Pricing an index call when variance does not scale

Using daily, weekly and monthly index returns, you find that estimated return variance grows faster than in proportion to the measurement horizon. Which volatility would you use to price a one-month index call, and what does the pattern tell you?

Use the volatility estimated from monthly returns, matching the option's horizon. Super-linear variance scaling means returns are positively autocorrelated at short horizons (for an index, largely from stale prices of thinly traded constituents), so annualising daily variance understates the one-month variance and would underprice the call. It also signals that the random-walk assumption behind Black-Scholes does not hold exactly.