected asset return equals $r$; risk-neutral valuation is a consequence of replication under these assumptions.
For a European call, impose $V(S,T)=(S-K)^+$, with $V(0,t)=0$ and $V(S,t)\sim S-Ke^{-r(T-t)}$ for large $S$. Its writer hedges the short option by holding $\Delta=\partial V/\partial S$ shares and investing or borrowing the residual cash $V-\Delta S$, rebalancing as $S$ and time change. In the constant-parameter model $\Delta=N(d_1)$, where $d_1=[\log(S/K)+(r+\sigma^2/2)(T-t)]/[\sigma\sqrt{T-t}]$ and $N$ is the standard normal cumulative distribution. Continuous rebalancing is an ideal limit: transaction costs, discrete trading, jumps and changing volatility create residual hedge risk. Writing the self-financing gains equation explicitly avoids incorrectly adding an unfinanced $S\,d\Delta$ term to the strategy.
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