= Integrated variance for linear-in-time volatility
{title2=$\int_0^1\alpha^2(1+t)^2dt=\frac73\alpha^2$}
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 $\alpha\neq0$. 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>.
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