= Solution
Assume a frictionless market with no taxes or transaction costs, unrestricted <short selling> and borrowing/lending at the same constant rate $r$, continuous trading, and absence of <arbitrage>. The non-dividend-paying <stock> has <geometric Brownian motion> dynamics $dS=\mu Sdt+\sigma SdW$ with constant $\sigma>0$, and the <bank account> satisfies $dB=rBdt$. A <contingent claim> with no interim cash flows has sufficiently regular value $f(t,S)$, meaning $f\in C^{1,2}$ before maturity; terminal <financial payoff> kinks can be handled by the smooth value for earlier times. Trading strategies are <adapted>, <self-financing portfolios>, and <admissible trading strategies> so that doubling strategies are excluded.
The <Itô formula> gives
$$
df=\left(f_t+\mu Sf_S+\frac12\sigma^2S^2f_{SS}\right)dt+\sigma Sf_SdW.
$$
Choose $\Delta=f_S$ shares and put the remaining value $f-Sf_S$ in the <bank account>. Its self-financing gain is
$$
\Delta\,dS+r(f-S\Delta)dt
=\{\mu Sf_S+r(f-Sf_S)\}dt+\sigma Sf_SdW.
$$
Matching the <contingent claim>'s gain to this replicating gain cancels the random increment and equates the drifts:
$$
\boxed{f_t+\frac12\sigma^2S^2f_{SS}+rSf_S-rf=0.}
$$
This is the <Black-Scholes equation>. The argument uses the self-financing gain $\Delta\,dS$, not a product differential that incorrectly treats a changing $\Delta$ as free cash. Rebalancing purchases are financed from the <bank account>. Conversely, a smooth solution with suitable growth and <boundary conditions> defines a <delta hedge> whose value replicates the <contingent claim>; <law of one price> identifies this value with the <contingent claim> price. Here $S$ is the current <stock> price, $t$ is time, $\sigma$ is proportional volatility, $r$ is the risk-free rate, and $f$ is the <contingent claim> price. The physical <expected return> $\mu$ cancels because exposure to price risk has been hedged.
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