 Integrated variance for linear-in-time volatility

Deterministic time-varying volatility enters the European Black-Scholes formula through its integrated square, not its arithmetic mean. For $\sigma (t)=\alpha (1+t)$ over one year, the arithmetic mean is $3\alpha /2$ but its square is only $9{\alpha }^2/4$, strictly below the actual integrated variance by ${\alpha }^2/12$ when $\alpha \ne 0$. Positive option vega therefore makes the arithmetic-mean-volatility call price too small. If the deterministic volatility path is known exactly in the complete frictionless model, the pricing difference can be locked in by buying the cheaper claim and selling its correctly computed replicating strategy.