 Solution

Assume a non-dividend-paying tradable asset follows geometric Brownian motion $dS=\mu Sdt+\sigma SdW$, 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 givesReplicate it by $\Delta$ shares and a cash account of value $b=V-\Delta S$. Self-financing requires $dV=\Delta dS+rbdt$. Matching the stochastic term fixes $\Delta =V_S$. Matching the remaining drift and cancelling the physical drift $\mu$ then givesThis 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 $V(S,t)\sim S-K{e}^{-r(T-t)}$ 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 $Sd\Delta$ term to the strategy.