An index carries 18% annualised volatility. What is the one week standard deviation using 52 weeks, and how does that square with the 252 trading day convention?

An index carries 18% annualised volatility. What is the one week standard deviation using 52 weeks, and how does that square with the 252 trading day convention?

Approach: Divide by the square root of the number of weeks, obtaining that root from a nearby square, then reconcile the weekly and daily conventions through the trading days per week.

2.50%. The square root of time rule divides the annual volatility by sqrt(52), and sqrt(52) sits just above 7.2 since 7.2^2 = 51.84 and the remaining 0.16 divided by 14.4 adds 0.011, giving 7.211. So 18/7.211 is 2.496%, which rounds to 2.50%. Dividing by 7.2 outright is accurate to 0.15% here and is the version to use under a clock. The two conventions reconcile through the calendar, since 252/52 = 4.85 trading days a week, so one weekly move is sqrt(4.85) = 2.20 daily moves. Checking against the daily figure of 18/15.87 = 1.134% gives 1.134 * 2.20 = 2.50%. The annualisation constant has to match the data, so weekly returns take sqrt(52) and daily returns take sqrt(252).

Follow-up: Implied volatility for a one week option is quoted at 26% while the realised weekly move has been 2.50%. What is the annualised gap in volatility points?

Key concepts: square root of time, weekly volatility, annualisation.