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Let $\tau =T-t>0$ and $I_{t,T}={\int }_t^T{\sigma }_s^{2}ds$. The stochastic exponential solution givesBoundedness of the predictable process $\sigma$ implies the Novikov condition on every finite horizon, so this exponential has conditional expectation $\mathbb{E}[M_{t,T}\mid {\mathcal{F}}_t]=1$. The pathwise bounds $a^2\tau \le I_{t,T}\le b^2\tau$ and positivity of $M_{t,T}$ giveThis proves the required bound in the independent-volatility case, and the exponential argument in fact does not need independence. In the usual filtration generated by Brownian motion and an independent volatility history, one can alternatively condition on the whole volatility path: the Brownian integral is then conditionally Gaussian with variance $I_{t,T}$. The result isthe conditional square-root price under independent volatility. The outer conditional expectation cannot in general be removed.

On $\{S_t>0\}$, these positive price bounds justify the unique square-root stock implied volatilityThere is a genuine zero-price qualification: if $S_t=0$, the nonnegative stock is absorbed at zero and both claim prices are zero for every volatility parameter. The price equation then does not determine a unique $\Sigma$. One may set $\Sigma =a$ on that event to retain the bounds; identification by inversion requires $S_t>0$.

The coefficient is $\boxed{k=1/8}$, but independence alone does not make the printed right-hand side ${\mathcal{F}}_t$-measurable. The future variance integral need not be known at time $t$. This is a genuine missing information assumption in the PDF.

The Itô formula or explicit stochastic exponential givesIf the volatility path is fixed at time zero and independent of $W$, conditioning on that path makes the first factor an exponential of a centered Gaussian variable with the compensating half-variance, so its conditional expectation is one. More generally, this conditioning works when enlarging the filtration by the entire independent volatility path preserves the Brownian property. For the usual joint filtration of Brownian history and independent volatility history, the valid conditional square-root price under independent volatility isWhen the integral is already ${\mathcal{F}}_t$-measurable, the outer conditional expectation can be removed, giving the intended printed formula. In particular this holds for deterministic volatility or an independent path disclosed initially.

For a counterexample to the unqualified printed assertion, take an independent fair Bernoulli variable $A\in \{0,1\}$, disclose it at time $1$, and letWith the filtration generated by the Brownian history and this disclosure, $W$ is Brownian and $\sigma$ is bounded, continuous, adapted, and independent of $W$. At $t=0,T=2$, the variance integral is $A/3$, while ${\mathcal{F}}_0$ is trivial. Direct Gaussian conditioning givesa constant. The proposed factor $e^{-A/24}$ is random, so cannot equal that conditional expectation. The formula requires knowledge of future integrated variance; independence by itself is insufficient.