= Solution
The boundedness of $a'$ makes $a$ globally Lipschitz, so the <stock> <stochastic differential equation> has a nonexplosive strong solution. Since $a$ is bounded, $S_t-S_0=\int_0^ta(S_u)dW_u$ is a square-integrable <martingale> on every finite horizon. Apply <Itô formula> to the option price:
$$
dV(t,S_t)=\left(V_t+\frac12a(S_t)^2V_{SS}\right)dt+a(S_t)V_S(t,S_t)dW_t
=V_S(t,S_t)dS_t.
$$
Thus both the <stock> and the option are <local martingales> under the original probability measure, while cash is constant. The relevant no-arbitrage theorem is the sufficient direction of the <fundamental theorem of asset pricing>: a common equivalent measure under which all numéraire-denominated traded prices are <local martingales> excludes <arbitrage> by <admissible trading strategies>. In this case the original measure itself is an <equivalent local martingale measure>. Hence \b[the augmented market has no admissible <arbitrage>]. The nonnegative option-value assumption also makes the replicating wealth below admissible.
Set
$$
\boxed{U(t,S)=V_S(t,S),\qquad\pi_t=U(t,S_t).}
$$
Differentiating the pricing equation in the stock-price variable gives
$$
U_t+a(S)a'(S)U_S+\frac12a(S)^2U_{SS}=0.
$$
The limiting terminal derivative is $\mathbf1_{\{S>K\}}$ away from the strike. Its value at the strike may be assigned as the stated $\mathbf1_{\{S\ge K\}}$: the smooth positive diffusion coefficient gives a continuous terminal distribution, so that point has probability zero. This is the <call delta equation for a driftless local volatility diffusion>.
Hold $\pi_t$ shares and $\eta_t=V(t,S_t)-\pi_tS_t$ units of cash. The <portfolio> value is $V(t,S_t)$, and the preceding Itô calculation gives $dV=\pi_t dS_t+\eta_t dB_t$, with $dB_t=0$. It is therefore a <self-financing portfolio>, with initial wealth $V(0,S_0)$ and terminal wealth $(S_T-K)^+$. This proves \b[replication by the stated <delta hedge>].
A further application of <Itô formula>, now to $U$, yields
$$
d\pi_t=-a(S_t)a'(S_t)U_S(t,S_t)dt+a(S_t)U_S(t,S_t)dW_t.
$$
The density process is the strictly positive <stochastic exponential>
$$
Z_t=\exp\left(\int_0^ta'(S_u)dW_u-\frac12\int_0^ta'(S_u)^2du\right).
$$
Its integrand is bounded, so the <Novikov condition> proves that it is a true <martingale>. The product formula for $M=Z\pi$ gives
$$
\begin{aligned}
dM_t&=Z_td\pi_t+\pi_tdZ_t+d[Z,\pi]_t,\\
d[Z,\pi]_t&=Z_ta'(S_t)a(S_t)U_S(t,S_t)dt.
\end{aligned}
$$
The drift in $Z\,d\pi$ cancels this <quadratic covariation> term, leaving
$$
\boxed{dM_t=Z_t\bigl(a(S_t)U_S(t,S_t)+a'(S_t)U(t,S_t)\bigr)dW_t.}
$$
Thus $M$ is a <local martingale>; the expression also identifies its <stochastic integral> integrand explicitly.
Under the stated assumption that $M$ is a true <martingale> on $[0,T]$, its terminal value gives
$$
Z_t\pi_t=\mathbb E[Z_T\mathbf1_{\{S_T\ge K\}}\mid\mathcal F_t].
$$
Since $0\le Z_T\mathbf1_{\{S_T\ge K\}}\le Z_T$ and $\mathbb E[Z_T\mid\mathcal F_t]=Z_t>0$, division gives
$$
\boxed{0\le\pi_t\le1\quad\text{a.s.}}
$$
This is the <derivative-weighted call delta martingale> argument. It uses true martingality through maturity, not just local martingality before the terminal date.
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